Combinations Of Order Statistics Research Articles

Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas.

On analogues of the berry?esseen inequality for L-statistics

In this paper we present estimates for the rate of convergence of distributions of linear combinations of order statistics (L-statistics) to the normal law, having an optimal order under light moment assumptions regarding the initial distribution.

$L$-Statistics in Complex Survey Problems

We study linear combinations of order statistics ($L$-statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$-statistics are considered: smooth $L$-statistics with weights generated by a smooth function and nonsmooth $L$-statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$-statistics or functions of smooth $L$-statistics used in survey problems. It is shown that under weak conditions the smooth $L$-statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing. For the sample quantiles, their asymptotic normality requires more conditions on the finite population distribution functions. Consistent estimators for the asymptotic variances of the sample quantiles are derived. Asymptotic validity of the Woodruff's confidence intervals for population quantiles is also proved.

On Analogues of Berry–Esseen Inequality for Truncated Linear Combinations of Order Statistics

On Analogues of Berry–Esseen Inequality for Truncated Linear Combinations of Order Statistics

Asymptotic distributions οf linear combinations of order statistics

We study the asymptotic distributions of linear combinations of order statistics (L-statistics) which can be expressed as differentiable statistical functionals and we obtain Berry-Esseen type bounds and the Edgeworth series for the distribution functions

A smooth adaptive estimator of the mean of a symmetric or asymmetric distribution

We study an adaptive estimate (SA) of the mean of a continuous distribution F based on a quasi-linear combination of order statistics with weights which smoothly adapt to the extremeness of the tail order statistics. SA is based in part on the selector statistic (HG2) of Hogg and Lenth (1984), which has been used widely for estimating the location when F is symmetric. We compare SA to HG2 via Monte Carlo for both symmetric and asymmetric F. We find that SA dominates or ties HG2 for symmetric F with t 5 or lighter tails, and enjoys robustness to asymmetric F (not possessed by HG2), for which it competes well with the sample mean.

Strong Moderate Deviation Theorems

Strong moderate deviation theorems are concerned with relative errors in the tails caused by replacing the exact distribution function by its limiting distribution function. A new approach for deriving such theorems is presented using strong approximation inequalities. In this way a strong moderate deviation theorem is obtained for statistics of the form $T(\alpha_n)$, where $T$ is a sublinear functional and $\alpha_n$ is the empirical process. The basic theorem is also applied on linear combinations of order statistics, leading to a substantial improvement of previous results.

Constrained adaptive LMS L-filters

Two novel adaptive nonlinear filter structures are proposed which are based on linear combinations of order statistics. These adaptive schemes are modifications of the standard LMS algorithm and have the ability to incorporate constraints imposed on coefficients in order to permit location-invariant and unbiased estimation of a constant signal in the presence of additive white noise. The convergence in the mean and in the mean square of the proposed adaptive nonlinear filters is studied. The rate of convergence is also considered. It is verified by simulations that the independence theory provides useful bounds on the rate of convergence. The extreme eigenvalues of the matrix which controls the performance of the location-invariant adaptive LMS L-filter are related to the extreme eigenvalues of the correlation matrix of the ordered noise samples which controls the performance of other adaptive LMS L-filters proposed elsewhere. The proposed filters can adapt well to a variety of noise probability distributions ranging from the short-tailed ones (e.g. uniform distribution) to long-tailed ones (e.g. Laplacian distribution).

Which measures of skewness and kurtosis are best?

Indices of distributional shape based on linear combinations of order statistics have recently been described by Hosking. Their usefulness as tools for practical data analysis is examined. They are found to have several advantages over the conventional indices of skewness and kurtosis (square root of b1 and b2) and no serious drawbacks. It is proposed, therefore, that they should replace square root of b1 and b2 in routine data analysis. To implement this suggestion, action by the developers of standard statistical software is needed.

Efficient computation of the performance of bootstrap and jackknife estimators of the variance ofL-statistics

Calculation of the bootstrap and the jackknife estimators of the variance of a statistic often relies on approximation techniques because the exact values are difficult if not impossible to obtain analytically. For the special case where the statistic is a linear combination of order statistics we propose to calculate the exact values combinatorically, thus completely eliminating the second-stage simulation error.

Quantile inequalities for linear combinations of order statistics from ordered families of distributions

Quantile inequalities for linear combinations of order statistics from ordered families of distributions

Adaptive filters based on order statistics

An adaptive filter structure which is based on linear combinations of order statistics is proposed. An efficient method to update the filter coefficients is presented, which is based on the minimal mean-square error criterion and which is similar to the Widrow algorithm for the linear adaptive filters. Another method for coefficient update is presented, which is similar to the recursive least squares (RLS) algorithm and which has faster convergence properties. The proposed-filter can adapt well to a variety of noise probability distributions, including impulsive noise. It also performs well in the case of nonstationary signals and, therefore, it is suitable for image-processing applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive robust estimation of location for stable distributions

An adaptive procedure based on using two selector statistics and ,where Xp is the pth sample quantile, is described and used to choose among some robust estimators of location with respect to a class V of symmetric stable distributions with a characteristic exponent a ranging from 0.5 to 2.0 with steps of 0.1 or 0.05. The estimators considered are special symmetrical linear combinations of order statistics. Asymptotic variances of these estimators are computed using the characteristic function. Optimum asymptotic efficiencies of these estimators for the entire class V as well as for several subclasses of it are given.

L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics

SUMMARY L-moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis described in Section 2, and new methods of parameter estimation for several distributions. The theory of L-moments parallels the theory of (conventional) moments, as this list of applications might suggest. The main advantage of L-moments over conventional moments is that L-moments, being linear functions of the data, suffer less from the effects of sampling variability: L-moments are more robust than conventional moments to outliers in the data and enable more secure inferences to be made from small samples about an underlying probability distribution. L-moments sometimes yield more efficient parameter estimates than the maximum likelihood estimates.

Kernel Quantile Estimators

Abstract For an estimator of quantiles, the efficiency of the sample quantile can be improved by considering linear combinations of order statistics, that is, L estimators. A variety of such methods have appeared in the literature; an important aspect of this article is that asymptotically several of these are shown to be kernel estimators with a Guassian kernel, and the bandwidths are identified. It is seen that some implicit choices of the smoothing parameter are asymptotically suboptimal. In addition, the theory of this article suggests a method for choosing the smoothing parameter. How much reliance should be placed on the theoretical results is investigated through a simulation study. Over a variety of distributions little consistent difference is found between various estimators. An important conclusion, made during the theoretical analysis, is that all of these estimators usually provide only modest improvement over the sample quantile. The results indicate that even if one knew the best estimator ...

Kernel Quantile Estimators

Abstract For an estimator of quantiles, the efficiency of the sample quantile can be improved by considering linear combinations of order statistics, that is, L estimators. A variety of such methods have appeared in the literature; an important aspect of this article is that asymptotically several of these are shown to be kernel estimators with a Guassian kernel, and the bandwidths are identified. It is seen that some implicit choices of the smoothing parameter are asymptotically suboptimal. In addition, the theory of this article suggests a method for choosing the smoothing parameter. How much reliance should be placed on the theoretical results is investigated through a simulation study. Over a variety of distributions little consistent difference is found between various estimators. An important conclusion, made during the theoretical analysis, is that all of these estimators usually provide only modest improvement over the sample quantile. The results indicate that even if one knew the best estimator ...

Inequalities for the partial sums of elliptical order statistics related to genetic selection

AbstractLet X1,…,Xn be exchangeable normal variables with a common correlation p, and let X(1) &gt; … &gt; X(n) denote their order statistics. The random variable σni=n—k+1xi, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.

On the Rate of Convergence to the Normal Law of Finitely Truncated Linear Combinations of Order Statistics

Previous article Next article On the Rate of Convergence to the Normal Law of Finitely Truncated Linear Combinations of Order StatisticsN. V. GribkovaN. V. Gribkovahttps://doi.org/10.1137/1133111PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Steinar Bjerve, Error bounds for linear combinations of order statistics, Ann. Statist., 5 (1977), 357–369 58:24666 0356.62015 CrossrefGoogle Scholar[2] R. Helmers, A Berry-Esseen theorem for linear combinations of order statistics, Ann. Probab., 9 (1981), 342–347 82i:62074 0474.62039 CrossrefGoogle Scholar[3] R. Helmers, Edgeworth expansions for linear combinations of order statistics, Mathematical Centre Tracts, Vol. 105, Mathematisch Centrum, Amsterdam, 1982iv+137 84f:62029 0485.62017 Google Scholar[4] V. A. Egorov and , V. B. Nevzorov, Certain estimates of the rate of convergence of sums of order statistics to the normal law, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 41 (1974), 105–128, 167, (In Russian.) 51:2148 Google Scholar[5] N. V. Gribkova, An estimate for the rate of convergence to the normal law of truncated linear combinations of order statistics, Mat. Zametki, 42 (1987), 739–746, 764, (In Russian.) 89b:60056 0634.62016 Google Scholar[6] W. R. van Zwet, A Berry-Esseen bound for symmetric statistics, Z. Wahrsch. Verw. Gebiete, 66 (1984), 425–440 86h:60063 0525.62023 CrossrefGoogle Scholar[7] H. A. David, Order statistics, John Wiley & Sons Inc., New York, 1970xi+272 44:3440 0223.62057 Google Scholar[8] Stephen M. Stigler, Linear functions of order statistics with smooth weight functions, Ann. Statist., 2 (1974), 676–693 51:9354 0286.62028 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 33, Issue 4| 1989Theory of Probability & Its Applications595-765 History Submitted:16 March 1987Published online:28 July 2006 InformationCopyright © 1988 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1133111Article page range:pp. 726-729ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Extension of the harrell-davis quantile estimator to finite populations

The quantile estimator formulated by Harrell and Davis (1982) as a linear combination of order statistics is extended to the case of sampling from a finite population. Sampling with and without replacement are considered. The relative efficiency of the estimator is discussed and then demonstrated with simulation evidence. A modified version of the estimator is discussed for the case of sampling without replacement. In the case of estimating the population median the proposed estimator is as much as 30% more efficient than the traditional sample median in the simulation experiments.

Stochastic comparisons of order statistics,with applications in relibility

This paper reviews recent developments in the stochastic comparison of order statistics. The results discussed are basically: (l) Stochastic comparisons of linear combinations of order statistics from distributions F and G where G−1 F is convex or starshaped. (2) Stochastic comparisons of individual order statistics and of vectors of order statistics from underlying heterogeneous distributions by the use of majorization and Schur function theory. (3) Stochastic comparison of random processes. Applications to reliability problems are presented illustrating the use and value of the theoretical results described