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on A1,0,0∪A2,0,0∪A3,0,0∪A0,2,0 [0,C42δ−1] : on A0,3,0∪A1,1,0∪A2,1,0∪A0,0,1∪A1,0,1 [0,C42δ−2] : on the sample space Ω. (59) Proof. We bound the value of W4on various events in (59). On the event A0,0,0∪A0,1,0∪A0,2,0∪A0,3,0:On this event we do not have any signal flips, but we do allow for moderate failures of reconstruc... | https://arxiv.org/abs/2501.17622v1 |
3.3(i), P(B1)≥1−2c6δ and P(B2)≥1−c22δ. Let us write A:=B1∩B2∩[ i+j+2k≤3Ai,j,k∪Bc 1∪Bc 2. Observe that by using (57) and the above bounds, P(A)=1−O(δ4). Together with (60), this yields E[eW41Ac]=O(1). Hence in order to show the first moment bound oneW4in (44), it su ffices to show that E[eW41A]≤C62. (62)... | https://arxiv.org/abs/2501.17622v1 |
implies that σy4= +1 as we are on the event B1. Since there is a single flip and no severe failure, we can break it into three sub-cases: (1)σw4= +1 and good reconstruction at w4; (2)σw4= +1 and moderate failure at w4; (3)σw4=−1. For the sub-case (1), we have σw4= +1 and|η4|≥1−O(δ). Then since there is no severe failur... | https://arxiv.org/abs/2501.17622v1 |
such that ηj≥1−O(δ). One the event B3:={η1,η2≥1−O(δ)}∪{η3,η4≥1−O(δ)}, we have W4=O(δ−1) by (58) and Proposition 6.8. Hence using the second equality in (60) and (57), we deduce EheW41[B1∩B2∩A0,1,1∩{σy=1}∩B3]i =O(δ−2)E[W41[B1∩B2∩A0,1,1∩{σy=1}∩B3]]=O(δ−3)P(A0,1,1)=O(1). It remains to consider the case where ηj≥1−O(δ) for... | https://arxiv.org/abs/2501.17622v1 |
generic bound on the denominator (74) R1≤1 (2c6δ)2=O(δ−2) using a generic bound on the denominator (again) (75) and R1≤1 (1−c23)2=O(1) whenever σw0η0∈[−c23,0] andξ0η0<0. (76) Using the above, we can further bound the second term in (71) as Eθ∗[R1]=1+O(δ−1)| {z } by (74)P(E1)+O(δ−2)P(E4)+O(δ−2)P(E2)| {z } both by (75)+O... | https://arxiv.org/abs/2501.17622v1 |
VID CLANCY , JR., HANBAEK LYU, AND SEBASTIEN ROCH Also, it is easy to see that if Zx≥0 and Zw,Zv<0 then ˆθfˆθgZwZy>0 and 1 +Zxˆθeq(ˆθgZw,ˆθfZv)≥ 1−ˆθeZx. This and the first inequality in (79) give ifZx≥0 and Zw<0,Zv<0 then ∂2 ∂ˆθe∂ˆθfℓ(ˆθ,σ) ≤1−(ˆθgZw)2 (1−ˆθeZx)2. (81) Note that the right-hand side satisfies 1−(ˆθgZw)... | https://arxiv.org/abs/2501.17622v1 |
is of O(1) and so we do not need to consider A∩{σwZw≥1−C22δ2}. By Claim 3.4 and Thm. 3.3 P(A∩{σwZw∈[−c23,1−O(δ2)], σ vZv<1−O(δ2)})=O(δ2) and so, using (85), the expectation of eR0onA∩{σvZv<1−O(δ2)}is also O(1). Note that onA′we haveσy= +1 and Zw<0. IfσwZw∈[0,1−O(δ2)) thenσw=−1 and there was a flip between wandy. By Cla... | https://arxiv.org/abs/2501.17622v1 |
a union bound and recalling that a flip has probability at most C5δby A1, Pθ∗(B\B′)≤5C5δ. 36 DA VID CLANCY , JR., HANBAEK LYU, AND SEBASTIEN ROCH Therefore by (88), Eθ∗ ∂2 ∂ˆθe∂ˆθfℓ(ˆθ,σ|L) 1B≤Eθ∗ ∂2 ∂ˆθe∂ˆθfℓ(ˆθ,σ|L) 1B′+O(1). Now onB′all signals have the same sign (as all the spins are the sam... | https://arxiv.org/abs/2501.17622v1 |
to get the generic upper bound ∂2 ∂ˆθe∂ˆθfℓ(ˆθ;σ|L) =O(δ−3). (91) More precisely, usingpˆθ3η3,η2,η1,η0,ξ0∈[−1+O(δ),1−O(δ)] a.s. by Cor. 3.5 and the fact thatpˆθ3=1−Θ(δ) under Assumption A1, we get ∂2 ∂ˆθe∂ˆθfℓ(ˆθ;σ|L) =1 1−(ˆθ1/2 3η3)2(1−(ˆθ1/2 3η3)2)(1−η2 2)(1−η2 1)(1−η2 0) (1+ξ3η3)2(1+ξ2η2)2(1+ξ1η1)2(1+ξ0η0)2 =O(δ−1)... | https://arxiv.org/abs/2501.17622v1 |
The paper is based upon work supported by the NSF under grant DMS-1929284 while one of the authors (SR) was in residence at the Institute for Computational and Experimental Re- search in Mathematics (ICERM) in Providence, RI, during the Theory, Methods, and Applications of Quantitative Phylogenomics semester program. S... | https://arxiv.org/abs/2501.17622v1 |
of evolution: a source of novel statistical problems. In Statistical decision theory and related topics , pages 1–27. Elsevier, 1971. [19] L.-T. Nguyen, H. A. Schmidt, A. von Haeseler, and B. Q. Minh. IQ-TREE: a fast and e ffective stochastic algo- rithm for estimating maximum-likelihood phylogenies. Molecular Biology ... | https://arxiv.org/abs/2501.17622v1 |
arXiv:2501.17722v3 [math.ST] 7 Mar 2025Fundamentals of non-parametric statistical inference for integrated quantiles Nadezhda V. Gribkova∗,1,2, Mengqi Wang†,3, and Riˇ cardas Zitikis‡,3 1Saint Petersburg State University, Saint Petersburg, 199034 Ru ssia 2Emperor Alexander I Saint Petersburg State Transport Universit y... | https://arxiv.org/abs/2501.17722v3 |
. . . . . . . . . . . . . . . . . 42 7.2 Downside tail-value-at-risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2.2 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2501.17722v3 |
(5.11) and (6.10) . . . . . . . . . . . . . . . . . . . . 46 8.2M-mixing and statements (3.24), (5.12) and (6.11) . . . . . . . . . . . . . . . . . . . 49 9L-functionals 53 10 Conclusion 59 A Technicalities 66 A.1 Auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.2 Proofs... | https://arxiv.org/abs/2501.17722v3 |
stat istics (e.g., Arnold et al. , 2008;David and Nagaraja ,2003), unlike empirical cumulative distribution functions (cdf’s) that are sums of (Bernoulli) random variables and can therefore be easily tackled using classical techniques of mathematical statistics and probability the ory (e.g., Shorack,2017). Inevitably, ... | https://arxiv.org/abs/2501.17722v3 |
Based on the observation, they set out to find a path that would lea d to desired asymptotic results without involving pdf’s. They succeeded in achieving this goal by introducing a technical tool that they called the Vervaat process, named afte r the Dutch mathematician Wim Vervaat, whose pioneering results ( Vervaat,19... | https://arxiv.org/abs/2501.17722v3 |
empirical upper-layer integral. Before we delve into the topics of the following sections, we present several notes that are useful for understanding and appreciating the results of this paper, and in particular the assumptions under which the results will be established. Note 2.1. Whenp= 0, integral ( 2.1) reduces to ... | https://arxiv.org/abs/2501.17722v3 |
the ge neralized (or absolute) 7 xF(x) x1 x20 p1 p2 1 (a) Cdf: The only p∈(0,1) that fails condition ( 2.2) isp=p2, provided that x1< x2. When x1=x2(no gap), every p∈(0,1) satisfies the condition.xF−1(x) 0 p1 p2 1x1 x2 (b) Quantile function: The only disconti- nuity of F−1is atp=p2, provided that x1< x2. When x1=x2(no j... | https://arxiv.org/abs/2501.17722v3 |
(so-called P&L variables), the underlying pop- ulation distribution may span the entire real line R, and thus both upper- and lower-layer integrals ( 2.1) and (2.3) may show up simultaneously. There are also problems that give rise to various combinations of the three integrals introduced above, a nd one of such instan... | https://arxiv.org/abs/2501.17722v3 |
1−Fn(x)/parenrightbig dx∈Rmeans/integraltext∞ F−1(p)/parenleftbig 1−Fn(ω,x)/parenrightbig dx∈Rfor almost all ω∈Ω. Furthermore, to enable ourselves to calculate, e.g., the expected values of integrals like /integraltext1 pF−1 n(u)duand/integraltext∞ F−1(p)/parenleftbig 1−Fn(x)/parenrightbig dx, we need joint measurabili... | https://arxiv.org/abs/2501.17722v3 |
Corollary 3.1. Example 3.1 (SRS).LetX1,...,X nbe SRS, that is, iid random variables. The empirical cdfFn,srsis defined by the equation Fn,srs(x) =1 nn/summationdisplay i=11{Xi≤x}, which is the arithmetic mean of nindependent copies of the Bernoulli random variable 1{X≤x}taking value 1 with probability F(x) and 0 otherwi... | https://arxiv.org/abs/2501.17722v3 |
for the main term in as ymptotic expansion ( 3.18) by using, for example, the empirical estimator F−1 n,srs(p) forF−1(p) and a kernel density estimator for f(e.g.,Prakasa Rao ,1983). For a discussion of the role of the density-quantile function p/ma√sto→f(F−1(p)) in statistics, and for a comprehensive statistical infer... | https://arxiv.org/abs/2501.17722v3 |
converges in probability to 0. Since Rem( p;F,Fn) is non-negative and does not exceed /vextendsingle/vextendsingleF−1 n(p)−F−1(p)/vextendsingle/vextendsinglesup x∈R/vextendsingle/vextendsingleFn(x)−F(x)/vextendsingle/vextendsingle, the statement AnRem(p;F,Fn) =oP(1) follows from conditions ( 3.21) and (3.22). This esta... | https://arxiv.org/abs/2501.17722v3 |
equation ( 3.13)) σ2 upper(p) = Var/parenleftBig/parenleftbig X−F−1(p)/parenrightbig+/parenrightBig = Var/parenleftBig/parenleftbig F−1(U)−F−1(p)/parenrightbig+/parenrightBig (3.28) are sometimes easier to use, where Udenotes a uniform on [0 ,1] random variable. 18 Note 3.8 (existence of uniform rv’s) .We often assume ... | https://arxiv.org/abs/2501.17722v3 |
( 3.29) and (3.30) give computationally-friendly formulas for the empirical upper- layer integral, which is connected to the left-trimmed mean (e.g., Stigler,1973) via equa- tion (3.30). 4 The upper-layer integral: simulated experiments In this section we present several experiments that empirically illust rate some of... | https://arxiv.org/abs/2501.17722v3 |
3.28) withF−1(u) = 2ufor all 0 ≤u≤1. Simulation results based on m= 10000 replications of ∆nare depicted in the form of a relative histogram in Figure 4.1, which supports the Δn−4 −3 −2 −1 0 1 2 3 40 0.1 0.2 0.3 0.4 Figure 4.1: The relative histogram of m= 10000 values of ∆ nin (4.1) whenn= 100000. limiting standard no... | https://arxiv.org/abs/2501.17722v3 |
two statements (see Appendix A.6.2) P/parenleftbig Z⌈n/2⌉:n>3/2/parenrightbig →1/2, (4.7) P/parenleftbig Z⌈n/2⌉:n<1/2/parenrightbig →1/2. (4.8) This explains non-consistency statement ( 4.4), which we have visualized in Figure 4.3using the formulas H−1 n,srs(1/2) =Z⌈n/2⌉:n(=Z(n/2):n), H−1(1/2) = 1/2 for various (even) ... | https://arxiv.org/abs/2501.17722v3 |
out the underlying theory. We continue our analysis of the lower-layer integral by noting that t he functional F− 1∋F/ma√sto→/integraldisplayp 0F−1(u)du∈R is not linear. It appears that when developing statistical inference it can be approximated by the linear one F− 1∋G/ma√sto→/integraldisplayF−1(p) −∞G(x)dx∈R. (5.1) ... | https://arxiv.org/abs/2501.17722v3 |
Vn,srs(p) =/integraldisplayp 0/parenleftbig F−1 n,srs(u)−F−1(u)/parenrightbig du+/integraldisplayF−1(p) −∞/parenleftbig Fn,srs(x)−F(x)/parenrightbig dx =/integraldisplayp 0/parenleftbigg/parenleftbig F−1 n,srs(u)−F−1(u)/parenrightbig +1 f(F−1(u))/parenleftbig Fn,srs(F−1(u))−u/parenrightbig /bracehtipupleft /bracehtipdo... | https://arxiv.org/abs/2501.17722v3 |
F∈ F− 1∩Tε, the empirical estimator/integraltextp 0F−1 n,srs(u)du of the lower-layer integral/integraltextp 0F−1(u)du, although being consistent by Example 5.1, has the positive bias Biaslower n,srs(p):=E/parenleftbig Rem(p;F,Fn,srs)/parenrightbig ∈[0,∞). (5.9) Note5.4. Theconditionthat F∈ F− 1∩Tεforsomeε >0canequivale... | https://arxiv.org/abs/2501.17722v3 |
dxdy. (5.15) Note 5.6 (dependent data) .Limiting distributions extending statement ( 5.14) to classes of dependent random sequences were studied by Davydov and Zitikis (2003),Davydov and Zitikis(2004), andDavydov et al. (2007). The knowledge of empirical processes and their asymptotic behaviour based on such random seq... | https://arxiv.org/abs/2501.17722v3 |
p1< p2<1 to 0≤p1< p2≤1. This is indeed possible by augmenting the definition of the remainder term Rem( p;F,G), which has so far been given only for p∈(0,1), by setting Rem( p;F,G) to 0 when p∈ {0,1}. If we agree with this augmentation, then it is also imperative to replace F−1(p1) by−∞whenp1= 0 in the 35 integral/integ... | https://arxiv.org/abs/2501.17722v3 |
Hence, we conclude from Corollary 6.2that when F∈ Tε, the estimator/integraltextp2 p1F−1 n,srs(u)duof the middle-layer integral/integraltextp2 p1F−1(u)du, although being consistent by Example 6.2, has the bias Biasmiddle n,srs(p1,p2) =E/parenleftbig Rem(p2;F,Fn,srs)/parenrightbig −E/parenleftbig Rem(p1;F,Fn,srs)/parenr... | https://arxiv.org/abs/2501.17722v3 |
5.17) for the two empirical lower-layer integrals on the right-hand side of equation ( 6.15). (All empirical integrals are finite, and so manipulations like those in equation ( 6.15) do not pose any technical issues.) In particular, we obtain the following equations /integraldisplayp2 p1F−1 n,srs(u)du=1 n⌈np2⌉/summation... | https://arxiv.org/abs/2501.17722v3 |
distribut ion but also uniform convergence overall p∈(0,1),whichallowsonetoestablishconfidencebandsforthefunction p/ma√sto→TVaR(p). Statement ( 7.3) is also a special case of Gribkova et al. (2022b) who consider the limiting distribution of the tail conditional allocation, which genera lizes TVaR( p). 7.2 Downside tail-... | https://arxiv.org/abs/2501.17722v3 |
we obtain from results of Section 5and equation ( 1.6) that (see Lemma A.3in Appendix A.3for a proof) √n/parenleftbig LCn,srs(p)−LC(p)/parenrightbigd→ N(0,σ2 lc(p)), (7.10) wheretheasymptoticvariance σ2 lc(p)isthesecondmomentofthemean-zerorandomvariable Ylc(p) =1 µ/integraldisplayF−1(p) −∞/parenleftbig 1{X≤x}−F(x)/pare... | https://arxiv.org/abs/2501.17722v3 |
al. (2011), respectively, and they cover many time series (linear and non- linear). Very importantly, it has also turned out that verifying S- andM-mixing conditions is often easier than verifying classical mixing conditions. Berkes et al. (2009,2011) provide illuminating discussions of these matters with accompanying ... | https://arxiv.org/abs/2501.17722v3 |
and so γm=m−Ain this case. We can now formulate the following corollaries to Theorem 8.1. Corollary 8.1. Let all the conditions of Theorem 8.1be satisfied. If F∈ F+ 1and the quantile function F−1is continuous at the point p, then statement (3.23)holds, that is, √n/parenleftBigg/integraldisplay1 pF−1 n,ts(u)du−/integrald... | https://arxiv.org/abs/2501.17722v3 |
the class Fp=F− p∩F+ p. As to the random variable Ψ(m) tpostulated in the condition, Berkes et al. (2011) offer several recipes for constructing it. The following example illustrates the notion and how to verify it. 50 Example 8.2 (AR(1) is Mp-mixing).Let (Xt)t∈Zbe the same AR(1) time series as in Example 8.1. In additi... | https://arxiv.org/abs/2501.17722v3 |
to Wang(1998), andJones and Zitikis (2003), with the latter paper being perhaps the first one to connect actuarial risk measures with L-functionals for the sake of developing statistical inference for the risk measures. To see the role ofL-functionals in the theory of distorted expectations, which are of particular inte... | https://arxiv.org/abs/2501.17722v3 |
2(x)F−1(u)du/parenrightbigg dx,(9.2) with the technical details justifying the right-most equation given in Appendix A.6.5. Equa- tion (9.2) paves a path that connects the results of Section 3with asymptotic properties of the difference Lw(Fn)−Lw(F), where Fn,n∈N, is a sequence of cdf’s that approximate Fwhennincreases.... | https://arxiv.org/abs/2501.17722v3 |
<1/2}/parenleftBig w11(u)−w12(u)/parenrightBig +1{1/2≤u <1}/parenleftBig w21(u)−w22(u)/parenrightBig (9.4) with the functions w11(u) =w(u) and w12(u) = 0, w21(u) = 0 and w22(u) =−w(u). Another way is to use equation ( 9.4) with the functions w11(u) = 6uandw12(u) = 6u2, w21(u) =−6(1−u)2andw22(u) =−6(1−u). Note that irre... | https://arxiv.org/abs/2501.17722v3 |
for inte grals of quantiles. The re- quired conditions are formulated in terms of general approximating sequences of cdf’s Fn, which are not attached to any specific sampling design or dependenc e structure. To maxi- mally illuminate the developed theory, we have illustrated the obtained results using simple random samp... | https://arxiv.org/abs/2501.17722v3 |
curves based on the empirical Lorenz cur ve.Journal of Statistical Planning and Inference , 74, 65–91. https://doi.org/10.1016/S0378-3758(98)00103-7 Cs¨ org˝ o,M.andZitikis, R.(1996).Strassen’sLILfortheLoren zcurve.Journal of Multivariate Analysis, 59, 1–12. https://doi.org/10.1006/jmva.1996.0050 Cs¨ org˝ o, M. and Zit... | https://arxiv.org/abs/2501.17722v3 |
consistency of the M≪Nbootstrap approximation for a trimmed mean. Theory of Probability and Its Applications, 55, 42–53. https://doi.org/10.1137/S0040585X97984607 Gribkova, N., Su, J. and Zitikis, R. (2022a). Empirical tail conditiona l allocation and its consistency under minimal assumptions. Annals of the Institute o... | https://arxiv.org/abs/2501.17722v3 |
472–477. https://doi.org/10.1214/aos/1176342412 Stigler, S.M.(1974).Linearfunctionsoforderstatisticswithsmoo thweightfunctions. Annals of Statistics , 2, 676–693. https://doi.org/10.1214/aos/1176342756 Vervaat, W. (1972a). Success Epochs in Bernouli Trials: with Applications in Num ber The- ory.MathematicalCentre Tract... | https://arxiv.org/abs/2501.17722v3 |
p/parenleftbig F−1(s)−F−1(p)/parenrightbig ds =/integraldisplay1 pF−1(s)ds−(1−p)F−1(p). To prove the second half of the lemma, we first note that in view of eq uation (A.4), the random variable Yupper(p) has a finite second moment whenever F∈ F+ 2. To prove equation ( 3.27), we use Fubini’s theorem and have Var/parenleft... | https://arxiv.org/abs/2501.17722v3 |
results need to include, as a special case, the empirical cd fFn,srs, which is a step- wise function and thus has flat regions as well as jumps, good care ( recallWacker(2023)) needs to be taken to maintain the largest possible class of cdf’s. Hen ce, only moment-type assumptions are required, as specified in the formula... | https://arxiv.org/abs/2501.17722v3 |
/integraldisplayp2 p1/parenleftbig G−1(u)−F−1(u)/parenrightbig du=p2/parenleftbig G−1(p2)−F−1(p2)/parenrightbig −p1/parenleftbig G−1(p1)−F−1(p1)/parenrightbig −/parenleftbigg/integraldisplayG−1(p2) G−1(p1)G(x)dx−/integraldisplayF−1(p2) F−1(p1)F(x)dx/parenrightbigg , 73 which can be rewritten as /integraldisplayp2 p1/pa... | https://arxiv.org/abs/2501.17722v3 |
equation ( A.21) is zero when w−1(x)≤a. Hence, continuing with 77 equation ( A.21) for only those xthat satisfy a < w−1(x), we have /integraldisplayb aF−1(u)1{w(u)<0}w(u)du=−/integraldisplay0 −∞1{a < w−1(x)}/parenleftBigg/integraldisplayb aF−1(u)1{u < w−1(x)}du/parenrightBigg dx =−/integraldisplay0 −∞1{a < w−1(x)}/pare... | https://arxiv.org/abs/2501.17722v3 |
A.27) follows if we establish at least one of statements ( 4.7) and (4.8), but for completeness of the argument, we shall next establish both of them. Using a well-known formula (e.g., David and Nagaraja ,2003, eq. (2.1.3)) for the cdf of order statistics, we have P/parenleftbigg Z⌈n/2⌉:n>3 2/parenrightbigg = 1−n/summa... | https://arxiv.org/abs/2501.17722v3 |
Universal Inference for Incomplete Discrete Choice Models∗ Hiroaki Kaido Boston University hkaido@bu.eduYi Zhang Jinan University yzhangjnu@outlook.com January 31, 2025 Abstract A growing number of empirical models exhibit set-valued predictions. This paper develops a tractable inference method with finite-sample valid... | https://arxiv.org/abs/2501.17973v1 |
regularity conditions or tuning parameters. We propose a novel inference method for conducting hypothesis tests and constructing confidence sets (or intervals) in incomplete models. Specifically, we consider testing the composite hypotheses: H0:θ∈Θ0v.s. H0:θ∈Θ1, (1.1) for subsets Θ 0and Θ 1of a parameter space Θ. Our p... | https://arxiv.org/abs/2501.17973v1 |
to apply a straightforward conditioning argument and a Chernoff-style bound developed in Wasserman et al. (2020), yielding a simple yet non-trivial critical value. The method proposed here offers a tractable way for practitioners to perform inference on various models while avoiding ad-hoc assumptions. It is particular... | https://arxiv.org/abs/2501.17973v1 |
a finite-sample valid method for incomplete models using the idea of Monte Carlo tests. Our proposal differs from theirs mainly in two respects. First, we focus on composite hypothesis tests with inference on subvectors and counterfactual objects 4Each of the properties has been studied somewhat separately. For (i), ot... | https://arxiv.org/abs/2501.17973v1 |
of the recently developed moment-based inference methods study subvector inference and attain computational tractability by focusing on spe- cific classes of models or testing problems (Cox and Shi, 2022; Andrews et al., 2023). This paper focuses on the class of incomplete discrete choice models, which is not nested by... | https://arxiv.org/abs/2501.17973v1 |
(Bresnahan and Reiss, 1990, 1991; Tamer, 2003). Each player may either choose y(j)= 0 or y(j)= 1. Let x(j)andu(j)be player j’s observable and unobservable characteristics. The payoff of player jis π(j)=y(j) x(j)′δ(j)+β(j)y(−j)+u(j) , j= 1,2 (2.6) where y(−j)∈ {0,1}is the opponent’s action. Let θ= (β′, δ′)′, and assum... | https://arxiv.org/abs/2501.17973v1 |
ˆθ1be any estimator of θcomputed from sample D1. Step 2: Using D0, construct a tailor-made likelihood function L0(θ) =Y i∈D0qθ(Yi|Xi), θ∈Θ0∪ {ˆθ1}, (2.11) where qθ(y|x) is the LFP-based density we introduce below. Let ˆθ0be the restricted maximum likelihood estimator (RMLE) based on D0: ˆθ0∈arg max θ∈Θ0L0(θ); (2.12) St... | https://arxiv.org/abs/2501.17973v1 |
we find a positive density p(·|x) compatible with the unrestricted estimator, by solving the following linear feasibility problem: Find p(·|x)∈∆Y(2.18) s.t.X y∈Ap(y|x)≥νˆθ1(A|x), A∈ C. Any solution can be used as long as p(y|x)>0 for all y∈ Y. We then set qˆθ1=p. One may view qˆθ1as a representative density in the unre... | https://arxiv.org/abs/2501.17973v1 |
this can be done for various models and provides a graph-based algorithm to obtain such inequalities (called the smallest core-determining class (CDC)).10 The second approach is to add a restriction to the model. For example, in discrete games, grouping players into several types helps reduce the number of inequalities... | https://arxiv.org/abs/2501.17973v1 |
with the degrees of freedom. This difference arises because we use the out-of-sample likelihood L0(ˆθ1) whose behavior differs from the in-sample likelihood Lfull(ˆθ1).13 Confidence regions for functions of θcan be constructed in a simple manner. Let φ: Θ→Rdφ. Examples of φ(θ) are subvectors of θand counterfactual obje... | https://arxiv.org/abs/2501.17973v1 |
testing H0:β(j)= 0, j= 1,2v.s. H 1:β(j)<0,for some j. (3.12) For this hypothesis, we let Θ 0={θ= (β′, δ′)′:β= 0}.16One can also construct confidence intervals for counterfactual probabilities. Let Y(j)(x(j), y(−j))≡1{x(j)′δ(j)+β(j)y(−j)+U(j)≥0} (3.13) be the potential entry decision by player jwhen the covariates and t... | https://arxiv.org/abs/2501.17973v1 |
of the random set Γ( X, U;θ): νθ(A) =Z XZ U1{Γ(x, u;θ)⊂A}dFθ(u|x)dPX(x). The set function νθbelongs to a class of two-monotone capacities whose properties have proven powerful for conducting robust inference (Huber, 1981).20For this class, the rejection 18To be precise, they apply Markov’s inequality to Tn, which can b... | https://arxiv.org/abs/2501.17973v1 |
test is at most α. The argument above fixed θ1, but an estimator ˆθ1is used in practice. In the proof of Theorem 1, we use a conditioning argument to handle the randomness of ˆθ1. Remark 4.1. Our analysis requires Pθ,xto be the core of a two-monotone capacity. In some applications, one may want to work with a set Mθ,xo... | https://arxiv.org/abs/2501.17973v1 |
0.060 0.235 0.522 0.794 0.948 0.988 0.997 1 1 1 1 1 1 LR-test (moment-based ˆθ1) 0.002 0.008 0.066 0.246 0.521 0.776 0.940 0.988 0.996 1 1 1 1 1 1 Table 2: Size and Power of the Cross-Fit Tests for testing H0:δ(j)= 0, j= 1,2. n Size Power (values of hbelow) 0.105 0.211 0.316 0.421 0.526 0.632 0.737 0.842 0.947 1.053 1.... | https://arxiv.org/abs/2501.17973v1 |
cross-fit LR test takes about 14 seconds (without any parallelization), which is significantly below the computation time required for the moment-based test with parallelization. In sum, the simulation results show that the cross-fit LR test has considerable power, even in small samples. In large samples for which exis... | https://arxiv.org/abs/2501.17973v1 |
Theorems Proof of Theorem 1. We present a version of the proof for the split-sample statistic Tnfirst. Letθ∈Θ0and let Pn∈ Pn θ. Let PDjbe the marginal distribution of PnonDj, and let Pn(·|D1) be the conditional distribution given {(Yi, Xi), i∈D1}. By Markov’s inequality, Pn(Tn>1 α)≤αEPn[Tn] =αEPD1[EPn[Tn|D1]]. (A.4) [2... | https://arxiv.org/abs/2501.17973v1 |
where Λn=dQn 1/dQn 0. The LFP consists of the product measures: Qn 0=nO i=1Q0,i,and Qn 1=nO i=1Q1,i, (B.5) where, for each i∈N,(Q0,i, Q1,i)∈ Pθ0,i× P θ1,iis the LFP in the i-th experiment; The result above states for distinguishing θ0against θ1, the least favorable pair consists of product measures. If ( Xi, Ui) are id... | https://arxiv.org/abs/2501.17973v1 |
are independently distributed across i. Therefore, conditioning on ( Xi, Ui), i∈D1does not provide additional information on the observations from D0through selection across the two subsamples. Below, we condition on D1and treat ˆθ1as fixed. Let Qn ˆθ1∈˜Pn ˆθ1. Consider a minimax testing problem between ˜Pn θ and{Qn ˆθ... | https://arxiv.org/abs/2501.17973v1 |
Identified Parameters in Moment Inequality Models,” Quantitative Economics , 8, 1–38. Canay, I. A. and A. M. Shaikh (2017): Practical and Theoretical Advances in Inference for Partially Identified Models , Cambridge University Press, 271–306, Econometric Society Monographs. Chamberlain, G. (2010): “Binary Response Mode... | https://arxiv.org/abs/2501.17973v1 |
Parameters in Panel Dynamic Discrete Choice Models,” Econometrica , 74, 611–629. [32] Horowitz, J. L. and S. Lee (2023): “Inference in a Class of Optimization Problems: Con- fidence Regions and Finite Sample Bounds on Errors in Coverage Probabilities,” Journal of Business & Economic Statistics , 41, 927–938. Huber, P. ... | https://arxiv.org/abs/2501.17973v1 |
Robust Mean Estimation With Auxiliary Samples Barron Han∗, Danil Akhtiamov∗, Reza Ghane∗, Babak Hassibi Department of Electrical Engineering Caltech Pasadena, California ∗Equal contribution Email: {bshan, dakhtiam, rghanekh, hassibi}@caltech.edu Abstract —In data-driven learning and inference tasks, the high cost of ac... | https://arxiv.org/abs/2501.18095v2 |
distribution? We aim to address such questions by analyzing the fundamental problem of mean estimation. Robust estimation using auxiliary data has a rich history in prior literature. The “data contamination" setting considers the auxiliary data as contaminated samples from the true distribution. The samples are not lab... | https://arxiv.org/abs/2501.18095v2 |
problem with mean square loss, [10] determined the exact min-max risk, the worst-case Wasserstein shift, and the least squares estimator. The remainder of this paper is organized as follows. In Section II, we formalize the auxiliary data model, which produces nsamples from the true distribution and Nsamples from an aux... | https://arxiv.org/abs/2501.18095v2 |
objective represents a ratio (up to a constant factor) between the MSE of the estimator f, and the error of an estimator that utilizes samples from the true distribution. For instance, the sample mean from the true distribution achieves a MSE of1 nTr(ΣX). We make the following linearity assumption on the estimator: Ass... | https://arxiv.org/abs/2501.18095v2 |
R∗, except for introducing a different dependence on the dimension of the problem d. In general, the weighting factor approaches s→1, implying the estimate becomes more sensitive to the true samples while disregarding the auxiliary samples, if ϵ2≫δ2 n. This condition suggests that the auxiliary distribution provides a ... | https://arxiv.org/abs/2501.18095v2 |
> 1the optimization in siis quadratic with a single constraint s1≥si≥sdand we have: si=s:= 1 1+N nifs1≥1 1+N n≥sd s1ifs1<1 1+N n sdifsd>1 1+N n Consider the cases above one by one: •Ifs1≥1 1+N n≥sd, then sd= 1 +N n=sas well because this choice of sdminimizes both (1−sd)2ands2 d n+(1−sd)2 N under the given const... | https://arxiv.org/abs/2501.18095v2 |
compared to just averaging over the true distri- bution. However, the improvement is marginal unless one of the following conditions holds: the auxiliary data distribution matches the original one closely, the original data is very noisy or there is a very limited number of samples from the original distribution. Poten... | https://arxiv.org/abs/2501.18095v2 |
at: min A∥A−I∥2 opϵ2 δ2+∥ATA n+(I−A)T(I−A) N∥op Taking the SVD and diagonalizing the objective again similar to the proof of Theorem 1, we arrive at min s1≥···≥ sd≥0max i=1,...,d(si−1)2ϵ2 δ2+ max i=1,...,ds2 i n+(1−si)2 N Hence, the minimum is achieved when s1=···=sdand the objective turns into: min s1≥···≥ sd≥0max i... | https://arxiv.org/abs/2501.18095v2 |
OPTIMAL SURVEY DESIGN FOR PRIVATE MEAN ESTIMATION A P REPRINT Yu-Wei Chen Department of Statistics Purdue University West Lafayette, IN 47907 chen4357@purdue.edu Raghu Pasupathy Department of Statistics Purdue University West Lafayette, IN 47907 pasupath@purdue.edu Jordan A. Awan Department of Statistics Purdue Univers... | https://arxiv.org/abs/2501.18121v1 |
to protect individual respondents, especially when sensitive questions are asked. Furthermore, another key motivation for incorporating DP is as a technique to reduce responsearXiv:2501.18121v1 [stat.ML] 30 Jan 2025 Optimal Survey Design for Private Mean Estimation A P REPRINT bias—also known as answer bias—which often... | https://arxiv.org/abs/2501.18121v1 |
integer-programming problem, identifying its alignment within the framework of DOE. •We establish strong convexity for a general variance objective, covering important cases such as A-optimality (minimizing the trace of the covariance matrix) and population mean estimation, under three common additive DP mechanisms: La... | https://arxiv.org/abs/2501.18121v1 |
agent profiles, Wang et al. [2017] establish a lower bound on the l1-induced norm of the covariance matrix for minimum-variance unbiased estimators when the agents’ profiles are ϵ-DP. Li et al. [2023a] expose how output poisoning attacks can manipulate and deteriorate mean and variance estimation under local DP. Amin e... | https://arxiv.org/abs/2501.18121v1 |
trusted curator, but gives the same DP guarantee to external adversaries and allows for asymptotically negligible noise to be added [Smith, 2011, Barber and Duchi, 2014]. Definition 3 (Central Differential Privacy: Dwork et al. [2006]) .LetXnbe the set of possible datasets with sample sizenanddH(·,·)be the Hamming dist... | https://arxiv.org/abs/2501.18121v1 |
Diwith a total sample size of η, the constrained minimization problem of interest becomes arg minkX i=1α2 i ni σ2 i+γ2(ni, Ni, ϵ) , (1) subject toPni=η, where niis searched over N,αiis weight, and ηis a pre-determined total sample size. This problem is classified as nonlinear integer programming, where both the objec... | https://arxiv.org/abs/2501.18121v1 |
+ni Ni) (exp ( ϵ/∆f)−1)2# . (5) •The Truncated-Uniform-Laplace mechanism for local ϵ−DP is Zij=Yij+Kij+Uijwhere Kij∼ DLap (pi= exp(1 /si)). and Uij∼Uniform (−1 2,1 2). Then, the variance objective becomes kX i=1αi2 ni" σ2 i+1 12+ 2ni Ni(exp ( ϵ/∆f)−1 +ni Ni) (exp ( ϵ/∆f)−1)2# . (6) By using an exhaustive search, the co... | https://arxiv.org/abs/2501.18121v1 |
naive stratified sampling design, also known as the Neyman allocation [Neyman, 1934] or the optimal allocation [Kempf-Leonard, 2004], where the sample size allocated to each group is proportional to both the variability and the size of the group. In contrast, (6)has a regularization effect on its sample sizes that resu... | https://arxiv.org/abs/2501.18121v1 |
we propose Algorithm 1 which starts with the continuous solution, identifies a small set of candidate integer solutions, and then identifies the integer-optimal design within this smaller set. Algorithm 1 Integer-Optimal Design Input: x∗(the optimal continuous solution) and Hessian matrix of g:Hg(x∗) fori= 1, ..., k−1d... | https://arxiv.org/abs/2501.18121v1 |
under the Laplace mechanism, the naive subsampling variance can be up to 2.5 times larger than the optimal design variance within 1< ϵ < 10. Under TuLap, the variance ratio can reach as high as 4. Note that DLap gives the same design for population mean. For the A-optimal case, it reveals a similar trend for the Laplac... | https://arxiv.org/abs/2501.18121v1 |
as kincreases the computation time exhibits exponential growth, which becomes large especially when k≥14. In practice, if r >1.5andkis large, we recommend identifying the nearest integer design ninit., corresponding to the first half of the algorithm. In this simulation, it maintains an optimality gap of less than 10−4... | https://arxiv.org/abs/2501.18121v1 |
10 Optimal Survey Design for Private Mean Estimation A P REPRINT Héber H. Arcolezi, Jean-François Couchot, Bechara Al Bouna, and Xiaokui Xiao. Random sampling plus fake data: Multidimensional frequency estimates with local differential privacy. In Proceedings of the 30th ACM International Conference on Information & Kn... | https://arxiv.org/abs/2501.18121v1 |
for differential privacy. In 2016 14th Annual Conference on Privacy, Security and Trust (PST) , pages 664–673, 2016. doi:10.1109/PST.2016.7906954. Cecilia Ferrando, Shufan Wang, and Daniel Sheldon. Parametric bootstrap for differentially private confidence intervals. InInternational Conference on Artificial Intelligenc... | https://arxiv.org/abs/2501.18121v1 |
2023a. Xiaoyu Li, Yang Cao, and Masatoshi Yoshikawa. Locally private streaming data release with shuffling and subsampling. In2023 IEEE 39th International Conference on Data Engineering Workshops (ICDEW) , page 125–131, April 2023b. doi:10.1109/ICDEW58674.2023.00026. Shurong Lin, Mark Bun, Marco Gaboardi, Eric D Kolacz... | https://arxiv.org/abs/2501.18121v1 |
0, it follows thatd2h(x) dx2>0for all x >0. By the same argument, knowing thatdh(x) dx x=0= 0, we conclude thatdh dx>0 for all x >0. Finally, since h(0) = 0 , we establish that h(x)>0for all x >0. Theorem 8 (Strong Convexity) .Letαi,Ni,σi, andηbe given for all i= 1, . . . , k . Then, the continuous relaxations of the v... | https://arxiv.org/abs/2501.18121v1 |
ν) =1 (PNi)2(kX i=1Ni2 xiτ2 i+ 2xi+ (exp ( ϵ/∆f)−1)Ni (exp ( ϵ/∆f)−1)2) +ν(X xi−η). The KKT condition implies that for all 1≤i≤k, ( ∂L ∂xi=1 (PNi)2h −Ni2 xi2τ2 i+2 (exp ( ϵ/∆f)−1)2i +ν= 0 ∂L ∂ν=Pxi−η= 0. Then, ν(PNi)2+2 (exp ( ϵ/∆f)−1)2=Ni2 xi2τ2 iimplies xi∝τiNi, as the LHS of the equation is constant for all i. The... | https://arxiv.org/abs/2501.18121v1 |
arXiv:2501.18204v1 [math.ST] 30 Jan 2025A theory of shape regularity for local regression maps Jérémy Bettinger, François Portier, Adrien Saumard jeremy.bettinger@ensai.fr ;francois.portier@ensai.fr ;adrien.saumard@ensai.fr Department of Statistics, University of Rennes, ENSAI, CNRS, CREST-UMR 9194, F-35000 Rennes, Fra... | https://arxiv.org/abs/2501.18204v1 |
success or failure of d ifferent tree constructions and to illustrate the success of random forest regression. Among p urely random trees, Mondrian trees, as introduced in [ LRT14 ], indeed achieve the optimal rate of convergence [ MGS17 ] for Lischitz functions while, in contrast, centered trees fail to reach i t [Bia1... | https://arxiv.org/abs/2501.18204v1 |
convergence for a large class of local regression estimators, that inclu des previously mentioned partitioning estimators. More precisely, in a random design regression w ith heteroscedastic sub-Gaussian noise framework, the theory allows the localization method to be general as it may depend on a different source of ra... | https://arxiv.org/abs/2501.18204v1 |
the SID condition is expressed through the b ehavior of the unknown regression function and covariates distribution, and cannot hold for a ny regression function. In contrast, our shape-regularity condition does not depend on the regre ssion function g, neither on the covariates distribution, and only imposes restricti... | https://arxiv.org/abs/2501.18204v1 |
restricted to partition based estimator. For any local map V, the associated regression estimator is given by ∀x∈SX,ˆgV(x) =/summationtextn i=1Yi1V(x)(Xi)/summationtextn i=11V(x)(Xi), with the convention 0/0 = 0.Local maps Vdepending on the sample (X1,Y1),...,(Xn,Yn)are of particular interest. This is indeed the case f... | https://arxiv.org/abs/2501.18204v1 |
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