Asymptotic properties of sample quantiles from a finite population

In this paper we consider the problem of estimating quantiles of a finite population of size N on the basis of a finite sample of size n selected without replacement. We prove the asymptotic normality of the sample quantile and show that the scaled variance of the sample quantile converges to the asymptotic variance under a slight moment condition. We also consider the performance of the bootstrap in this case, and find that the usual (Efron’s) bootstrap method fails to be consistent, but a suitably modified version of the bootstrapped quantile converges to the same asymptotic distribution as the sample quantile. Consistency of the modified bootstrap variance estimate is also proved under the same moment conditions.     

Asymptotic properties of rank tests under discrete distributions

Summary This paper is devoted to problems of rank tests when samples are drawn from purely discrete distributions. There are considered two ways of treatment of ties, and the distribution of the respective test statistics is derived under the hypothesis of randomness and under the contiguous alternative. Furthermore, their asymptotic power and efficiency are established.     

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

Asymptotic properties of conditional quantiles of the Cauchy distribution in Hilbert space

In this paper, we consider the conditional distributions that are induced by finite-dimensional projections of a σ-additive Cauchy measure defined in a real Hilbert space. In the case where the dimension of projections tends to infinity, we establish the almost sure convergence of “conforming” sequences of finite-dimensional conditional quantiles and prove the strong law of large numbers for the triangular array scheme applied to a family of conditional distributions. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

On Bahadur's representation of sample quantiles

We extend the well known transformation technique for order statistics to get less restrictive conditions for the Bahadur representation of sample quantiles.     

Asymptotic properties of solutions to two discrete airy equations

We construct two finite difference models for the Airy differential equation. In one model, the form of the complete asymptotic representation of the solution can be found. However, this is not the case for the second model which is based on the use of a nonstandard difference scheme. This scheme leads to a second-order, linear difference equation that is not of a form for which the theorems of Poincaré and Perron can be directly applied to obtain the asymptotic behavior of the solutions.     

Asymptotic properties of symmetric stable distributions with small index

The family of symmetric stable distributions is considered. A local limit theorem is established which takes account of large deviations as the characteristic exponent α tends to zero. On the basis of this theorem the asymptotics of Fisher information with respect to α is obtained. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Tests of significance using selected sample quantiles

Large sample tests of significance for the location parameter, the scale parameter, and quantiles for a location-scale family of distributions based on a few optimally chosen sample quantiles are considered.     

Some characterizations of almost sure bounds for weighted multidimensional empirical distributions and a Glivenko-Cantelli theorem for sample quantiles

Summary Characterizations of almost sure bounds and a Glivenko-Cantelli theorem are obtained for certain weighted m-dimensional empirical distributions. These results constitute generalizations and extensions of the work of Shorack and Wellner (1978) and Wellner (1977, 1978). Also as an example of the potential use of the techniques developed in this paper a Glivenko-Cantelli type theorem is proven for sample quantiles.     

可加Lévy 过程的轨道渐近性质

本文主要研究有限个相互独立的从属过程之和的样本轨道的渐近性质. 给出了样本轨道在零点 附近和无穷远处的渐近增长率的上下极限, 并且得出了在零点附近渐近增长率的一致下极限.     

Asymptotic properties of optimal trajectories of discrete economic and demographic models

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 37–48, November–December, 1988.     

Asymptotic properties of posterior distributions in nonparametric regression with non-Gaussian errors

We investigate the asymptotic behavior of posterior distributions in nonparametric regression problems when the distribution of noise structure of the regression model is assumed to be non-Gaussian but symmetric such as the Laplace distribution. Given prior distributions for the unknown regression function and the scale parameter of noise distribution, we show that the posterior distribution concentrates around the true values of parameters. Following the approach by Choi and Schervish (Journal of Multivariate Analysis, 98, 1969–1987, 2007) and extending their results, we prove consistency of the posterior distribution of the parameters for the nonparametric regression when errors are symmetric non-Gaussian with suitable assumptions.     

On the uniform asymptotic joint normality of sample quantiles

Summary Uniform (or type (B) _d ) asymptotic normality of the joint distribution of an increasing number of sample quantiles as the sample size increases is investigated in both cases where the basic distributions are equal and are unequal. Under fairly general assumptions, sufficient conditions are derived for the asymptotic normality of sample quantiles. Type (B) _d asymptotic normality is a strictly stronger notion than the usual one which is based on the convergence in law, and the results obtained in this article will be helpful to widen the applicability of results on asymptotic normality of sample quantiles to related statistical inferences.     

Asymptotic stability of discrete cocycles

We introduce the notion of asymptotic stability of sequences of multifunctions associated with discrete cocycles. Some sufficient conditions for existence of attracting sets are given. The use of the topological (Kuratowski's) limits, as less complicated as commonly used Hausdorff metric, let us to weaken many standard assumptions. We show that in considered case existence of attractor is a property of a cocycle mapping itself and does not depend on properties of a parameter nor a state space. The obtained results generalize earlier on iterated function systems and can be applied for non-autonomous as well as random dynamical systems.     

Asymptotic dependence between random central quasi-ranges and random empirical quantiles

The asymptotic dependence between the central quasi-ranges and empirical quantiles was studied. The asymptotic dependence are obtained when the sample size is a positive integer valued random variable (r.v.). The dependence conditions and limit forms are obtained under generl conditions such as: the interrelation of the basic variables (the original random sample) and the random sample size is not restricted. In addition the normalizing constants do not depend on the random size.     

Asymptotic quantization of probability distributions

We give a brief introduction to results on the asymptotics of quantizatlon errors. The topics discussed in-clude the quantization dimension, asymptotic distributions of sets of prototypes, asymptotically optimalquantizations, approximations and random quantizations.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.