Aug 2026· Theory of Probability & Its Applications· 0 citations· 10 references
Abstract
We study asymptotic properties of statistics in goodness-of-fit tests under the simple null hypothesis of a multinomial distribution. It is also assumed that the statistics are based on $\phi$-divergences, the sample size $n$ tends to infinity, and the number of cells in the group $k$ is fixed. It is known that in this case the limiting law is the chi-square distribution, and according to the classical theory, the distribution function of the test statistic converges to the limiting one at rate $O(n^{-1/2})$. A distinctive feature of the present paper is the use of an approach in which the original problem is reduced to a well-known problem in number theory, namely, to the generalized Gauss problem on the number of “integer” points in an expanding convex set with smooth boundary. This makes it possible to refine the order of convergence to $ O(n^{-1+\alpha(k)})$, where $\alpha(k) > 0$ and $\alpha(k)$ decreases to zero with increasing $k$. Thus, a result previously known only for the family of power divergence statistics is extended to a wider class of statistics based on $\phi$-divergences.
We introduce a distribution-free goodness-of-fit test, termed the omega-1 test, which naturally complements the Kolmogorov--Smirnov test and Cram\'{e}r--von Mises test and can be viewed as their (piecewise) linear analog. Defined as an $\mathrm{L}^{1}$-functional of the empirical process, the test statistic improves on...
The null distribution of distance covariance is usually approximated by permutation, which is prohibitive when very small p-values are needed, or by matching a few moments to a parametric family, which is inaccurate in the tails. A third option is to approximate the limiting distribution, a weighted sum of chi-square v...
We study from a finite-sample viewpoint the problem of building tests and simultaneous confidence bands for cumulative distribution functions (CDFs), continuous or discrete. We emphasize procedures based on reweighted empirical distribution function (EDF) with shrinking bandwidths in the tails of the distribution. Sinc...
Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+\lambda/n$ in dim...
We study how the response of a Bayesian posterior statistic to future observations changes as information accumulates. For a non-decreasing function $T$, define $\Pi_n^T=\E[T(\Theta)\vert \mathcal F_n]$, where $\Theta$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditionin...
We derive convergence rates for the approximation of the Fr\'echet distribution $\mathcal{F}(\alpha)$ with parameter $\alpha>0$ by sequences of renormalized maxima in the extreme value theorem. Our proofs rely on the application of the infinitesimal generator approach to Stein's method to max-stable distributions, usin...
B. Costacèque, Nicolas Privault· 0 citations
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