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Register a new userComplex sampling designs: uniform limit theorems and applications
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Added by
Qiyang Han
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz-Thompson empirical process arising from complex sampling designs. Global theorems such as Glivenko-Cantelli and Donsker theorems, and local theorems such as local asymptotic modulus and related ratio-type limit theorems are proved for both the Horvitz-Thompson empirical process, and its calibrated version. Limit theorems of other variants and their condition
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For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the Hajek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.
The minimum aberration criterion has been frequently used in the selection of fractional factorial designs with nominal factors. For designs with quantitative factors, however, level permutation of factors could alter their geometrical structures and statistical properties. In this paper uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion. We show that minimum aberration designs have low discrepancies on average. An efficient method for constructing uniform minimum aberration designs is proposed and optimal designs with 27 and 81 runs are obtained for practical use. These designs have good uniformity and are effective for studying quantitative factors.
This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $Pr(n^{-1/2}sum_{i=1}^n X_iin A)$ where $X_1,dots,X_n$ are independent random vectors in $mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_nto infty$ as $n to infty$ and $p gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadurs representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that [sqrt{n}Biggl(frac{1}{n}sum_{i=1}^nphibigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}bigr)-bar{gamma}Biggr)=frac{1}{sqrt{n}}sum_{i=1}^nZ_{n,i}+mathrm{o}_P(1)] as $nrightarrowinfty$, where $bar{gamma}$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.
We consider the problem of optimal transportation with general cost between a empirical measure and a general target probability on R d , with d $ge$ 1. We extend results in [19] and prove asymptotic stability of both optimal transport maps and potentials for a large class of costs in R d. We derive a central limit theorem (CLT) towards a Gaussian distribution for the empirical transportation cost under minimal assumptions, with a new proof based on the Efron-Stein inequality and on the sequential compactness of the closed unit ball in L 2 (P) for the weak topology. We provide also CLTs for empirical Wassertsein distances in the special case of potential costs | $bullet$ | p , p > 1.
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