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20ac1d2db33180a2d173fdcfe0c0fc2b86231c69 | subsection | 4 | 5 | Interface challenges | For example, when one of the authors contacted a ministry of health for more detailed epidemiological data, the data were offered with a five-page data request form that significantly restricted use and sharing of the data. Furthermore, it stated that it would take “up to three months” to be released because of the rev... | {
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} | 10.3389/fpubh.2018.00336 | 1805.00445 | Epidemiological data challenges: planning for a more robust future
through data standards | [
"Geoffrey Fairchild",
"Byron Tasseff",
"Hari Khalsa",
"Nicholas Generous",
"Ashlynn R. Daughton",
"Nileena Velappan",
"Reid Priedhorsky",
"Alina Deshpande"
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23ae5beb5823c712e70d38a5bcd52d86fedca4be | abstract | 0 | 25 | Abstract | We present the first scattered-light images of two debris disks around the F8
star HD 104860 and the F0V star HD 192758, respectively $\sim45$ and $\sim67$
pc away. We detected these systems in the F110W and F160W filters through our
re-analysis of archival Hubble Space Telescope NICMOS data with modern
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} | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
scattered-light with HST | [
"É. Choquet",
"G. Bryden",
"M. D. Perrin",
"R. Soummer",
"J. -C. Augereau",
"C. H. Chen",
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"E. Gofas-Salas",
"J. B. Hagan",
"D. C. Hines",
"D. Mawet",
"F. Morales",
"L. Pueyo",
"A. Rajan",
"B. Ren",
"G. Schneider",
"C. C. Stark",
"S. Wolff"
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e4c3609509ccac353463180352eceb237086fdd1 | subsection | 1 | 25 | Introduction | Debris disks are extrasolar system components evolving around main-sequence stars. They are composed of kilometer-sized planetesimals formed during the earlier protoplanetary stage of the system, and of dust particles generated by colliding bodies through a destructive grinding cascade stirred by secular perturbations ... | {
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afbe07513c64e8e3a1f584c073d8d0cb85bc49eb | subsection | 2 | 25 | Introduction | Using the measured radii of a sample of Herschel-resolved disks, interestingly showed that the typical grain size in these disks does not directly scale with the radiative pressure blowout particle size, but decreases with stellar luminosity. This may indicate that other mechanisms are at work in debris disks that limi... | {
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84d07f4c0d3fb9f72fb2fddf69563953bbfc0251 | subsection | 3 | 25 | Datasets | The data on HD 104860 and HD 192758 were obtained as part of two surveys with the near-IR NICMOS instrument on HST that aimed at resolving a selection of debris disks identified from their infrared excess, respectively with the Spitzer Space Telescope (HST-GO-10527, PI: D. Hines) and with IRAS/Hipparcos (HST-GO-11157, ... | {
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096109e3c7ba53ce577bae35c601eb70d1616100 | subsection | 4 | 25 | Data Processing | lrrr
Observing and processing parameters
Parameters HD 104860 HD 192758 HD 192758(F110W) (F110W) (F160W)
UT date 2006-03-20 2c2007-07-032c2008-06-04# orientations 2 2c3Orient difference () 30 2c30; 2; 28Filter F110W F110W F160W\lambda _p (\mu m) 1.116 1.116 1.600F_{\nu } (\mu Jy.s.DN^{-1})1.211211.21121 1.49585# comb... | {
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32cf593373554db6ec6636295d0e9cb4fcd454fa | subsection | 5 | 25 | Data Processing | The noise maps were computed from the pixel-wise standard deviation across these sets of processed reference star images.lrr
System properties
Properties HD 104860 HD 192758
RA (J2000) 12 04 33.731 20 18 15.790DEC (J2000) +66 20 11.715 -42 51 36.297Spectral Type F8 F0VJ (mag) 6.822 (1) 6.387 (1)H (mag) 6.580 (1) 6.2... | {
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e508ebf5f09daada0f80ababdc1273d09376f508 | subsection | 6 | 25 | Disk detections | We detect faint and resolved dust emission around both stars. Fig. REF presents the images of the two disks and their respective S/N maps.
We list some properties of the two systems in Table REF .
[Figure: Debris disks detected around HD 104860 (F110W filter) and HD 192758 (F110W and F160W filters) by re-analyzing arch... | {
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} | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
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99c68077bd2eeb516127c7db290685719549473c | subsection | 7 | 25 | HD 104860 | HD 104860 is a F8 field star at 45.0\pm 0.5 pc . Its age was estimated, based on its chromospheric activity, to 32 Myr by , and to 19–635 Myr by . The system has a significant infrared excess at 70~\mu m identified with Spitzer with fractional infrared luminosity of L_{dust}/L_{\star }\sim 6.3\times 10^{-4} . Its SED i... | {
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10a6e80dcdb32eb277c9e2749ffe3d2ccb4a1064 | subsection | 8 | 25 | HD 104860 | The East side is significantly brighter than the West side, indicative of anisotropic scattering from the dust, and showing the near-side of the disk assuming grains preferentially forward-scattering.
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d54a5bbb2ce0d951fd9a12d8511ab40be0fec330 | subsection | 9 | 25 | HD 192758 | HD 192758 is a F0V star at 67\pm 2 pc . reported a 50% probability for the star to be a member of the IC 2391 supercluster, of age 50\pm 5 Myr . On the other hand, found the system to be a field star of isochronal age \sim 830 Myr.
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af70f1eff905d9392b6b3398a6e6c4d43332af73 | subsection | 10 | 25 | Disk modeling | PSF subtraction with algorithms that solve the least-square problem of minimizing the residuals between the science image and a set of eigen-images systematically involves some level of over-subtraction of circumstellar materials, along with the PSF , , . This effect biases both the morphology and the photometry of cir... | {
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74a9e0e32b199b5e56181abd88f39602db2ade34 | subsection | 11 | 25 | Forward modeling method | Assuming that the morphology of the astrophysical source is known, the post-processing throughput can be inferred though forward modeling. To constrain the morphology and photometry of the disks detected around HD 104860 and HD 192758, we used the same methodology as in , using parametric modeling and the analytical fo... | {
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"J. -C. Augereau",
"C. H. Chen",
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"E. Gofas-Salas",
"J. B. Hagan",
"D. C. Hines",
"D. Mawet",
"F. Morales",
"L. Pueyo",
"A. Rajan",
"B. Ren",
"G. Schneider",
"C. C. Stark",
"S. Wolff"
] | [
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dbbf16188e64ff3d5c390b4f0fce5dd43ae0d9c1 | subsection | 12 | 25 | Forward modeling method | Furthermore, given the inclinations of the disks, a Henyey-Greenstein scattering phase function (SPF) model will describe the surface brightness variations in the disks over a range of scattering angles limited by the disk inclinations, but may not properly describe the actual SPF over all angles .We estimated the good... | {
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} | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
scattered-light with HST | [
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3f759fa9e5b04cf00cb746e4f70ae2de140e8d8f | subsection | 13 | 25 | HD 104860 analysis | lrrr|rr
Parameter grid and best model for HD 104860
Param. Min. Max. N_{val} Best Model Best Model(in grid) (interpolated)a
R_0 (au) 106 122 5 114 114\pm 6|g| 0.0 0.4 5 0.2 0.17\pm 0.13\theta () -5 7 5 1 1\pm 5i () 52 64 5 58 58\pm 5\alpha _{in} 2 10 5 10 \ge 4.5\alpha _{out} -6 -2 5 -4 -3.9\pm 1.6du (au) -10 10 5 0... | {
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} | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
scattered-light with HST | [
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"R. Soummer",
"J. -C. Augereau",
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00f739ae748847ec9cc8bf931d26b3be83c5057d | subsection | 14 | 25 | HD 104860 analysis | The 1\sigma uncertainties are computed accordingly from the noise map.][Figure: Scattering albedo computed under the Mie theory as a function of grain size for a disk with HD 104860's best fit morphology, assuming different grain compositions: pure ice (blue), dirty ice (red), silicates (purple), and different porositi... | {
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f101b38b12b3f1537fd312419ef85ba2161db77e | subsection | 15 | 25 | HD 104860 analysis | Sharp inner edges in debris disk can be sculpted by planets orbiting within the disk, confining the dust out of a chaotic zone through mean motion resonances , , while unperturbed systems have smoother inner edges filled by small grains due to Poyting-Robertson drag. We do not find significant offsets of the ring with ... | {
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scattered-light with HST | [
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f01a78ff321be8d7fd2ebe38cfa656332cd6bc1b | subsection | 16 | 25 | HD 104860 analysis | The true albedo of the dust can only be recovered with assumptions on the disk's SPF.
Assuming that our scattering efficiency measurement integrates the light scattered by the grains responsible for the thermal emission, and using the infrared fractional luminosity of f_{emit}=6.4\times 10^{-4} reported for the outer d... | {
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... | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
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9f296431d88caa1391145bf9200bd87bbcfa190d | subsection | 17 | 25 | HD 192758 analysis | lrrr|rr|rr
Parameter grid and best models for HD 192758
F110W F110W F160W F160WParam. Min. Max. N_{val} Best Model Best Model Best Model Best Model(in grid) (interpolated)a (in grid) (interpolated)a
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} | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
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896d2f5f344efeb5ee085cab440698f65b034133 | subsection | 18 | 25 | HD 192758 analysis | The best fit PA differs by 8 between the two datasets. This may be due to starlight residuals from the telescope spider, at a comparable orientation to the disk major axis, which may bias the disk position angle. The mean PA between both fits is \theta =-89\pm 12 East of North, using conservative error bars encompassin... | {
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46cc803bbe01c53b62f13795f1d6ca54effd6440 | subsection | 19 | 25 | HD 192758 analysis | After normalizing by the stellar contribution, we find that the disk around HD 192758 has scattering efficiencies of f_{scat}^{F110W}=(85\pm 2)\times 10^{-6} and f_{scat}^{F160W}=(98\pm 4)\times 10^{-6}, respectively in the two NICMOS filters.From these scattering efficiency measurements in two different NICMOS filters... | {
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2ae9392fbe19fc2a60ffbcc1bc62f94f54b15266 | subsection | 20 | 25 | Discussion | These two detections add to a growing population of debris disks resolved in scattered light. To date, 41 of such have been imaged around stars from \sim 10 Myr to a few Gyr, over a large range of spectral types (See Fig. REF ). Yet, considering the numerous attempts to image debris disks around systems with large infr... | {
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81e5e4eb545dacda844c52c3d560c9c7dfc25a9e | subsection | 21 | 25 | Discussion | These combined results seem to indicate that there is an underlying population of debris disks much fainter than the population of bright debris disks discovered so far, indicative of low scattering albedos. A rigorous statistical analysis estimating the completeness of previous surveys to debris disks as function of t... | {
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885066fb4eac995e168ad93d62db89c838c7958c | subsection | 22 | 25 | Discussion | A better characterization of these systems would be needed to discriminate one scenario from the other, for instance by constraining their dust size distribution with multi-band imaging.lcrrrrr
Scattering albedos of \sim 60-inclination debris disks
System Spectral Age Inc. Scattering \lambda Ref.Type (Myr) () Albedo ... | {
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ead03faa20b0c7121950ac166032832f52a84589 | subsection | 23 | 25 | Conclusion | To conclude, we have detected two debris disks in scattered-light, around HD 104860 and HD 192758. The former disk has previously been imaged in thermal emission with Herschel but never in scattered-light, and the latter disk has never been imaged before. These disks were found in archival HST-NICMOS data in the near-i... | {
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... |
60ce50d11df27279661721223cfbcf2dc32e2cc4 | subsection | 24 | 25 | Conclusion | This research has used archival data from HST programs HST-GO-10527 (PI: D. Hines) and HST-GO-11157 (PI: J. Rhee), and from Keck programs. We thank the anonymous referee for her or his comments which made the paper much clearer.HST(NICMOS). | {
"cite_spans": []
} | 10.3847/1538-4357/aaa892 | 1801.05424 | HD 104860 and HD 192758: two debris disks newly imaged in
scattered-light with HST | [
"É. Choquet",
"G. Bryden",
"M. D. Perrin",
"R. Soummer",
"J. -C. Augereau",
"C. H. Chen",
"J. H. Debes",
"E. Gofas-Salas",
"J. B. Hagan",
"D. C. Hines",
"D. Mawet",
"F. Morales",
"L. Pueyo",
"A. Rajan",
"B. Ren",
"G. Schneider",
"C. C. Stark",
"S. Wolff"
] | [
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] | 2,018 | en | Physics | [
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cd1adfa8ad4ffe11806c5a3132e1765568582026 | abstract | 0 | 68 | Abstract | This paper addresses an estimation problem of an additive functional of
$\phi$, which is defined as $\theta(P;\phi)=\sum_{i=1}^k\phi(p_i)$, given $n$
i.i.d. random samples drawn from a discrete distribution $P=(p_1,...,p_k)$ with
alphabet size $k$. We have revealed in the previous paper that the minimax
optimal rate of... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
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] | 2,018 | en | Computer Science | [
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0e40d83a9287859a40eb907050ec6d6fba138de4 | subsection | 1 | 68 | Introduction | Let P be a probability measure with alphabet size k, where we use a vector representation of P; P=(p_1,...,p_k) for p_i=P{i}. Let \phi be a mapping from [0,1] to . Given a set of i.i.d. samples S_n={X_1,...,X_n} \sim P^n, we deal with the problem of estimating an additive functional of \phi . The additive functional \t... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
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] | 2,018 | en | Computer Science | [
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c4cafdb145b8b26df5921a232055f40cce11a2a5 | subsection | 2 | 68 | Introduction | For fixed k, asymptotic efficiency and minimax optimality were proved if we employ the plugin or the maximum likelihood estimator, in which the estimated value is obtained by substituting the empirical mean of the probabilities P into \theta van2000asymptotic. However, the plugin estimator suffers from a large bias if... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
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] | 2,018 | en | Computer Science | [
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cebeb89383e42a83eebc30d945919824b07ff764 | subsection | 3 | 68 | Related Work | Mane researchers have been dealing with the estimation problem of the additive functional and provides many estimators and analyses in decades past. The plugin estimator or the maximum likelihood estimator (MLE) is the simplest way to estimate \theta , in which the empirical probabilities \tilde{P} = (N_1/n,...,N_k/n) ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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90f1211653b2e26a37440c46e2f4263c33aacb1a | subsection | 4 | 68 | Related Work | For \alpha \in [1/2,1), the optimal rate was obtained as\frac{k^2}{(n\ln n)^{2\alpha }} + \frac{k^{2-2\alpha }}{n}.However, the minimax optimal rate for \alpha \ge 1 still remains as an open problem.Although it is a special case, jiao2015minimax,wu2016minimax revealed the minimax optimal rate for \alpha \ge 1; the dive... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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ac5676acba71c4475880854898f0cd217e8814a4 | subsection | 5 | 68 | Our Contribution | In this paper, we derive the minimax optimal rate of the additive functional estimation and construct minimax optimal estimators for any \alpha > 1. The results are summarized in tbl:results-summary. This table shows the minimax optimal rate (third column) and the estimator that achieves the optimal rate (fourth column... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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7cce2b58ae3a0c5f9ab6ca78e4edc23cc61eb751 | subsection | 6 | 68 | Our Contribution | For any positive real sequences {a_n} and {b_n}, a_n \gtrsim b_n denotes that there exists a positive constant c such that a_n \ge c b_n. Similarly, a_n \lesssim b_n denotes that there exists a positive constant c such that a_n \le c b_n. Furthermore, a_n \asymp b_n implies a_n \gtrsim b_n and a_n \lesssim b_n. For an ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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e650433dd2b08cdfa10a957cf21554501994fe2c | subsection | 7 | 68 | Poisson Sampling | We employ the Poisson sampling technique to derive upper and lower bounds for the minimax risk. The Poisson sampling technique models the samples as independent Poisson distributions, while the original samples follow a multinomial distribution. Specifically, the sufficient statistic for P in the Poisson sampling is a ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0d093a87907029506db53a50892eb101d0d2ef7f | subsection | 8 | 68 | Best Polynomial Approximation | cai2011testing presented a technique of the best polynomial approximation for deriving the minimax optimal estimators and their lower bounds for the risk. Let {P}_L be the set of polynomials of degree L. Given a function \phi defined on an interval I \subseteq [0,1] and a polynomial g, the L_\infty error between \phi a... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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2735cb4cd15c841bf3e48f90f9fff3deeaa2c9a1 | subsection | 9 | 68 | Basic Estimator Construction for | In the line of literature DBLP:conf/soda/AcharyaOST15,jiao2015minimax,wu2016minimax,DBLP:journals/corr/FukuchiS17, the basic construction of the optimal estimator for the additive functional has a common part. Here, we describe the common methodology to construct the optimal estimator. For simplicity, we assume that we... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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d8b960cc537ba18cedf653f1167e9abd764a3962 | subsection | 10 | 68 | Best Polynomial Estimator | Let {a_m}_{m=0}^L be coefficients of the polynomial that achieves the best approximation of \phi by a degree-L polynomial with range I=[0,\frac{4\Delta _{n,k}}{n}]; that is, the best approximation polynomial of \phi is written as\phi _L(p_i) =& \sum _{m=0}^L a_m p_i^m.We utilize the factorial moments to construct an un... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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37b96717ab98931182b844a8b0701cee61a9dc14 | subsection | 11 | 68 | Second Order Bias Correction Estimator | In DBLP:journals/corr/FukuchiS17, we employ the plugin estimator with Miller's bias correction miller1955nbi as \phi _{\mathrm {plugin}}. The bias correction offsets the second order bias, which is obtained as follows.*{\phi *{\frac{\tilde{N}}{n}} - \phi (p)} \approx & *{\frac{\phi ^{(2)}(p)}{2}*{\frac{\tilde{N}}{n} - ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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79cc3ea8b62fdb0af11b172f1a7c374505943f46 | subsection | 12 | 68 | Second Order Bias Correction Estimator | To ensure smoothness of \phi _2, we set L = 4. Then, the estimator is given as follows.\phi _{\mathrm {plugin}}(\tilde{N}_i) = H_{4,\frac{\Delta _{n,k}}{n}}[\phi ]*{\frac{\tilde{N}_i}{n}} - \frac{\tilde{N}_i}{2n^2}H^{(2)}_{4,\frac{\Delta _{n,k}}{n}}*{\phi }*{\frac{\tilde{N}_i}{n}}. | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0593367c1a2c993d8a604928e6194d3252aaf3a8 | subsection | 13 | 68 | Main Results | Our main results are revealing the minimax optimal rate of the additive functional estimation in characterizing with the divergence speed. We derive the minimax optimal rates for each range of \phi ; \alpha \in (1,3/2) and \alpha \in [3/2,2]. First, we derive the minimax optimal rate for \alpha \in (1,3/2). In this cas... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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a7c8a7b05edaffdbe2b045f694018315b47462a8 | subsection | 14 | 68 | Main Results | For a real \bar{\alpha }\in (0,1], a function \phi is \bar{\alpha }-Hölder continuous if{\phi }_{C^{0,\bar{\alpha }}} = \sup _{x \ne y \in I}\frac{*{\phi (x) - \phi (y)}}{*{x - y}^{\bar{\alpha }}} < \infty .In particular, 1-Hölder continuous is called as Lipschitz continuous. If the divergence speed of \phi is p^\alpha... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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58db4fe5c3e97867ceead15ee6a7236ac330b198 | subsection | 15 | 68 | Estimators and Upper Bound Analysis | In this section, we introduce the minimax optimal estimators for each range of \alpha . For \alpha \in (1,3/2), we employ the basic construction described in sec:basic-construction. We thus describe construction of \phi _{\mathrm {plugin}} in sec:forth-order. Besides, we analyze the bias and the variance of \phi _{\mat... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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f11f5d12ffefc88fde6e07706d896c18d6f90186 | subsection | 16 | 68 | Fourth order bias correction for | As mentioned before, the second order bias correction offsets the second order approximation of bias. In analogy with that, the fourth order bias correction offsets the fourth order approximation of bias. By the Taylor approximation, the bias of \phi _2 is obtained as*{\phi _2*{\frac{\tilde{N}}{n}} - \phi (p)} \approx ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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a28b7cc50982e7c6bd7a048b7ca73a5612db9a2e | subsection | 17 | 68 | Fourth order bias correction for | Hence, the estimator is obtained as\phi _{\mathrm {plugin}}(\tilde{N}_i) = H_{6,\frac{\Delta _{n,k}}{n}}[\phi ]*{\frac{\tilde{N}_i}{n}} - \frac{\tilde{N}_i}{2n^2}H^{(2)}_{6,\frac{\Delta _{n,k}}{n}}*{\phi }*{\frac{\tilde{N}_i}{n}} \\ - \frac{2\tilde{N}_i}{3n^3}H^{(3)}_{6,\frac{\Delta _{n,k}}{n}}*{\phi }*{\frac{\tilde{N}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0... | |
5067ad8ca0acbd2244865e801cae00d534e5505d | subsection | 18 | 68 | Fourth order bias correction for | Then, we have*{\bar{\phi }_{4,\Delta }*{\frac{\tilde{N}}{n}}} \lesssim \frac{1}{n^{2\alpha }} + \frac{p}{n}.We set the parameter \Delta as \Delta \asymp \frac{\ln n}{n}. With this \Delta , we can see that the bias and the variance do not exceed the rate in thm:optimal-rate-1-3/2. | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.04906526580452919,
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-0.00336... | |
d4739593031c7bb18cf1d88f99b18da91012b821 | subsection | 19 | 68 | Best polynomial error analysis for | Here, we analyze the bias and the variance of the best polynomial estimator \phi _{\mathrm {poly}}. For the variance, we can use the following lemma shown in [Lemma 5]DBLP:journals/corr/FukuchiS17.
[[Lemma 5]DBLP:journals/corr/FukuchiS17]
Let \tilde{N} \sim (np). Given an integer L and a positive real \Delta \gtrsim \f... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.04882211983203888,
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0.03487730026245117,
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0.0... | |
43fe3c9dbf0d7d9512feb961fd2c902001e19a87 | subsection | 20 | 68 | Best polynomial error analysis for | Then, we haveE_L(\phi ,[0,\Delta ]) \lesssim *{\frac{\Delta }{L^2}}^\alpha .To prove lem:best-error, we use the Jackson’s inequality which gives a bound on the best trigonometric polynomial approximation error by using the first order moduli of smoothness, defined as\omega _1(f,t) = \sup _{x, y \in (-\pi ,\pi )}{*{f(x)... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.024935821071267128,
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0.034397393465042114,
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8a42d58b45765d6917338f7c7c8e37b23394338f | subsection | 21 | 68 | Analysis of plugin estimator with | For \alpha \in [3/2,2], the plugin estimator is a minimax optimal estimator in which the optimal rate is 1/n. To prove this, we analyze the bias and the variance of the plugin estimator. The variance is easily proved as follows.
If \phi is Lipschitz continuous,*{\sum _{i=1}^n\phi *{\frac{N_i}{n}}} \lesssim \frac{1}{n}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.05436617136001587,
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0.03904229775071144,
-0.029075676575303078,
0.0... | |
6c951e27b4372bc9bf9c282806db715ddaf457e1 | subsection | 22 | 68 | Analysis of plugin estimator with | By utilizing lem:bias-moduli,lem:moduli-bound, we prove the bias.
Suppose \phi :[0,1]\rightarrow is a function of which second divergence speed is p^{\alpha } for \alpha \in [3/2,2) such that \phi (0)=0. Then, we have*{\sum _i\phi *{\frac{N_i}{n}} - \theta (P)} \lesssim \frac{1}{n^{\alpha -1}}.With the bias-variance d... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.0549735389649868,
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0.0034663700498640537,
0.0198863185942173,
-0.... | |
d0d0bd10562afd2679c0e15cb7295a5b9724f851 | subsection | 23 | 68 | Lower Bound Analysis | We here describe lower bound analyses and prove the lower bound of thm:optimal-rate-1-3/2,thm:optimal-rate-3/2. First, we prove the \frac{1}{n} term, which is accomplished by applying the LeCam's two point method with appropriate construction of two probability vectors. The precise claim is given by the following theor... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06474968045949936,
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0.0436038114130497,
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0.023037703707814217,
-0.002271349774673581,
... | |
7d35d83d0d062dce16b411439ee30c5a380e32f1 | subsection | 24 | 68 | Lower Bound Analysis for | Here, we will prove the following lower bound.
Suppose \phi :[0,1]\rightarrow is a function of which second divergence speed is p^{\alpha } for \alpha \in (1,3/2). If n \gtrsim k^{1/\alpha }/\ln k,R^*(n,k;\phi ) \gtrsim \frac{k^2}{(n\ln n)^{2\alpha }}.The proof of thm:lower2 basically follows the same manner of wu2016... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06278851628303528,
0.019831163808703423,
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0.03322482481598854,
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0.008214677684009075,
-0.021493930369615555,
... | |
10a933f5a55440133b17818bf6011cd70f1c9ba9 | subsection | 25 | 68 | Lower Bound Analysis for | Then, there exists an universal constant c > 0 such that\limsup _{L \rightarrow \infty , \gamma \rightarrow 0 : \gamma \le 1/2L^2}\gamma ^{1-\alpha }E_L*{\phi _\gamma ^\star , [\gamma ,2L^2\gamma ]} > c.We can choose such family *{\phi _\gamma } because the minimax risk is invariant among \phi _{c,c^{\prime }}(x) = \ph... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.07502169907093048,
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0.035740602761507034,
-0.008103442378342152,
... | |
63815fe02fc080e645bab2d69a99c027db35c678 | subsection | 26 | 68 | Discussion | In this paper, we reveal that the divergence speed characterizes the minimax optimal rate for \alpha \in (1,3/2) and \alpha \in [3/2,2]. Combining the previous result in DBLP:journals/corr/FukuchiS17, the minimax rate is characterized by the divergence speed for any \alpha \in (0,2] except \alpha = 1. The Shannon entro... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.04678884148597717,
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0.019075686112046242,
-0.030597399920225143,
0.... | |
d1c24b655916804803af2452581b722034249ec1 | subsection | 27 | 68 | Proof of lem:div-speed-holder | [Proof of lem:div-speed-holder]
The Lipschitz continuousness is proved by showing there exists an universal constant C > 0 such that\sup _{p \in (0,1)}*{\phi ^{(1)}(p)} \le C.For any p \in (0,1), the absolutely continuousness of \phi ^{(1)} gives*{\phi ^{(1)}(p)} =& *{\int _0^p\phi ^{(2)}(s)ds} \\
\le & \int _0^p*{\phi... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.09216376394033432,
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0.00563053460791707,
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-0.008430542424321175,
0.03640773892402649,
0.02... | |
61280af84de5903cd261d07b9eae6cb46dffacc6 | subsection | 28 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | We firstly prove that the smoothing procedure ensures p^\beta *{H^{(\ell )}_{L,\Delta }[\phi ](p)} \lesssim \Delta ^{\alpha +\beta -\ell }.
For \alpha , let \phi :[0,1]\rightarrow be a function of which Lthe divergence speed is p^\alpha , where L > \alpha is an universal constant. For \ell \le L and \beta such that \e... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.04412258416414261,
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0.042230743914842606,
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0.024776997044682503,
-0.004279524553567171,
... | |
4f965a8bff8f27ceb5b835c3f60ba2e8e054b7ca | subsection | 29 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | Besides, we have&*{\frac{p}{2n}\phi ^{(2)}(p)-\frac{\tilde{N}}{2n^2}H^{(2)}_{6,\Delta }[\phi ]*{\frac{\tilde{N}}{n}}} \\
=& *{\frac{1}{2n}\phi ^{(2)}(p)*{p-\frac{\tilde{N}}{n}}+\frac{\tilde{N}}{2n}*{\phi ^{(2)}(p)-H^{(2)}_{6,\Delta }[\phi ]*{\frac{\tilde{N}}{n}}}} \\
=& \begin{}[t]
\frac{p}{2n^2}\phi ^{(3)}(p) + \frac{... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.013040782883763313,
0.035955630242824554,
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0.029942674562335014,
-0.0249980129301548,
0... | |
f9302bea56d73a58fb5edbfec8624bb8e9d1ae12 | subsection | 30 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | Hence,& *{\bar{\phi }_{4,\Delta }*{\frac{\tilde{N}}{n}}-\phi (p)} \\
\le & \begin{}[t]
*{*{\frac{2p}{3n^2}\phi ^{(3)}(p)-\frac{2\tilde{N}}{3n^3}H^{(3)}_{6,\Delta }[\phi ]*{\frac{\tilde{N}}{n}}}}
+*{*{\frac{7p}{24n^3}\phi ^{(4)}(p)-\frac{7\tilde{N}}{24n^4}H^{(4)}_{6,\Delta }[\phi ]*{\frac{\tilde{N}}{n}}}} \end{}\\
+*{*{... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.03662314638495445,
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0.010315519757568836,
-0.04089584946632385... | |
faa5cd106326d065737b042315c130967e6af8a1 | subsection | 31 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | Thus, the first term in eq:plugin-var1 is bounded above as*{*{\frac{2p}{3n^2}\phi ^{(3)}(p)-\frac{2\tilde{N}}{3n^3}H^{(3)}_{6,\Delta }[\phi ]*{\frac{\tilde{N}}{n}}}} \lesssim \frac{\Delta ^{\alpha -3}}{n^3}.Similarly, the second and third terms in eq:plugin-var1 is bounded above as*{*{\frac{7p}{24n^3}\phi ^{(4)}(p)-\fr... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.012955251149833202,
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-0.020584963262081146,
... | |
5798642ed742b1bd53df93370d66a1ac9e38345b | subsection | 32 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | Hence, we have*{*{R_5*{\frac{\tilde{N}}{n};H_{6,\Delta },p}}} \lesssim & \frac{\Delta ^{\alpha -3}}{n^3}+\frac{\Delta ^{\alpha -3}}{n^4p}+\frac{\Delta ^{\alpha -3}}{n^5p^2} \\
\lesssim & \frac{\Delta ^{\alpha -3}}{n}.Let G(x)=(\hat{p}-x)^4/x^3. For a fourth time differentiable function g, the Taylor theorem and the mea... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.026668990030884743,
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-0... | |
e74a4ea4b52c3c92a9800a93915055c4b2b6d98a | subsection | 33 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | Hence,*{*{\frac{\tilde{N}}{2n^2}R_3*{\frac{\tilde{N}}{n};H^{(2)}_{6,\Delta },p}}} \lesssim \frac{\Delta ^{\alpha -3}}{n^3}.[Proof of lem:plugin-var-1-3/2]
By the triangle inequality, we have& \frac{1}{5}*{\bar{\phi }_{4,\Delta }*{\frac{\tilde{N}}{n}}} \\
\le & \frac{1}{5}*{*{\bar{\phi }_{4,\Delta }*{\frac{\tilde{N}}{n}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.03854833170771599,
0.05087891221046448,
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0.022524677217006683,
-0.01877056434750557,
0.... | |
207a22bc05d254dadaa8a51fe51018a0ba42cbc0 | subsection | 34 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | Hence,*{*{H_{6,\Delta }[\phi ]*{\frac{\tilde{N}}{n}} - H_{6,\Delta }[\phi ](p)}^2} \le & \sup _{\xi > 0}*{*{H^{(1)}_{6,\Delta }[\phi ](\xi )*{\frac{\tilde{N}}{n} - p}}^2} \\
\lesssim & *{\frac{\tilde{N}}{n}} = \frac{p}{n}.Next, we derive a bound on the second term in eq:plugin-var1. Let \hat{p}=\frac{\tilde{N}}{n}, g(p... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.031028730794787407,
0.01170060783624649,
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0.021265512332320213,
0.015163499861955643,
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-0.013996490277349949,
-0.03231015428900719,... | |
44bf5fe777cf410e7b225b9938926d2f79fb155d | subsection | 35 | 68 | Detailed Analysis of Fourth Order Bias Corrected Plugin Estimator | With the same manner of the second term, we have*{*{g(\hat{p})-g(p)}^2} \le & \frac{4\sup _{\xi > 0}\xi *{g^{(1)}(\xi )}^2}{n}.From lem:hermite-bound, we have \sup _{\xi > 0}\xi *{H^{(3)}_{6,\Delta }[\phi ](\xi )+\xi H^{(4)_{6,\Delta }[\phi ](\xi )}}^2 \lesssim \Delta ^{2\alpha -5}, \sup _{\xi > 0}\xi *{H^{(4)}_{6,\Del... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.03888688609004021,
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0.0011265063658356667,
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-0.007375230547040701,
-0.026494361460208893,
... | |
ba5871a9c8291d754b3b862c6ddd983e23fd08c8 | subsection | 36 | 68 | Upper Bound Analysis for | Here, we prove the upper part of thm:optimal-rate-1-3/2. In this section, we denote \hat{\theta } as an estimator with the basic construction where \phi _{\mathrm {poly}} is the best polynomial estimator and \phi _{\mathrm {plugin}} is the fourth order bias corrected plugin estimator. We prove the following theorem.
S... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.023889686912298203,
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... | |
cb54231e6cc865847a93928f94d285530a7fe93e | subsection | 37 | 68 | Upper Bound Analysis for | If the sixth divergence speed of \phi is p^\alpha for \alpha \in (1,3/2), the worst-case risk of \hat{\theta } is bounded above by\sup _{P \in {M}_k} *{*{\hat{\theta }{\tilde{N}} - \theta (P)}^2} \lesssim \frac{k^2}{(n\ln n)^{2\alpha }} + \frac{1}{n}.To prove thm:upper-bound, we use the following bounds on the bias and... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0.023767616599798203,
-0.0028622588142752647,... | |
afe96fd310bd241a88820358b4c83011cfb7dfda | subsection | 38 | 68 | Upper Bound Analysis for | For \phi _{\mathrm {plugin}}, we can check from the last truncation and Chebyshev alternative theorem that {\phi _{\mathrm {plugin}}(\tilde{N}_i) - \phi (p_i) } \lesssim 1 + E_L(\phi ,[0,\Delta _{n,k}/n]) \lesssim 1. For \phi _{\mathrm {poly}}, application of lem:hermite-bound yields the claim.[Proof of thm:upper-bound... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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-0.008307000622153282,
0.0... | |
338d7429dc9ec35b9f00934831a6fc455e6984a2 | subsection | 39 | 68 | Proof of lem:hermite-bound | [Proof of lem:hermite-bound]
It is clear that p^\beta *{H^{(\ell )}_{L,\Delta }[\phi ](p)} \lesssim p^{\alpha +\beta -\ell } for p \in [\Delta ,1] because of the divergence speed assumption. For p \ge 2, *{H^{(\ell )}_{L,\Delta }[\phi ](p)} = 0 by definition. For p \in (1,2), we have& p^\beta *{H^{(\ell )}_L*{p;\phi ,1... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0.008713257499039173,
0.01991383731365204,
0.... | |
38432a9f219227cafce1a2d83863ea319e2fb785 | subsection | 40 | 68 | Proof of lem:hermite-bound | Similarly, for p < \Delta , we have& p^\beta *{H^{(\ell )}_L*{p;\phi ,\Delta ,\frac{\Delta }{2}}} \\
\le & \begin{}[t]
(L+1)\sum _{u=0}^\ell \binom{\ell }{u}\sum _{m=1\vee u}^{L}\frac{*{\phi ^{(m)}(\Delta )}}{m!}*{\Delta -p}^{m-u}\sum _{s=0}^{L-m}*{\prod _{w=1}^{\ell -u-1}(L+s+1-w)}\end{}\\\sum _{w=\ell -u-s}^{\ell -u}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06615126878023148,
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0.005938510410487652,
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0.0144171928986907,
-0.012456759810447693,
0.01... | |
102e605853e1bbc977d7b1b0e98bdacd33381ada | subsection | 41 | 68 | Proof of lem:first-moduli-bound | [Proof of lem:first-moduli-bound]
From the divergence speed assumption, we have for x,y \in (-1,1),&*{\phi ^{(2)}_\Delta (x)-\phi ^{(2)}_\Delta (y)} \\
\le & 2\Delta *{\phi ^{(1)}(\Delta x^2)-\phi ^{(1)}(\Delta y^2)}+4\Delta ^2*{x^2\phi ^{(2)}(\Delta x^2)-y^2\phi ^{(2)}(\Delta y^2)} \\
\le & 4\Delta ^2*{\int _y^xs\phi ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.050746798515319824,
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-0.025068247690796852,
0.0011204804759472609,
0.009337655268609524... | |
1f3bc5a1e79bf82dd938c67b1c3c866e43a6f908 | subsection | 42 | 68 | Proof of thm:variance-lipschitz | We utilize the concentration result of the bounded difference.
[see e.g., boucheron2013concentration]
Suppose that X_1,...,X_n are independent random variables on {X}. For a function f:{X}^n\rightarrow , suppose there exist universal constants c_1,...,c_n such that for any i \in [n],\sup _{x_1,...,x_n,x^{\prime }_i}*{f... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06090383604168892,
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0.03014678880572319,
0.023998429998755455,
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-0.013036655262112617,
-0.01323498971760273,
... | |
8855bf447451c0a3140d92a6c0e0515f59da199f | subsection | 43 | 68 | Proof of thm:bias-lipschitz | [Proof of thm:bias-lipschitz]
Application of the Taylor theorem yields there exists \xi _1,...,\xi _k such that& *{\sum _{i=1}^k*{\frac{N_i}{n}} - \theta (P)} \\
=& *{*{\sum _{i=1}^k*{\phi ^{(1)}(p_i)*{\frac{N_i}{n}-p_i} + \frac{\phi ^{(2)}(\xi _i)}{2}*{\frac{N_i}{n}-p_i}^2}}} \\
\le & *{\sum _{i=1}^k\frac{*{\phi ^{(2)... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.04904303327202797,
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0.03341762349009514,
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0.013199198991060257,
0.02877883054316044,
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0.019913241267204285,
-0.027207134291529655,
-0.0034886321518570185,
-0.03421110287308693,
... | |
1476f344c42ecece331e6fe15d65152ae0eab71d | subsection | 44 | 68 | Proof of thm:bias-plugin | [Proof of thm:bias-plugin]
We divide the alphabets into two cases; p_i \le 1/n and p_i > 1/n.Case p_i \le 1/n. Since \phi (0) = 0, we have from the Taylor theorem that there exists \xi _i between \frac{N_i}{n} and p_i such that& *{*{\sum _{i : p_i \le 1/n}*{\phi *{\frac{N_i}{n}} - \phi (p_i)}}} \\
\le & \sum _{i : p_i ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.026495952159166336,
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0.03223470598459244,
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0.04822995886206627,
-0.01548395399004221,
-0.021932421252131462,
-0.037210330367088... | |
0a800e9681e67bd04ba262638084ebe49c4ecb85 | subsection | 45 | 68 | Proof of thm:bias-plugin | Combining lem:bias-moduli,lem:moduli-bound, we have& *{*{\sum _{i : p_i > 1/n}*{\phi *{\frac{N_i}{n}} - \phi (p_i)}}} \\
\le & \sum _{i : p_i > 1/n}*{\phi *{\frac{N_i}{n}} - \phi (p_i)} \\
\lesssim & \sum _{i : p_i > 1/n}\frac{p_i^{\alpha /2}}{n^{\alpha /2}}.Since \sup _{P \in {M}_k}\sum _{i : p_i > 1/n}p_i^{\alpha /2}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.007932649925351143,
0.006510111968964338,
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0.006868606898933649,
-0.006586387753486633,
... | |
2afdf3b0f2cf2d9e5992a8120b6fc0de7e9b173f | subsection | 46 | 68 | Proof of thm:lower1 | We use the LeCam's two point method LeCam:1986:AMS:20451. Let P and Q be two probability vectors in {M}_k. Then, the lower bound is given by
[LeCam:1986:AMS:20451]
The minimax lower bound is given asR^*(n,k;\phi ) \ge \frac{1}{4}*{\theta (P) - \theta (Q)}^2e^{-n(P,Q)},where denotes the KL divergence.
From lem:le-cam-t... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.0668768510222435,
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0.01932777464389801,
-0.0009753509657457471,
0.01... | |
87d35d4be9e4b0bf8727541a1a3c392f12982725 | subsection | 47 | 68 | Proof of thm:lower1 | Thus, \phi ^{(2)} has same sign in p \in (0,p_0]. Hence, for sufficiently large p and q such that 1-p,1-q < p_0,*{\phi ^{(1)}(\xi _1)} =& *{\int _0^{\xi _1}\phi ^{(2)}(s)ds} \\
\ge & \int _0^{\xi _1}*{W_2s^{\alpha -2}-c^{\prime }_2}ds \\
=& \frac{W_2}{\alpha -1}\xi _1^{\alpha -1}-c^{\prime }_2\xi _1 > 0.Also, we have*{... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.05799415707588196,
0.025929491966962814,
-0.051126427948474884,
-0.018909147009253502,
-0.018100280314683914,
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0.021702023223042488,
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0.006280156783759594,
0.018252898007631302,
... | |
7023611de47e258d140ed63f6dfc0b4e68e5c523 | subsection | 48 | 68 | Detailed Analysis of Lower Bound for | First, we derive the association between the minimax risk and the approximated minimax risk defined below. For \epsilon \in (0,1), define the approximated probabilities as{M}_k(\epsilon ) = *{(p_1,...,p_k) \in ^k_+ : *{\sum _{i=1}^kp_i - 1} \le \epsilon }.With this definition, we define the approximated minimax risk as... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.039071224629879,
0.02821980230510235,
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0.005761479493230581,
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0.03537777066230774,
0.04438246786594391,
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0.029913906008005142,
-0.03202008828520775,
0.0191... | |
6e2a852017ac89eb9b5834b798c060ae46c3d9f4 | subsection | 49 | 68 | Detailed Analysis of Lower Bound for | If [V^j] = [V^{\prime j}], j = 1,...,L and L > 2eM, then([(V)], [(V^{\prime })]) \le *{\frac{2eM}{L}}^L.Combining thm:lower-approximated,thm:approx-tv-lower,lem:tv-poi-bound gives the following corollary.
For \alpha \in (1,2), suppose \phi is Lipschitz continuous, and \phi ^{(1)} is (\alpha -1)-Hölder continuous. Let ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.08322975784540176,
0.03514687716960907,
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0.00944267213344574,
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0.014095362275838852,
-0.013050413690507412,
0.... | |
e1404020d6c41f217c584b3c21a914d6ff0e5c63 | subsection | 50 | 68 | Detailed Analysis of Lower Bound for | For any given integer L > 0, \eta > 0, and \gamma \in (0,1) such that \gamma \le \eta , there exists two probability measures \nu _0 and \nu _1 on [0,\gamma /\eta ] such that_{X \sim \nu _0}[X] = _{X \sim \nu _1}[X] = \gamma , \\
_{X \sim \nu _0}[X^m] = _{X \sim \nu _1}[X^m], m=2,...,L+1, \\
_{X \sim \nu _0}[\phi (X)] ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06971905380487442,
0.03409640118479729,
-0.020421210676431656,
-0.011278980411589146,
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0.026419369503855705,
-0.031807027757167816,
0.02217639796435833,
-0.00744046363979578,
0.0... | |
70c2c01b62c99288af8a0beeb28e0e45fd36624c | subsection | 51 | 68 | Detailed Analysis of Lower Bound for | Then, we have\tilde{R}^*(n/2,k;\phi ) \ge \frac{c^2k^2\gamma ^{2\alpha }}{8}[\Bigg ]{\frac{7}{8} - k*{4enL\gamma }^L - \frac{32*{\phi ^{(1)}}_{C^{0,\alpha -1}}^2L^{4\alpha }}{c^2k} \\
- \frac{32*{\phi }_{C^{0,1}}^2e^{-n/32}}{c^2k^2\gamma ^{2\alpha }} - \frac{512*{\phi }_{C^{0,1}}^2L^4\gamma ^{2-2\alpha }}{c^2k}}.It is ... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.056520964950323105,
0.042909543961286545,
-0.036714211106300354,
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0.012703482992947102,
-0.022492414340376854,
0.02897767536342144,
-0.010330640710890293... | |
10e08af9d123eca17f11c6a5408e3274b3c39c5e | subsection | 52 | 68 | Detailed Analysis of Lower Bound for | Letting \delta L = \alpha + \delta ^{\prime } for \delta ^{\prime } > 0, we havek*{4enL\gamma }^L =& k*{\frac{4eC_1{C_2\ln n}^2}{n^\delta \ln n}}^{{C_2\ln n}} \\
\lesssim & \frac{k\ln n}{n^{\alpha +\delta ^{\prime }}} = o(1) \because k \lesssim (n\ln n)^\alpha .Also, for sufficiently small \delta , we have\frac{32*{\ph... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.0700024738907814,
0.04620041325688362,
-0.031247835606336594,
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0.017042886465787888,
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0.003179329913109541,
-0.018644949421286583,
0.... | |
2d9b6f8d7fdb1a50c27cfc438e0ef8f15baf2d14 | subsection | 53 | 68 | Proof of thm:lower-approximated | [Proof of thm:lower-approximated]
This proof is following the same manner of the proof of [Lemma 1]wu2016minimax. Fix \delta > 0. Let \hat{\theta }(\cdot ,n) be a near-minimax optimal estimator for fixed sample size n, i.e.,\sup _{P \in {M}_k}*{*{\hat{\theta }(N,n) - \theta (P)}} \le \delta + R^*(k,n;\phi ).For an arbi... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06303679198026657,
0.024393955245614052,
-0.019695183262228966,
-0.0018917139386758208,
0.007604995276778936,
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-0.001868830295279622,
0.02146485075354576,
0.023127729073166847,
0.061267122626304626,
-0.036491770297288895,
0.007086299359798431,
-0.05013041943311691,
... | |
adcc944f5c0c69fe6882399d4e6a4fa0c627f0b7 | subsection | 54 | 68 | Proof of thm:lower-approximated | From the triangle inequality, \bar{\alpha }-Hölder continuousness of \phi , and lem:bound-sum-alpha, we have& \frac{1}{2}*{\tilde{\theta }(\tilde{N}) - \theta (P)}^2 \\
\le & \frac{1}{2}*{*{\tilde{\theta }(\tilde{N}) - \theta *{\frac{P}{\sum _ip_i}}} + *{\theta *{\frac{P}{\sum _ip_i}} - \theta (P)}}^2 \\
\le & \frac{1}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.019129853695631027,
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0.005983799695968628,
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... | |
6088b70106e45dc8e84240d5792df8731b87157e | subsection | 55 | 68 | Proof of thm:lower-approximated | Therefore, we have*{\tilde{\theta }(\tilde{N}) - \theta *{\frac{P}{\sum _{i=1}^k p_i}}}^2
=& \sum _{m = 0}^\infty *{*{\tilde{\theta }(\tilde{N},m) - \theta *{\frac{P}{\sum _{i=1}^k p_i}}}^2 | n^{\prime } = m}{n^{\prime } = m} \\
\le & \sum _{m = 0}^\infty \tilde{R}^*(m,k;\phi ){n^{\prime } = m} + \delta .From \alpha -H... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.04581838846206665,
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0.007799199316650629,
-0.03269253298640251,
0.009... | |
0ecd5446def321e514723a4d0c218897d60bab31 | subsection | 56 | 68 | Proof of thm:approx-tv-lower | [Proof of thm:approx-tv-lower]
The proof follows the same manner of the proof of [Lemma 2]wu2016minimax expect eq:approx-tv-expect below. Let \beta = [U] = [U^{\prime }] \le 1. Define two random vectorsP = *{\frac{U_1}{k},...,\frac{U_k}{k}, 1-\beta }, P^{\prime } = *{\frac{U^{\prime }_1}{k},...,\frac{U^{\prime }_k}{k},... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.0676480233669281,
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0.03980732336640358,
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0.017446231096982956,
-0.03281661868095398,
0.02... | |
f767254e4657ba43af49c8f5798d73b1700d0a7d | subsection | 57 | 68 | Proof of thm:approx-tv-lower | Hence, the Taylor theorem indicates that there exists \xi _i between 0 and U_i/k such that^c \le & \frac{1}{16} + \frac{16\sum _i*{*{U_i*{\phi ^{(1)}(\xi _i) - \phi ^{(1)}(0)}/k}^2}}{d^2}.From Hölder continuousness of \phi ^{(1)}, we obtain^c \le & \frac{1}{16} + \frac{16\sum _i*{{\phi ^{(1)}}_{C^{0,\alpha -1}}U_i\xi ^... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.06824390590190887,
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0.003101648762822151,
-0.022371547296643257,
-0... | |
04003959ae424983ab78aa29cd7c25b50cee7cde | subsection | 58 | 68 | Proof of thm:approx-tv-lower | By the definition of events ,^{\prime } and triangle inequality, we obtain that under \pi ,\pi ^{\prime },*{\theta (P) - \theta (P^{\prime })} \ge \frac{d}{2}.By triangle inequality, we have the total variation of observations under \pi ,\pi ^{\prime } as(P_{\tilde{N}|}, P_{\tilde{N}^{\prime }|^{\prime }})
\le & (P_{\t... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.011354909278452396,
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0.015048307366669178,
-0.021397288888692856,
... | |
a8b83b45cdd7074f68ac7f8c8ecf25b0a898c64a | subsection | 59 | 68 | Proof of lem:best-approx-solution | [Proof of lem:best-approx-solution]
From lem:prior-construction, there exists a pair of probability measures \rho _0 and \rho _1 on [\eta ,1] such that_{X \sim \rho _0}[X^m] = _{X \sim \rho _1}[X^m], m=0,...,L, \\
_{X \sim \rho _0}[\phi (X)] - _{X \sim \rho _1}[\phi (X)] = 2E_L(\phi , [\eta ,1]).For X \sim \rho _i, let... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.014283154159784317,
0.016572121530771255,
-0.047457918524742126,
-0.015343708917498589,
0.001087259384803474,
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0.005165435839444399,
0.06531186401844025,
0.025590652599930763,
0.014359453693032265,
-0.0168162789195776,
0.005134915933012962,
-0.004078176338225603,
-... | |
f62b0fa10819e2a96ba9b6f1361654eab8c34ff1 | subsection | 60 | 68 | Proof of thm:lower-poly-approx | [Proof of thm:lower-poly-approx]
Letting \phi ^\star _{\eta ,\gamma }(x) = \phi ^\star (\gamma \frac{1+\eta +(1-\eta )x}{2\eta }), we have E_L(\phi ^\star ,[\gamma ,\gamma /\eta ]) = E_L(\phi ^\star _{\eta ,\gamma },[-1,1]). We utilize the first-order Ditzian-Totik modulus of smoothness ditzian2012moduli defined as\ome... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.022145353257656097,
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0.0684964507818222,
-0.025075683370232582,
0... | |
a766341a63e508d741197849511a8d7968bfe7d6 | subsection | 61 | 68 | Proof of thm:lower-poly-approx | Hence,\omega ^1_\varphi (\phi ^\star _{\eta ,\gamma },t) \ge & \sup _{x \in [-1,1]}*{*{\phi ^\star _{\eta ,\gamma }(x)-\phi ^\star _{\eta ,\gamma }(-1)} : 0 \le 1 + x \le \frac{4}{t^{-2}+1}} \\
=& \sup _{x \in [0,1]}[\Bigg ]{*{\phi ^\star *{\gamma *{1+\frac{(1-\eta )x}{\eta }}}-\phi ^\star (\gamma )} : 0 \le x \le t^2}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
-0.026857519522309303,
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-0.030275749042630196,
-0.... | |
f0aa7149b6af390258cbcaf7659931ebf0a25b08 | subsection | 62 | 68 | Proof of thm:lower-poly-approx | For sufficiently small \gamma such that \gamma (1+\frac{(1-\eta )t^2}{\eta }) \le p_0, we have& *{\int _\gamma ^{\gamma *{1+\frac{(1-\eta )x}{\eta }}}\int _0^s\phi ^{(2)}(s^{\prime })ds^{\prime }ds} \\
\ge & *{\int _\gamma ^{\gamma *{1+\frac{(1-\eta )x}{\eta }}}\int _0^sW_2(s^{\prime })^{\alpha -2} - c^{\prime }_2 ds^{... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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-0.0... | |
1f7b9658f3ae981b33f0140066571ced194ca8ac | subsection | 63 | 68 | Proof of thm:lower-poly-approx | From the Jackson inequality, we haveE_L(\phi ^{\star }_\gamma ,[-1,1]) \lesssim \omega _1(\phi ^{\star }_\gamma ,L^{-1})For any x,y \in (-1,1), we have&*{\phi ^{\star }_\gamma (x)-\phi ^{\star }_\gamma (y)} \\
\le & *{\int _y^x 4\gamma L^2s\phi ^{\star (1)}(\gamma (1+2L^2s^2))ds} \\
\le & *{\int _y^x\frac{4L^2s}{1+2L^2... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0.01155116781592369,
-0.035370681434869766,
-... | |
c93cba5eeb44aa200cdc7cf8998ba959e277a1ea | subsection | 64 | 68 | Proof of thm:lower-poly-approx | For the first term, we have&*{\int _y^x\frac{4L^2s}{1+2L^2s^2}\int _0^{\gamma (1+2L^2s^2)}*{\phi ^{(2)}(s^{\prime })}ds^{\prime }ds} \\
\le & *{\int _y^x\frac{4L^2s}{1+2L^2s^2}\int _0^{\gamma (1+2L^2s^2)}W^{\prime }_2(s^{\prime })^{\alpha -2}ds^{\prime }ds}\\
=& 2W^{\prime }_2\gamma ^{\alpha -1}*{\int _y^x\frac{2L^2s(1... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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-0... | |
4c8fc7ea08b9711f1707979688798fdb1c1f3f4e | subsection | 65 | 68 | Proof of thm:lower-poly-approx | Also, we have the converse result \frac{1}{L}\sum _{m=0}^LE_m(f, [-1,1]) \gtrsim \omega ^1_\varphi (f,L^{-1}) ditzian2012moduli. Let L^{\prime } be an integer such that L^{\prime } = c_\ell L where c_\ell > 1. Then, we have& E_L(\phi ^\star ,[\gamma ,2L^2\gamma ])\\
\ge & \frac{1}{L^{\prime } - L}\sum _{m = L+1}^{L^{\p... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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... | |
989d1f15e623d51c6d28c687714d8f76af22e6fb | subsection | 66 | 68 | Proof of thm:lower-poly-approx | Applying the converse result and the fact E_L(\phi ^{\star }_\gamma ,[-1,1]) \lesssim \gamma ^{\alpha -1} yields that there are constants C > 0, C^{\prime } > 0, and C^{\prime \prime } > 0 such that& E_L(\phi ^\star ,[\gamma ,2L^2\gamma ])\\
\ge & C^{\prime }\gamma ^{\alpha -1} - \frac{C^{\prime }\gamma ^{\alpha -1}}{L... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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... | |
34b50d14588f235d525f4fa0bcc3c3febfce51d0 | subsection | 67 | 68 | Helper Lemmas | Given \alpha \in [0,1], \sup _{P \in {M}_k} \sum _{i=1}^k p_i^\alpha = k^{1-\alpha }.
[Proof of lem:bound-sum-alpha]
If \alpha = 1, the claim is obviously true. Thus, we assume \alpha < 1. We introduce the Lagrange multiplier \lambda for a constraint \sum _{i=1}^n p_i = 1, and let the partial derivative of \sum _{i=1}... | {
"cite_spans": []
} | 1801.05362 | Minimax Optimal Additive Functional Estimation with Discrete
Distribution: Slow Divergence Speed Case | [
"Kazuto Fukuchi",
"Jun Sakuma"
] | [
"cs.IT",
"math.IT",
"math.ST",
"stat.TH"
] | 2,018 | en | Computer Science | [
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0... | |
a002beac008c919ab17493682070fc12016b7abb | abstract | 0 | 17 | Abstract | In this paper we list all possible degrees of a faithful transitive
permutation representation of the group of symmetries of a regular map of types
$\{4,4\}$ and $\{3,6\}$ and we give examples of graphs, called CPR-graphs,
representing some of these permutation representations. | {
"cite_spans": []
} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
"Claudio Alexandre Piedade"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... | |
a1e2ad325e8ca37040a6c66c9177718452bce311 | subsection | 1 | 17 | Introduction | The classification of highly symmetric objects, particularly regular maps and polytopes, is a problem that attracts both geometers and algebraists.
The idea of using permutation representations to classify regular maps and polytopes is not new but in 2008 the concept of a graph associated with a regular polytope, calle... | {
"cite_spans": []
} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
"Claudio Alexandre Piedade"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
530ce2a81fe8b723254146a047d50fd72f268461 | subsection | 2 | 17 | Introduction | The problem is extensible to regular hypertopes of rank 4 with toroidal rank 3 residues and we strongly believe that CPR graphs of regular toroidal maps may play an important role in the classification of regular hypertopes with toroidal rank 3 residues. Indeed, the group theoretical conditions for an incidence system ... | {
"cite_spans": []
} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
"Claudio Alexandre Piedade"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.033393993973731995,
-0.008295080624520... | |
ee9ab7ce596d5f51e668f72fb6ee4ebf6c498d72 | subsection | 3 | 17 | Toroidal regular maps | Consider the regular tessellations of the plane by identical squares, triangles or hexagons. These tesselations are infinite regular 3-polytopes whose full symmetry groups are the Coxeter groups [4, 4], [3,6] or [6,3], respectively, generated by three reflections \rho _0,\, \rho _1,\rho _2, as shown in Figures REF and ... | {
"cite_spans": []
} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
"Claudio Alexandre Piedade"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.012255366891622543,
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0.006245963741093874,
0.01938270963728428,
-... | |
5441ff77bd225dfd18cfb33f4f4007609462840b | subsection | 4 | 17 | Toroidal Map | Consider the toroidal map \lbrace 4,4\rbrace _{(s,t)} having V=s^2+t^2 vertices, 2V edges and V faces, that is the obtained identifying opposite sides of the parallelogram with vertices (0,0), (s,t), (s-t,s+t) and (-s,t), as shown in Figure REF .
[Figure: Toroidal map of type \lbrace 4,4\rbrace .]The product of two ref... | {
"cite_spans": []
} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
"Claudio Alexandre Piedade"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.004048644099384546,
-0.0067693633027374744,
-0.002680652542039752,
-0.017110386863350868... | |
e00c01442ab722006c6af69f9e35823b8f7a6c6f | subsection | 5 | 17 | Toroidal Map | Indeed the group of \lbrace 4,4\rbrace _{(s,0)} is isomorphic to a subgroup of index 2 of \lbrace 4,4\rbrace _{(s,s)} and the group of the map \lbrace 4,4\rbrace _{(s,s)} is also isomorphic to a subgroup of index 2 of the group of the map \lbrace 4,4\rbrace _{(2s,0)}.For the map \lbrace 4,4\rbrace _{(s,0)} consider the... | {
"cite_spans": []
} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
"Claudio Alexandre Piedade"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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