Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.2116

Exact Computation of Minimum Sample size for Estimation of Poisson Parameters

Xinjia Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we develop an approach for the exact determination of the minimum sample size for the estimation of a Poisson parameter with prescribed margin of error and confidence level. The exact computation is made possible by reducing infinite many evaluations of coverage probability to finite many evaluations. Such reduction is based on our discovery that the minimum of coverage probability with respect to a Poisson parameter bounded in an interval is attained at a discrete set of finite many values.

Explore related subjects

math.STmath.PRstat.MEstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xinjia Chen. 2008-06-19. Exact Computation of Minimum Sample size for Estimation of Poisson Parameters. https://arxiv.org/abs/0707.2116

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test

In this article, we introduce the novel concept of the second maximum of a Gaussian random field defined on a Riemannian manifold: Once the field reaches its global maximum at a point $t_1$, the second maximum is the largest value of the field away from $t_1$, after the value and the gradient of the field at $t_1$ have been removed. This statistic serves as a powerful tool for characterizing the distribution of the global maximum. By employing an ad-hoc Kac-Rice formula, we derive the explicit distribution of the maximum, conditioned on the second maximum and the independent part of the Riemannian Hessian (the part of the Hessian at $t_1$ that is independent of the value and the gradient of the field at $t_1$). This approach yields an exact test based on the spacing between these maxima, which we refer to as the spacing test. We investigate the applicability of this test for detecting sparse alternatives in Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Theoretical results are supported by numerical experiments that illustrate the calibration and power of the proposed procedures. More generally, we provide a framework for applying the spacing test to continuous sparse regression for any Gaussian random field on a Riemannian manifold. Furthermore, when the variance-covariance function is known only up to a scaling factor, we derive an exact Studentized version of the procedure, coined the t-spacing test. We show that this test is perfectly calibrated under the null hypothesis and exhibits high power for detecting sparse alternatives. As a corollary, we derive an exact, non-asymptotic test for the spiked tensor model without requiring prior knowledge of the noise level.

math.ST↗
arXiv2026

Robustness for free: asymptotic size and power of max-tests in high dimensions

Allowing for adversarial contamination and heavy tails, we study testing whether the mean of a high-dimensional random vector equals zero. Because standard max-tests based on sample averages are highly non-robust, we propose a max-test based on quantile-winsorized observations. The test controls asymptotic size under adversarial contamination and only requires $m>2$ moments, while allowing dimension to grow exponentially with sample size. We fully characterize its asymptotic power function. Comparing with the standard max-test, for which we also derive a power characterization as a benchmark, we show that robustness is obtained for free at first order: under the stronger conditions that make the standard max-test valid, our robust test has identical asymptotic power. We show that the decay condition on the contamination-rate imposed is necessary for this no-first-order-loss property. Finally, we also study the role of bootstrap critical values, showing that their use never decreases power, can strictly improve asymptotic power in extremely correlated designs, but often has no first-order asymptotic effect.

math.ST↗
arXiv2026

A proximal gradient algorithm for composite log-concave sampling

We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $α_f$-convex $f$ and a restricted Gaussian oracle (RGO) for $α_g$-convex $g$, where $α_g \ge 0$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance using $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ oracle calls where $κ:= 1 \vee β/α$, with $0\le α_g \le β$ and $α:= α_f + α_g > 0$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.

math.ST↗