Sample Quantiles Research Articles

On the most powerful quantile test of the scale parameter

The most powerful test of the null hypothesisH 0:σ=σ 0 versus the alternative hypothesisH 1:σ=σ 1 based on a few selected sample quantiles is proposed here where σ is the scale parameter of the distribution and the location parameter μ is known. The quantiles are chosen from a large sample that is either complete or censored (singly-censored or doubly-censored). The relationship between the proposed test and the asymptotically best linear unbiased estimate (ABLUE) of the scale parameter is discussed.

Uniform asymptotic joint normality of a set of increasing number of sample quantiles

Type (B) d asymptotic normality of the joint distribution of sample quantiles is investigated when the number of sample quantiles increases as the sample size increases. This paper aims at a refinement of the original results by Ikeda and Matsunawa [3].

Prediction intervals for survival data.

This paper concerns large sample prediction intervals for the survival times of a future sample based on an initial sample of censored survival data. Simple procedures are developed for obtaining non-parametric and exponential prediction intervals for the future sample quantiles; the non-parametric interval results from inversion of an appropriate test statistic. A simulation study performed under various conditions evaluates the accuracy of the proposed intervals. An adjuvant chemotherapy study of breast cancer patients illustrates the methodology.

Rate of strong uniform convergence of k-NN density estimates

Abstract Let f n (x) be the univariate k -nearest neighbor ( k -NN) density estimate proposed by Loftsgaarden and Quesenberry (1965). By using similar techniques as in Bahadur's representation of sample quantiles (1966), and by the recent results on the oscillation of empirical processes by Stute (1982), we derive the rate of strong uniform convergence of f n (x) on some suitably chosen interval J δ . Some comparison with the kernel estimates is given, as well as the choice of the bandwidth sequence relative to the sample size.

Sample Size for Estimating the Quantiles of Endothelial Cell-Area Distribution

The estimation of corneal endothelium mean cell area (and, hence, mean cell density) is an important problem in clinical ophthalmology. Mitotic division of these cells is not known to occur, and cell deaths are followed by the enlargement of adjacent cells. As a consequence, cell-area distributions change drastically as functions of age and disease. Changes in cell-area distributions, in particular multimodality and skewness due to aging, are observed, and give rise to some difficult sampling problems. In this paper, sample quantiles are investigated as an alternative to the use of the sample mean. Asymptotic approximations are provided for the sample sizes required to estimate population quantiles with a desired precision. Asymptotic sample sizes are then compared with those obtained from tolerance limits. Empirical sample quantiles that can be used as benchmarks to compare corneas of normal individuals against corneas with unknown cell-area distributions are also presented. Aspects that merit further investigation are noted.

A family of minimum quantile distance estimators for the three-parameter Weibull distribution

A family of minimum quantile distance estimators, based on a subset of the sample quantiles, is proposed for the parameters of the three-parameter Weibull distribution. The estimation procedure is applicable to either complete or censored samples and, through use of the associated distance measure, provides a goodness-of-fit test for the Weibull model. The proposed estimators are both consistent and asymptotically normal and, in a particular instance, are optimal over the class of all estimators based on the same quantile subset. The problem of optimal quantile selection is also considered.

On selecting from k finite populations the population with the largest α -quantile

Several procedures for ranking populations according to the quantile of a given order have been discussed in the literature. These procedures deal with continuous distributions. This paper deals with the problem of selecting a population with the largest α-quantile from k ≥ 2 finite populatins, where the size of each population is known. A selection rule is given based on the sample quantiles, where he samples are drawn without replacement. A formula for the minimum probability of a correct selection for the given rule, for a certain configuration of the population α-quantiles, is given in terms of the sample numbers.

Quantile interval estimation

As an alternative to the conventional sample quantile, Kaigh and Lachenbruch (1982) propose a certain U-statistic as a non-parametric estimator of a continuous population quantile. Other related quantile estimators are introduced and studied here. The jackknife procedure is applied to obtain sample variance estimates for the construction of symmetric asymptotic confidence intervals for population quantiles and Monte Carlo studies are employed to investigate validity for small samples.

On r-quick limit sets for empirical and related processes based on mixing random variables

The r-quick limit points of normalized sample paths and empirical distribution functions of mixing processes are characterized. An r-quick version of Bahadur-Kiefer-type representation for sample quantiles is established, which yields the r-quick limit points of quantile processes. These results are applied to linear functions of order statistics. Some results on r-quick convergence of certain Gaussian processes are also established.

On the Computation of Optimal Designs for Certain time Series Models with Applications to Optimal Quantile Selection for Location or Scale Parameter Estimation

Using the results of Chow (Ph.D. dissertation, Texas A & M Univ., 1978) on the optimal placement of knots in the approximation of functions by piecewise polynomials, we present an algorithm for the computation of optimal designs for certain time series models considered by Eubank, Smith and Smith (Ann. Statist., 9 (1981), pp. 486–493), (Tech. Rep. 150, Southern Methodist Univ., 1981). The ideas underlying this algorithm form a unified approach to the computation of optimal spacings for the sample quantiles used in the asymptotically best linear unbiased estimator of a location or scale parameter.

Asymptotic Expansion for a Class of Sample Quantiles

Let X k: n be the kth order statistic (1 ⩽ k ⩽ n) for a random sample of size n from a population with the distribution function F. Let {α n}, { βn} ( βn > 0) be sequences of real numbers and let { kn} ( kn ⩽ n) be a sequence of positive integers. The present article explores the various choices of α n, βn and kn such that under some mild regularity conditions on F, L( Yn) → n (0,1) as n→∞, where Yn = ( Xkn:n + α n)⁄ βn. It is further shown that under some additional conditions on F, standard asymptotic expansion (in Edgeworth form) for the distribution of Yn can be derived.

A generalized quantile estimator

An alternative to the conventional sample Is proposed as a nonparametrlc estimator of a continuous population quantlle.The alternative estimator Is a generalized sample quantlle obtained by averaging an appropriate subsample over all subsamples of .a fixed size.Since the resulting statistic is a U-statistic with representation also as a linear combination of order statistics, known results are employed then to establish asymptotic normality.The alternative estimator is shown to be asymptotically efficient in the class of nonparametrlc models specified by Pfanzagl (1975).Analytic results and Monte Carlo studies with a moderate sample size for a variety of distributions Indicate that the proposed estimator usually provides mean square error of estimation less than that of the conventional sample quantile.

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

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.

On the Asymptotic Accuracy of Efron's Bootstrap

In the non-lattice case it is shown that the bootstrap approximation of the distribution of the standardized sample mean is asymptotically more accurate than approximation by the limiting normal distribution. The exact convergence rate of the bootstrap approximation of the distributions of sample quantiles is obtained. A few other convergence rates regarding the bootstrap method are also studied.

Estimation of the parameters and quantiles of the logistic distribution by linear functions of sample quantiles

Abstract This paper considers the problems of estimating the mean, standard deviation, and quantiles for the logistic distribution. A large sample estimation procedure is utilized which employs the asymptotically best linear unbiased estimators based on sample quantiles. An approximate procedure for optimal spacing selection for the sample quantiles is presented and its properties are discussed.

One‐Sample Probability Plots

SummaryA graphical procedure based on the sample distribution function is shown to be useful for assessing the symmetry of the underlying population, both from a descriptive and a computational point of view. To some extent, it complements graphical techniques based on the sample quantiles.

A Density-Quantile Function Approach to Optimal Spacing Selection

In this paper design techniques for continuous parameter time series regression analysis are employed to develop a general approach to optimal spacing selection for the linear estimation of location and scale parameters by sample quantiles from uncensored or censored samples. The spacings derived from this approach are asymptotically optimal in the sense that they result in near optimal asymptotic relative efficiencies for large values of $k$, the number of spacing elements. A comparison with the optimum efficiencies for several distribution types indicates that the asymptotically optimum spacings perform well for $k \geq 7$. The regression framework is also utilized to develop sufficient conditions for optimal spacing unicity and to obtain asymptotically optimal spacings for quantile estimation.

Large-sample quantile estimation in pareto laws

A large-sample method of estimation for the parameters of Pareto laws is investigatedo The estimates are derived by using a small subset of k sample quantiles out of the original observations. The optimum spacing of the k quantiles is also examined. A Monte Carlo study compares this method with the method of moments and that of maximum likelihood for a selected set of parameter values and sample sizes.

Chi-Square Tests of Fit for Type II Censored Data

The theory of general chi-square statistics for testing fit to parametric families of distributions is extended to samples censored at sample quantiles. Data-dependent cells with sample quantiles as cell boundaries are employed. Asymptotic distribution theory is given for statistics in which unknown parameters are estimated by estimators asymptotically equivalent to linear combinations of functions of order statistics. Emphasis is placed on obtaining statistics having a chi-square limiting null distribution. Examples of such statistics for testing the fit of Type II censored samples to the negative exponential, normal, two-parameter uniform and two-parameter Weibull families are given.

Robust Estimation of Location Using Optimally Chosen Sample Quantiles

Robust estimation of the location parameter α based on selected order statistics is considered. The distribution function is only known to belong to a subset of a set D of distributions consisting of Tukey's lambda family of symmetric distributions with the inverse distribution function of the form α + β/γ) [λγ – (1 – λ)γ], 0 ≤ λ ≤ 1, and the normal, Cauchy and double exponential distributions. The scale parameter β can be unknown. In the set D, distributions with tail length varying from short to extremely long, when the shape parameter γ varies, are included. The asymptotically best linear estimate (ABLE) based on k (k = 2(1)5) optimally chosen symmetrical sample quantiles is considered. It can be used as a robust estimate and is shown to compete favorably with optimally trimmed and Winsorized means, in the sense of giving a higher guaranteed relative asymptotic efficiency (GRAE) for subsets of D. Tables are provided so that the robust k-ABLE giving the highest GRAE for any subset of D can easily be obtained. An adaptive procedure of trying to identify the prototype by means of estimating the tail length is suggested.