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distribution in the exponential family in [24]. These articles also prove strong consistency even if the distribution used in the likelihood is mis-specified ( e.g.the use of a negative binomial MLE on data coming from Poisson autoregession). We point out that [28] and [27] prove the strong consistency for general auto... | https://arxiv.org/abs/2504.02649v1 |
by ˜A∗0=δkIand ˜A∗(k+1)=˜A∗˜A∗k. When can then write for any t∈Z |xt| ⪯ ˜A∗ |x| t+|K|. For any given m∈N, taking the convolution by ˜A∗mto the right and the left and rearranging the terms yield ˜A∗m∗ |x| t− ˜A∗(m+1)∗ |x| t⪯ ˜A∗m∗ |K| t. (24) Summing the last inequality for mranging from 0 to some n∈Nyields |xt... | https://arxiv.org/abs/2504.02649v1 |
autoregression. Journal Of Time Series Analysis . 45, 584-612 (2024), https://onlinelibrary.wiley.com/doi/abs/10.1111/jtsa.12728 [5] Knight, M., Leeming, K., Nason, G. & Nunes, M. Generalized Network Autoregres- sive Processes and the GNAR Package. Journal Of Statistical Software .96, 1-36 (2020), https://www.jstatsoft... | https://arxiv.org/abs/2504.02649v1 |
[28] Ahmad, A. & Francq, C. Poisson QMLE of count time series models. J. Time Series Anal. .37, 291-314 (2016), https://doi.org/10.1111/jtsa.12167 [29] Knight, M., Nunes, M. & Nason, G. Modelling, Detrending and Decorrelation of Network Time Series. (2016), https://arxiv.org/abs/1603.03221 [30] Knight, M., Leeming, K.,... | https://arxiv.org/abs/2504.02649v1 |
T. & Ghysels, E. Periodic Autoregressive Conditional Heteroscedasticity. Journal Of Business & Economic Statistics .14, 139-151 (1996), http://www.jstor.org/stable/1392425 [52] Gardner, W., Napolitano, A. & Paura, L. Cyclostationarity: Half a century of research. Signal Processing . 86, 639-697 (2006), https://www.scie... | https://arxiv.org/abs/2504.02649v1 |
Computing High-dimensional Confidence Sets for Arbitrary Distributions Chao Gao∗, Liren Shan†, Vaidehi Srinivas‡, Aravindan Vijayaraghavan§ Abstract We study the problem of learning a high-density region of an arbitrary distribution over Rd. Given a target coverage parameter δ, and sample access to an arbitrary distrib... | https://arxiv.org/abs/2504.02723v2 |
. . . . . . 10 2 Technical Overview 11 2.1 Proper Learning for Non-Worst-Case Instances . . . . . . . . . . . . . . . . . . . . . 12 2.2 Reducing Worst-Case Improper Learning to Non-Worst-Case Proper Learning . . . . 14 3 Proper Learning of Euclidean Balls 15 3.1 Completing the Proof of Theorem 3.1 . . . . . . . . . . ... | https://arxiv.org/abs/2504.02723v2 |
a rich body of work on robust statistics, where the remaining 1 −δfraction can be arbitrary outliers, but these works often make stronger model assumptions about the δfraction of points that are “inliers” (Huber, 1964; Diakonikolas and Kane, 2023). See Section 1.5 for applications in statistics like conformal predictio... | https://arxiv.org/abs/2504.02723v2 |
on the distribution. While there is a simple algorithm that is sample efficient (but computationally inefficient) when the VC-dimension of Cis bounded (by a polynomial), the algorithmic problem becomes challenging in high dimensions. The family of Euclidean Balls. Consider the basic setting where Cis the set of ℓ2balls... | https://arxiv.org/abs/2504.02723v2 |
more precisely, it satisfies Py∼D[y∈S]≥δ, and vol(S)1/d≤vol(C⋆ k)1/d 1 +Ok,δ d−1/2+o(1) ·O(log(k/γ)) γ1/d where C⋆ kis the minimum volume union of k-balls that achieves at least δ+γ+O(p kd2/n)coverage overD. See Corollary 5.5 for details. Our algorithms are surprisingly simple and conceptually different from exis... | https://arxiv.org/abs/2504.02723v2 |
algorithm (and the hard instances in Theorem 1.3), the points sampled from Dthat lie inside the optimal ball B⋆effectively lie on a lower-dimensional subspace. On the other hand, our algorithm can give a significantly stronger guarantee when the points inside B⋆are approximately isotropic. In the following theorem, we ... | https://arxiv.org/abs/2504.02723v2 |
rate 0< α < 1, and coverage slack factor 0≤γ≤1, outputs a set bC(not necessarily in C), such that for an unknown test example Yn+1∈ Y, (a) if Y1, . . . , Y n+1are exchangeable, then P[Yn+1∈bC≥1−α]. (b) if Y1, . . . , Y n+1are drawn i.i.d. from some (unknown) distribution D, and n= Ω(d2/γ2), then vol(bC)1/d≤ 1 +Oγ,δ d... | https://arxiv.org/abs/2504.02723v2 |
et al. (2005). Unlike traditional classification, here all the data points are generated from the same distribution, which leads to the name “one-class classification” in the literature. Compared with directly learning confidence sets through (3) or (4), the one-class classification perspective is more flexible, and ca... | https://arxiv.org/abs/2504.02723v2 |
1−η > 1/2 fraction of the points are inliers, while a η= 1−δ <1/2 fraction of the points are outliers (Huber, 1964; Diakonikolas and Kane, 2023). The setting when δ <1/2 has also been studied as list-decodable robust estimation (Charikar et al., 2017). Most relevant to us are the works in algorithmic robust statistics ... | https://arxiv.org/abs/2504.02723v2 |
and the shape of the bCare used as a robust location estimator and a robust scatter matrix estimator. Other applications of MVE are discussed in Van Aelst and Rousseeuw (2009). 3.Testing unimodality. The original motivation of Hartigan (1987) in learning a minimum volume convex set is to test whether a distribution is ... | https://arxiv.org/abs/2504.02723v2 |
strategy to work in the worst case. However, it turns out that for certain non-worst-case instances, it is indeed possible to recover B⋆within a better approximation ratio than in the worst case. Our first main technical insight is that when the variance of the points is well-spread (i.e., the sampled points do not hav... | https://arxiv.org/abs/2504.02723v2 |
is illustrated in Figure 1. Figure 1: The figure shows the points Y(in red) with mean µ, that are contained in ball Bwith center cand radius R. It is not necessarily the case that µis near c, asBcan be defined by just a few points. However, Chebyshev’s inequality tells us that most of the points Y(depicted here as the ... | https://arxiv.org/abs/2504.02723v2 |
.Suppose we are given a set of points Y⊆Rd,|Y|=n, and 0≤δ≤1,0≤γ≤1, such that there exists a subset Y⋆⊆Y,|Y⋆| ≥δ|Y|, that is contained in an unknown ball B⋆=B(c⋆, R⋆). Then we can find an ellipsoid bEsuch that |bE∩Y| ≥δ(1−γ)|Y|, and vol(bE)1/d≤vol(B⋆)1/d· 1 +O d−1/2+o(1)/γδ . Intuitively, the case where our non-wors... | https://arxiv.org/abs/2504.02723v2 |
space is bounded, i.e., the points in the transformed space are approximately isotropic: cM−1/2ΣYcM−1/2≼Obτ2R2 d Id. This allows us to bound the approximation ratio of applying Lemma 2.1 to the transformed points. (iii) At most d/bτ2eigenvalues of cMare set to d. This bounds the distortion of cM. That is, for any sha... | https://arxiv.org/abs/2504.02723v2 |
we can find a ballbBsuch that with high probability, Py∼Dh y∈bB|Yi ≥δ, and vol(bB)1/d≤vol(B⋆)1/d 1 +Olog log d γδlogd where B⋆is the minimum volume ball that achieves at least δ+γ+O(p d/n)coverage over D. Our approach starts with the following structural observation about the true high-density ball B⋆. It is not in... | https://arxiv.org/abs/2504.02723v2 |
case, we cannot expect that the variance is less than ( R′)2, for example if all of the variance of Y′is concentrated in one direction. However, there can be at most log ddirections with variance higher than ( R′)2/logd. Thus we can find a list of candidate centers for Y⋆by estimating the location of the center separat... | https://arxiv.org/abs/2504.02723v2 |
Let Y′ high= Π highY′ denote the projection of points in Y′onto the high variance directions. Note that the set Y′ highlies within the q-dimensional ball B′ high= Π highB′, which is centered at the origin and has a radius of at most R′. Next, consider the projection of Y∗onto the high variance directions, denoted by Y∗... | https://arxiv.org/abs/2504.02723v2 |
Input: a set of point Y∈Rd, target fraction δ∈(0,1), and slack γ∈(0,1) Output: an ball bB⊂Rd 1. Create a list of coarse balls B={B(y1,∥y1−y2∥)|y1, y2∈Y}and remove from B all balls that contain less than δnpoints in Y. 2. Set Rminto be the minimum radius among all balls in B. Then remove from Ball balls with a radius gr... | https://arxiv.org/abs/2504.02723v2 |
this ball can be done in time eO(|B2| ·n) =eO(n4d7 2). Thus the total runtime of our algorithm is bounded by eO(n4d7 2). Now we argue that B2contains a ball that is a good approximation to B⋆in volume. Consider the ball B′from (10). We have that B′contains all points in Y⋆, has radius R′≤2R⋆, and is in the list B. Let ... | https://arxiv.org/abs/2504.02723v2 |
the mean in the low-variance directions. This allows us to argue about the standard deviation of the (log d+ 1)th highest variance direction, which can be at most R⋆√logd(Lemma 3.7). This bound on the variance that we use in Theorem 3.1 is the best that our approach achieves, since it is possible that the variance of t... | https://arxiv.org/abs/2504.02723v2 |
there is a bB∈Bsuch that the radius bRofbBsatisfies R⋆≤bR≤2R⋆. Taking the minimum radius Rminover all balls in Bthat covers at least δnpoints gives us an estimate of R⋆such that R⋆≤Rmin≤bR≤2R⋆. Letµ⋆be the mean of the points in Y⋆. We know that Y⋆has covariance p.s.d. dominated byβ(R⋆)2 dI, which is in turn p.s.d. domi... | https://arxiv.org/abs/2504.02723v2 |
Lemma 3.6 for q= 0,λ(q+1)(ΣY′)≤σ2,σ2=bτ2(R′)2/d, slack factor γ. (e) Set bEi=cM1/2 ibB. 3. Return the ellipsoid bEwith the smallest volume in the list {bEi}. Figure 5: Algorithm Dense Ellipsoid for finding a small volume ellipsoid that contains at least δ′=δ(1−γ) fraction of points Proof of Theorem 2.2. We begin by fin... | https://arxiv.org/abs/2504.02723v2 |
vol(bE)≤expdlnd bτ2 ·vol(bBT) by (14) ≤expdlnd bτ2 ·cdbRd T cd: vol. of ddim. unit ball ≤expdlnd bτ2 ·cd (R⋆ T)2+σ2γ 1 +2(1−δ) δd/2 by (15) ≤expdlnd bτ2 ·cd (R⋆)2+σ2 γ 1 +2(1−δ) δd/2 R⋆≥R⋆ Tby (11) ≤expdlnd bτ2 ·cd(R⋆)d 1 +σ2 γ(R⋆)2 1 +2(1−δ) δd/2 ≤expdlnd bτ2 ·expd 2·σ2 γ(R⋆)2 1 +2(1−δ) δ ·... | https://arxiv.org/abs/2504.02723v2 |
phases. This means that the total volume of the sets output over all phases is bounded by Oα γ·log(k/γ) ·vol(C⋆). 28 Claim 5.2 (Marginal volume added in each phase is bounded) .In every phase j, the total marginal volume added is ≤O(α)· vol(C∗ 1) +···+ vol( C∗ k) . Proof. Lettjbe the first iteration in phase j, and... | https://arxiv.org/abs/2504.02723v2 |
|C(t)∩Yt|, that maximizes |C(t)∩Yt| vol(C(t)). Leti∗∈ {0, . . . ,⌈log2n⌉}be the value that satisfies 2i⋆≤ |C(t)∩Yt| ≤2·2i⋆. 30 Since C(t)achieves coverage at least2i⋆ n, we have that the set S(t) i⋆chosen in step (a) of the algorithm must have vol(S(t) i⋆)≤α·vol(C(t)) and |S(t) i⋆∩Yt| ≥(1−γ′)2i⋆ n. Thus, we have that |... | https://arxiv.org/abs/2504.02723v2 |
this construction in our reduction as follows. Let Gbe a ∆-regular graph with nvertices and medges. We construct a set of npoints Y⊂Rdwith dimension d=m3/εfor a small constant ε∈(0,1). Each point y∈Ycorresponds to a vertex vin graph G. The first mcoordinates of yare the m-dimensional row for vertex vin the incident mat... | https://arxiv.org/abs/2504.02723v2 |
on it going outside i.e.,|E(S, V\S)| ≥(1−η)|S|D. We prove the following theorem. Theorem 6.3 (Computational Intractability with Slack in Coverage) .For any constant γ > 0, there exists a constant δ∈(0,1), such that assuming the SSE hypothesis for any constant ε >0 there is no algorithm that given a set of points Y⊆Rdru... | https://arxiv.org/abs/2504.02723v2 |
the high-dimensional setting. Conformal prediction is the statistical problem of finding prediction intervals. That is, given training examples Y1, . . . , Y nlying in some space Y, and a target miscoverage rate α >0, our goal is to output a set C, such that for an unknown test example Yn+1∈ Y, P[Yn+1∈C]≥1−α, (16) assu... | https://arxiv.org/abs/2504.02723v2 |
Cto come from some class of bounded VC-dimension C. That is, we compete with min C∈Cvol(C) s.t. P y∼D[y∈C]≥1−α, which they term C-restricted volume optimality . In fact, in this setting, the problem essentially reduces to finding the minimum volume set in Cthat achieves coverage 1 −αassuming the samples are i.i.d., whi... | https://arxiv.org/abs/2504.02723v2 |
(a) when Y1, . . . , Y n+1are exchangeable, then we achieve coverage P[Yn+1∈bC≥1−α]. (b) if Y1, . . . , Y n+1are drawn i.i.d. from some (unknown) distribution D, and n= Ω( d2/γ2), then sincebCis in our nested set system, and Theorem 1.1 guarantees that bCachieves coverage ≥1−α, the conformal predictor will not output a... | https://arxiv.org/abs/2504.02723v2 |
Chandola, Arindam Banerjee, and Vipin Kumar. Anomaly detection: A survey. ACM computing surveys (CSUR) , 41(3):1–58, 2009. Moses Charikar, Jacob Steinhardt, and Gregory Valiant. Learning from untrusted data. In Proceed- ings of the 49th Annual ACM SIGACT Symposium on Theory of Computing , STOC 2017, page 11Technically ... | https://arxiv.org/abs/2504.02723v2 |
Estimation of a convex density contour in two dimensions. Journal of the American Statistical Association , 82(397):267–270, 1987. Peter J Huber. Robust estimation of a location parameter. The Annals of Mathematical Statistics , 35(1):73–101, 1964. P.J. Huber. Robust Statistics . Wiley Series in Probability and Statist... | https://arxiv.org/abs/2504.02723v2 |
for estimating a multivariate mode and isopleth. Journal of the American Statistical Association , 74(366a):329–339, 1979. Bernhard Sch¨ olkopf, John C Platt, John Shawe-Taylor, Alex J Smola, and Robert C Williamson. Estimating the support of a high-dimensional distribution. Neural computation , 13(7):1443–1471, 2001. ... | https://arxiv.org/abs/2504.02723v2 |
Universal Log-Optimality for General Classes ofe-processes and Sequential Hypothesis Tests Ian Waudby-Smith:, Ricardo Sandoval:, and Michael I. Jordan:; :University of California, Berkeley ;Inria, Paris April 4, 2025 Abstract We consider the problem of sequential hypothesis testing by betting. For a general class of co... | https://arxiv.org/abs/2504.02818v1 |
In the former case, larger values represent more power against P, while in the latter, larger values represent less power. While there has been a flurry of work on sequential hypothesis testing and “testing by betting” in recent years [see, e.g., 41, 51], less is known about useful general classes of nonparametric test... | https://arxiv.org/abs/2504.02818v1 |
that are all special cases of that displayed in (1). The properties of (1) are proven for more general test supermartingales discussed in Section 5. We nevertheless focus our main discussions on sequential tests that fall under (1) as it strikes a balance between generality and concreteness. We can now formulate our ge... | https://arxiv.org/abs/2504.02818v1 |
0 2 4 6 8 10 12 14 16 First Rejection TimeRegret-CO96 ONSRegret-OJ23Univ. Portfolio Log-optimal ( λ⋆ Q)log(1/α) /lscript⋆ Q Figure 2: Empirical growth rates (left) and distributions of rejection times (right) for various e-processes (see Section 1.1 for a precise definition). As will be discussed in Corollary 2.2, the ... | https://arxiv.org/abs/2504.02818v1 |
say that a stochastic process pMnq8 n“1onpΩ,Fqis aP-supermartingale if it is adapted to F—meaning that MnisFn-measurable for each nPN—and if @nPN,EPrMn|Fn´1sďMn´1P-almost surely. We say that this process is a P-supermartingale if this inequality holds for all PPP. We have analogous definitions for P- and P-martingales ... | https://arxiv.org/abs/2504.02818v1 |
best-in- hindsight constant rebalanced portfolio for processes of the form (1)) versus the logarithm of some other process W. One could think of Wnas being given by Wn“nź i“1´ p1´λiqep1q i`λiep2q i¯ for a sequence pλnq8 n“1where λndepends only on the stocks up until time n´1.2We note in passing that Wnmay also be some ... | https://arxiv.org/abs/2504.02818v1 |
paper, but it is worth noting that γONS nPr´1{2,1{2s for every nPNand hence if the maximizer of the expected log-wealth increment (i.e., the log-optimal choice of γ‹ Qunder Q) lies outside of this range, ONS cannot adapt to this fact, leading to suboptimal growth rates and expected rejection times; see Figure 2. Compar... | https://arxiv.org/abs/2504.02818v1 |
Wang, Wang, and Ziegel [58, Definition 3] in the context of the test supermartingale in (2) but they consider convergence in L1pQqrather than Q-almost surely. The authors do show that some explicit betting strategies satisfy this notion of L1pQq-log-optimality and we discuss one of them further in Section 7. However, w... | https://arxiv.org/abs/2504.02818v1 |
equivalence holds with a rate of pan{nq8 n“1. (iii) For every QPQ,Whas an asymptotic growth rate of ℓ‹ Q, meaning that lim nÑ81 nlogWn“ℓ‹ QQ-almost surely, and this is unimprovable in the sense that for any other W1PW, lim sup nÑ81 nlogW1 nďℓ‹ QQ-almost surely. 101102103104 n−0.15−0.10−0.050.000.050.10log(Wn) n Univ. P... | https://arxiv.org/abs/2504.02818v1 |
forms a test P-supermartingale (Appendix B.1) combined with the fact thatpλUP nq8 n“1is predictable and r0,1s-valued as can be deduced from its definition in (5). Furthermore, we have that WCO96 n andWOJ23 n both form P-e-processes as they are almost surely upper-bounded by WUP nfor every nPN. By construction, WCO96 n ... | https://arxiv.org/abs/2504.02818v1 |
and later discuss lower bounds for arbitrary portfolios (Proposition 3.2) and correspond- ing upper bounds for those with sublinear portfolio regret (Theorem 3.3). The following result shows that in the general setting presented in (1), any constant rebalanced portfolio λPr0,1shas an expected rejection time given by 1 ... | https://arxiv.org/abs/2504.02818v1 |
expected rejection time of sublinear portfolio regret e-processes) .LetpWnq8 n“1be anyP-e-process satisfying the portfolio regret bound Rnďrnfor some sublinear rn“opnq. Let QPQ be an element of the alternative hypothesis for which ρspλ‹ Qq:“EQ log´ p1´λ‹ QqEp1q`λ‹ QEp2q¯ ´ℓ‹ Q s ă8 for some są2. Then the expected rejec... | https://arxiv.org/abs/2504.02818v1 |
this reliance on boundedness are the stopping time bounds of Agrawal et al. [2] but their test supermartingales do not appear to be a special case of (1). With the main general results from Sections 2 and 3 in mind, we now turn our attention to some special testing problems that can be approached using an instantiation... | https://arxiv.org/abs/2504.02818v1 |
rejection times for Pď-supermartingales under the alternatives Qą. In the following section, we consider the slightly more complicated setting of equality nulls and two-sided alternatives where stronger optimality guarantees can be stated since the class of test martingales given by (12) (but with a larger allowable ra... | https://arxiv.org/abs/2504.02818v1 |
inequality logp1`yqěy´y2for all yPr´1{2,1{2s, we have that log p1`γpX1´µ0qqěγpX1´µ0q´γ2pX1´µ0q2 whenever γPr´1{2,1{2s, and it is not hard to check that the maximizer of the expectation of this lower bound is given (and can be further lower-bounded) by argmax γPr´1{2,1{2sEQ“ γpX1´µ0q´γ2pX1´µ0q‰ “∆Q 2pVarQpX1q`∆2 Qqě∆Q 2... | https://arxiv.org/abs/2504.02818v1 |
their use of ONS: WpD“q n :“nź i“1p1`γiDiq“nź i“1ˆ p1´λiqp1´Ziq 1{2`λiZi 1{2˙ , where λn“p1`γnq{2 yields a betting strategy taking values in r0,1s. Note that if ONS were employed, then λnwould have been restricted to the range r1{4,3{4s. Clearly, WpD“q n is an instantiation of the general test supermartingale in (1) fo... | https://arxiv.org/abs/2504.02818v1 |
we say that its regret is sQ-uniformly sublinear if supQPsQrnpQq“opnq. Fur- thermore, we say that θ‹ QPΘis thepΘ, Qq-numeraire portfolio if for all Θ-valued predictable sequences pθnq8 n“1, the process nź i“1` Eipθiq{Eipθ‹ Qq˘ (18) forms a nonnegative Q-supermartingale with EQrE1pθq{E1pθ‹ Qqsď1. The choice to refer to ... | https://arxiv.org/abs/2504.02818v1 |
larger than log pWnpθ‹ Qqqby definition. Combined with the first fact that log pWnpθ‹ Qqqwill even- tually exceed log pWnq, it must be the case that log pWnq, logpWnpθ‹ Qqq, and supθPΘlogpWnpθqqare all sandwiched within rnpQqof each other in finite time with Q-probability one. The formal details are in Appendix B.2. Le... | https://arxiv.org/abs/2504.02818v1 |
following equality lim αÑ0`EQrsταs logp1{αq“1 ℓ‹ Q. It is thus straightforward to derive both Proposition 3.2 and Theorem 3.3 from Lemma 5.2 as long as it can be shown that λ‹ Qis apr0,1s, Qq-numeraire portfolio. This fact was shown in the same paper that defined “numeraire portfolios”; see Long Jr [34, Appendix A]. On... | https://arxiv.org/abs/2504.02818v1 |
we provide a requisite definition. Definition 3 (Distribution-uniform asymptotic almost sure log-optimality and equivalence) .LetWbe a collection of P-e-processes. We say that W‹”pW‹ nq8 n“1isQ˝-uniformly and universally log-optimal inWif for any other WPW, it holds that @εą0,lim mÑ8sup QPQ˝PQˆ sup kěm1 klogpWk{W‹ kqěε... | https://arxiv.org/abs/2504.02818v1 |
large numbers and central limit theorems, respectively. See [8, 61, 52, 35] for an incomplete list of examples. Let us now provide a distribution-uniform bound on the expected rejection time. Theorem 6.2. LetQ˝ĎQbe a subset of alternative distributions for which the sthmoment of the log-wealth increments of Wpλ‹ Qqare ... | https://arxiv.org/abs/2504.02818v1 |
in (2): Wn:“nź i“1p1`λi¨pEi´1qq, and they study growth optimality properties when pEnq8 n“1are i.i.d. random variables. In particular, they show that ifpλnq8 n“1are chosen according to a follow-the-leader-type strategy—referred to as growth-rate for empirical e-statistics (GREE)—defined by λGREE n :“argmax λPr0,1s1 n´1... | https://arxiv.org/abs/2504.02818v1 |
None of the aforementioned works have exact matching lower and upper bounds for growth rates nor expected rejection times in the settings they consider. Moreover in the context of expected rejection time bounds, once taking αÑ0`, the aforementioned bounds do not match the lower bounds provided in Sections 3 and 4. The ... | https://arxiv.org/abs/2504.02818v1 |
show that they are in a sense optimal, but they did not consider the log-optimality and rejection time desiderata specific to testing that we focus on here. Our motivations have some overlap with those of Shekhar and Ramdas [53], but from a perspective of testing rather than estimation; moreover, our results and proof ... | https://arxiv.org/abs/2504.02818v1 |
Wang. Optimistic interior point methods for sequential hypothesis testing by betting. arXiv preprint arXiv:2502.07774 , 2025. 13, 16, 22, 23 [6] Brian M Cho, Kyra Gan, and Nathan Kallus. Peeking with PEAK: Sequential, nonparametric composite hypothesis tests for means of multiple data streams. In International Conferen... | https://arxiv.org/abs/2504.02818v1 |
confidence sequences. The Annals of Statistics , 2021. 22 [26] Kwang-Sung Jun and Francesco Orabona. Parameter-free online convex optimization with sub- exponential noise. In Conference on Learning Theory , pages 1802–1823. PMLR, 2019. 23 [27] Emilie Kaufmann and Wouter M Koolen. Mixture martingales revisited with appl... | https://arxiv.org/abs/2504.02818v1 |
David Siegmund. A class of stopping rules for testing parametric hypotheses. InProceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 4: Biology and Health , volume 6, pages 37–42. University of California Press, 1972. 22 26 [47] Herbert Robbins and David Siegmund. The expected s... | https://arxiv.org/abs/2504.02818v1 |
special class of functions Gwith the property that dGpPX, PYq“0if and only if PX“PY, which allows them to reduce the problem of testing Pversus Qto that of whether a supremum of means dGpPX, PYqis zero or non-zero. Of course, it is not yet immediately obvious if or how one can compute dGor an estimate thereof. The auth... | https://arxiv.org/abs/2504.02818v1 |
those processes that are upper-bounded by test P-supermartingales and thus it suffices to show that Wn:“nź i“1´ p1´λiqEp1q i`λiEp2q i¯ forms a test P-supermartingale under the assumption that the sequences pEp1q nq8 n“1andpEp2q nq8 n“1consist of i.i.d. P-e-values and that pλnq8 n“1is a betting strategy—i.e., predictabl... | https://arxiv.org/abs/2504.02818v1 |
proof of Property piiq. Proof of Property piiiq.By the strong law of large numbers, we have that with Q-probability one, 1 nlogWpQ‹q nÑEQ“ E1pθ‹ Qq‰ ”ℓ‹ Q. Appealing to Property piiqand sublinearity of rn, we have that 1 nlogpWnq“1 nlogpWn{WpQ‹q nq`1 nlogpWpQ‹q nqÑℓ‹ Q Q-almost surely for every QPsQ. To demonstrate uni... | https://arxiv.org/abs/2504.02818v1 |
with the upper bound and then later derive the lower bound. Upper bounding EQrsταs.Throughout, let τ”sτα:“inftnPN:Wně1{αuand define ℓ‹ Q:“ EQrlogE1pθ‹ Qqs. Let δPp0,1qbe arbitrary and define ε:“δ 1`δ¨ℓ‹ Qand m:“S logp1{αq ℓ‹ Q´εW “S p1`δqlogp1{αq ℓ‹ QW . 32 Writing out the expected stopping time under Qand partitioning... | https://arxiv.org/abs/2504.02818v1 |
5.3. Dividing both sides by ℓ‹ Qyields the desired result: EQpτqělogp1{αq ℓ‹ Q, which completes the proof of the lower bound and hence of Lemma 5.2. Lemma B.1 (A Chebyshev-Nemirovski concentration inequality) .LetX1, X2, . . . , X nbe independent random variables with mean zero for which EP|X1|să8 for some sě2. Then fo... | https://arxiv.org/abs/2504.02818v1 |
bound additionally satisfies rn“ opn1{pq, then1 nlogWnconverges Q˝-uniformly to ℓ‹ Qalmost surely at a rate of opn1{p´1q: @εą0,lim mÑ8sup QPQ˝PQˆ sup kěmk1´1{p 1 klogWk´ℓ‹ Q ěε˙ “0. We proceed by first proving Property piiq, thenpiiiq, and thenpiq. For brevity, let WpQ‹q k:“Wkpθ‹ Qq for every kPN. Proof of Property pii... | https://arxiv.org/abs/2504.02818v1 |
Dynamic Investment Strategies Through Market Classification and Volatility: A Machine Learning Approach∗ Jinhui Li†, Wenjia Xie‡, Luis Seco§ March 2025 Abstract This study introduces a dynamic investment framework to enhance portfolio management in volatile markets, offering clear advantages over traditional static str... | https://arxiv.org/abs/2504.02841v1 |
consideration for evolving mar- ket conditions. For instance, strategies like minimum variance or equally-weighted investment assume that past data can reliably predict future risks and returns, which is not always the case during market upheavals. This limitation is crucial, as it can lead to suboptimal asset allocati... | https://arxiv.org/abs/2504.02841v1 |
return and Sharpe ratio. For the second asset, the dy- namic portfolio outperforms all methods, including ERC. This paper extends the existing body of knowledge by integrating classical financial theories with cutting- edge machine learning techniques to create a more responsive and effective port- 3 folio management f... | https://arxiv.org/abs/2504.02841v1 |
parameters for the Dirichlet distribution, representing prior counts for transitions from each Markov state, are denoted as: αi= (αi1, αi2, . . . , α i10) (4) Here, αijrepresents the prior belief (or prior counts) about the transition from Markov state ito state j. Next, we count the number of posterior transitions fro... | https://arxiv.org/abs/2504.02841v1 |
[5]: (trel−1) log1 2ϵ ≤tmix(ϵ)≤trel1 2log1 πmin + log1 2ϵ (9) where trel=1 1−λ2is the relaxation time, λ2is the second-largest eigenvalue modulus of the transition matrix P, and πmin= min x∈Xπ(x) is the minimum probability in the stationary distribution. Note that for a reversible Markov transi- tion matrix, th... | https://arxiv.org/abs/2504.02841v1 |
with large numbers of assets and more complex constraints. 2.4.3 Maximum Diversification The Maximum Diversification strategy aims to maximize the diversification ratio of a portfolio. The diversification ratio (DR) is defined as the ratio of the weighted average of the volatilities of individual assets to the volatili... | https://arxiv.org/abs/2504.02841v1 |
maintain consistent performance without sig- nificant fluctuations. In addition, we evaluated the Sharpe ratio, which measures the risk-adjusted return of the portfolio. A higher Sharpe ratio signifies better performance rela- tive to the amount of risk taken, making it a crucial metric to compare different investment ... | https://arxiv.org/abs/2504.02841v1 |
four distinct methods: equal risk contribution (ERC), minimum variance (Min Var), maximum diversification (Max Div), and equal investment. The ERC strategy allocated portfolio weights so that each asset contributed equally to the overall portfolio risk, ensuring a balanced risk distribution. The Min Var strategy focuse... | https://arxiv.org/abs/2504.02841v1 |
the same state or transitioning to adjacent states. This behavior reflects the persistence of volatility regimes, in which the market is likely to stay in a particular volatility state or move to a state with similar characteristics rather than making abrupt transitions to vastly different states. This insight is cruci... | https://arxiv.org/abs/2504.02841v1 |
Div, Equal Investment, and Dynamic), the daily returns were calculated, which were then used to compute the annualized return, volatility, and Sharpe ratio. The cumulative return was derived using the compounded return method over the entire period: 2005 to 2024 for the first asset set and 2015 to 2024 for the second a... | https://arxiv.org/abs/2504.02841v1 |
-7.160476 -7.449646 2009 25.978954 40.837342 29.623158 29.966843 39.403275 2010 20.185768 15.832373 19.912697 18.846316 20.329096 2011 22.541652 35.265132 24.984848 24.270206 36.035022 2012 6.607280 -8.372604 4.040191 4.045298 -2.669597 2013 30.245351 39.651365 32.324376 33.255147 37.014991 2014 24.546323 27.533208 25.... | https://arxiv.org/abs/2504.02841v1 |
Asset Set Year ERC Sharpe Min VarSharpe Max DivSharpe Equal Sharpe Dynamic Sharpe 2015 7.587189 11.548392 -1.164784 8.238009 13.503713 2016 45.312471 38.175456 30.967458 43.211571 45.836624 2017 73.528922 79.006286 42.103345 72.026459 64.737642 2018 -13.881292 -13.780565 -14.466792 -13.893789 -15.036143 2019 34.711795 ... | https://arxiv.org/abs/2504.02841v1 |
Diversification (Max Div), and Equal Investment, across different market states. Utilizing a Bayesian approach to construct the Markov transition matrix, our objective was to dynamically allocate portfolio weights based on the probabil- ities of transitioning between these states. This approach aims to enhance future p... | https://arxiv.org/abs/2504.02841v1 |
allocation in anticipation of future market states, rather than reacting to past performance alone. For the first set of assets, the dynamic portfolio achieved the second-best total return of 4910% and the second-highest Sharpe ratio of 237.90 throughout the period. This indicates that the dynamic strategy was able to ... | https://arxiv.org/abs/2504.02841v1 |
a total return of 4910%, signif- icantly outperforming several static methods. For the second set of assets, the 26 dynamic strategy achieved a total return of 5993%, showing superior performance compared to the maximum diversification method, although it was slightly less than the equal investment, equal risk contribu... | https://arxiv.org/abs/2504.02841v1 |
arXiv:2504.02950v1 [math.ST] 3 Apr 2025KULLBACK -LEIBLER CONSISTENCY OF p-DIMENSIONAL PÓLYA TREE POSTERIORS AND DIFFERENTIAL ENTROPY ESTIMATION A P REPRINT Fernando Corrêa Institute of Mathematics and Statistics University of São Paulo São Paulo, SP fptcorrea@gmail.comRafael Bassi Stern Institute of Mathematics and Sta... | https://arxiv.org/abs/2504.02950v1 |
convergence with respect to the supremum norm [Walker, 2004, Ghosal and v an der Vaart, 2017, Barron et al., 1999]. In this paper, we present a novel technique for obtaining con sistency results on Pólya Trees. Let K(f,θ)be the Kullback-Leibler divergence between a density fandθsampled by a Pólya Tree. Our first main re... | https://arxiv.org/abs/2504.02950v1 |
of Gosh [2003] and restate resu lts from Ghosal and van der Vaart [2017] employing it. Throughout the paper, we adopt θas the symbol for a random density sampled by a suitable Pólya Tree. This choice emphasizes that the parameter of interest is the samp led density under the posterior, rather than the random measure it... | https://arxiv.org/abs/2504.02950v1 |
density of Pwith respect to the suitable Lebesgue Measure being conside red asθ|Xnand to the measure induced by P|XnasΠθ|Xn. Expectations with regard to Πθ|Xnshall be denoted Eθ[·|Xn]and expectations with regard to Xnshall be denoted EXn[·]. We shall switch notations between the random variable Nǫ=/summationtextn i=1IB... | https://arxiv.org/abs/2504.02950v1 |
on the prior) then for allǫ>0 EXn[Eθ[K(f0,θ)|Xn]]→0 Πθ|Xn(TV(f0,θ)>ǫ)→0 and Πθ|Xn(K(f0,θ)>ǫ)→0 inXnprobability. As much of the literature is concerned with such results, bef ore continuing we make a brief survey of related works. Schwartz [1965] establishes general conditions for consis tency in the weak topology, whic... | https://arxiv.org/abs/2504.02950v1 |
stat es the consistency of the entropy functional H(f) =−/integraltext f(t)logf(t)dt, which is unbounded. Theorem 3. Considerθ∼PT(B,a)parametrized by a canonical partition with respect to the Le besgue Measure, satisfying the sufficient condition for absolute continuit y ofPandXnan i.i.d. sample of some absolutely conti... | https://arxiv.org/abs/2504.02950v1 |
by al=l2+α. Thus, this condition is satisfied for althat grows as fast as many polynomials, and consequently is w eaker than the exponential growth rates suchal= 8lrequired for stronger forms of consistency [Barron et al., 1 999]. We are motivated to prove this result because a similar formu la holds for any f0with finit... | https://arxiv.org/abs/2504.02950v1 |
almost surely satisfied and therefore: /integraldisplay f0(t)logθ(t)dt=EX∼f0/bracketleftBigg log/parenleftBigg∞/productdisplay l=12Yǫ1(X)...ǫl(X)/parenrightBigg/bracketrightBigg =∞/summationdisplay l=1EX∼f0/bracketleftbig log/parenleftbig 2Yǫ1(X)...ǫl(X)/parenrightbig/bracketrightbig . (2) It follows that /integraldispl... | https://arxiv.org/abs/2504.02950v1 |
level of sample Xnon a Pólya Tree prior Π0. It consists of the first level along the partition tree such that all Nǫare either 1or0.L(Xn)is almost surely finite if Xis absolutely continuous, as the probability of ties is 0. AfterL(Xn), allYǫ|Xnthat constitute P|Xnare eitherBeta(aǫ+ 1,aǫ)orBeta(aǫ,aǫ) distributed. In this... | https://arxiv.org/abs/2504.02950v1 |
spaceX= [0,1]pwith respect to the Lebesgue measure. This construction is due to Hanson [2006]. The same idea appl ies to other compact sets. This construction is relevant for applications of Theorem 2 and Theorem 3 on arbitrary p-dimensional spaces. Whenp= 1the canonical partition with respect to the Lebesgue Measur e ... | https://arxiv.org/abs/2504.02950v1 |
, 20(3):1203–1221, 1992. Michael Lavine. Some Aspects of Polya Tree Distributions fo r Statistical Modelling. The Annals of Statistics , 20(3): 1222–1235, September 1992. ISSN 0090-5364, 2168-8966. doi :10.1214/aos/1176348767. Publisher: Institute of Mathematical Statistics. Charles H. Kraft. A Class of Distribution Fu... | https://arxiv.org/abs/2504.02950v1 |
distance between any pair of i.i.d. random points in Rd. Ann Inst Stat Math , 44(1):121–131, March 1992. ISSN 1572-9052. doi:10.1007/BF00048674. URL https://doi.org/10.1007/BF00048674 . A Proof of Lemma 1 We prove Lemma 1 by applying the following auxiliary results Proposition 5. [Marchal and Arbel, 2017] LetX∼Beta(a,a... | https://arxiv.org/abs/2504.02950v1 |
outline the steps of proof as: 1. Apply Theorem 1 to decompose EXnEθ[−/integraltext f0(t)log(θ(t))dt|Xn]as a converging series. 2. Estimate the size of the terms to obtain an upper bound of th e form H(f0)<EXn/bracketleftbigg Eθ/bracketleftbigg −/integraldisplay f0(t)log(θ(t))dt|Xn/bracketrightbigg/bracketrightbigg </s... | https://arxiv.org/abs/2504.02950v1 |
F(Bǫ)Eθ[log(2Yǫ)|Xn]/vextendsingle/vextendsingle/vextendsingle/vextendsingle= /vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay/parenleftbiggNǫ n−F(Bǫ)/parenrightbigg Eθ[log(2Yǫ)|Xn]/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/summationdisplay ǫ∈E/vextendsingle/vextendsingle/vextend... | https://arxiv.org/abs/2504.02950v1 |
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