Abstract

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized T02 statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples.

Highlights

  • In classical statistical inference, the number of observations is usually known

  • The random variable Nn ∈ N+ denotes the random size of the underlying sample, that is the random number of observations, depending on a parameter n ∈ N+

  • Chebyshev–Edgeworth expansions are derived for the distributions of various statistics from samples with random sample sizes

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Summary

Introduction

The number of observations is usually known. If observations are collected in a fixed time span or we lack observations the sample size may be a realization of a random variable. General transfer theorems for asymptotic expansions of the distribution function of statistics based on samples with non-random sample sizes to their analogues for samples of random sizes are proven in [9,10]. In these papers, rates of convergence and first-order expansion are proved for asymptotically normal statistics.

Statistical Models with a Random Number of Observations
Assumptions on Statistics Tm and Random Sample Sizes Nn
Transfer Proposition from Samples with Non-Random to Random Sample Sizes
Examples for Asymptotically Normally Distributed Statistics
Examples for Asymptotically Chi-Square Distributed Statistics
Chebyshev–Edgeworth Expansions for Distributions of Normalized Random
Limit Distributions of Statistics with Random Sample Sizes Using Different
Main Results
Formal Construction of the Expansions
Proof of Theorems
Conclusions
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