Sample Quantiles Research Articles

On the Distribution of a Studentized Quantile

SUMMARY The asymptotic variance of a sample quantile depends on the value of the population density at the population quantile. Therefore Studentizing a sample quantile involves density estimation, either explicitly or implicitly. One popular way of Studentizing is to use the Siddiqui–Bloch–Gastwirth estimator, whose construction depends crucially on the choice of a smoothing parameter m. We examine the effect which the selection of m has on the level error of tests or confidence intervals based on Studentized quantiles and show that, if we wish to minimize this error, m should be of a smaller order of magnitude than is recommended by squared error theory. The cases of one- and two-sided procedures are distinctly different, the former being less sensitive to the choice of m.

Discussion of the Papers by Hinkley and Diciccio and Romano

Discussion of the Papers by Hinkley and Diciccio and Romano

O-statistics and their applications

O-statistic distributional properties and nonparametric applications including numerical and graphical data summary, quantile point and interval estimation, and tests of uniformity are presented and discussed in a survey of both published and new results. Initially termed generalized sample quantiles and proposed as smoothed alternatives to the conventional sample quantile, the O-statistics generalize the usual sample order statistics and are obtained by averaging subsample quantiles over all subsamples of a fixed size. The resulting statistics form natural bases for the LU-statistics.

Comparison of nonparametric point estimators for interquantile differences in moderate sized samples

Non-parametric estimators for interquantile differences in moderate sized samples are identified, including a simple method based on sample quantiles, an extension of the Harrell and Davis (1982) method, an extension of the Kaigh and Lachenbruch (1982) approach, and methods based on Yang (1985). For those estimators compared via a simulation study (Harrell-Davis and sample quantile), using relative bias and relative efficiency as judgment criteria, the Harrell-Davis method is judged to be preferable for interquantile estimation with data sets of moderate size.

A note on bootstrapping the variance of sample quantile

Letmn, p denote thep-th quantile based onn observations and let λp denote the population quantile. In this paper consistency of the bootstrap estimate of variance of $$\sqrt n (m_{n,p} - \lambda _p )$$ is established.

Quantile Processes with Statistical Applications.

A Preliminary Study of Quantile Processes A Weak Convergence of the Normed Sample Quantile Process Strong Approximations of the Normed Quantile Process Two Approaches to Constructing Simultaneous Confidence Bounds for Quantiles Weak Convergence of Quantile Processes in Weighted Sup-Norm Metrics and Further Strong Approximations On Bahadur's Representation of Sample Quantiles and on Kiefer's Theory of Deviations Between the Sample Quantile and Empirical Processes Quadratic Forms of the Quantile Process. Weighted Spacings and Testing for Composite Goodness-of-Fit Strong Approximations of the Quantile Process of the Product-Limit Estimator An Invariance Principle for Nearest-Neighbor Empirical Density Functions A Nearest- Neighbor Estimator for the Score Function.

Rizvi and Sobel Type Nonparametric Selection Procedures: Review and Extensions

SYNOPTIC ABSTRACTThis paper reviews the area of nonparametric selection procedures based on sample quantiles; the underlying distributions are assumed to be continuous. Both the indifference zone and the (random-size) subset selection formulations are surveyed. The problem of comparing several experimental populations with a control is reviewed and some new results are obtained. The three-decision problem which allows nonselection with the assertion of equivalence and the problem of simultaneous selection and interval estimation of the largest α-quantile are also discussed.

On the accuracy of the bootstrap approximation of the joint distribution of sample quantiles

It is proved that the accuracy of the bootstrap approximation of the joint distribution of sample quantiles lies between O(n−1/4) and O(n−1/4 an), where (log(n))1/2=O(an). As an application, we investigated confidence intervals based on the bootstrap.

Simple consistent estimators of stable distribution parameters

The four parameters of a stable distribution may be estimated consistently from five pre-determined sample quantiles with the aid of the accompanying tables, for a in the range [0.6, 2.0] and g in the range [-1, 1]. The problem of the discontinuity of the traditional location parameter in the asymmetrical cases as a passes unity is resolved. The proposed estimators of a and c are similar to those of Fama and Roll, except that the small asymptotic bias in their estimators has been eliminated, and their restrictions that a be no less than 1.0 and that the distribution be symmetrical have been relaxed. The proposed estimators can provide good initialization values for other more efficient, but computer-intensive, methods.

A Smooth Nonparametric Estimator of a Quantile Function

Abstract A smooth alternative to the conventional sample quantile function as a nonparametric estimator of a population quantile function is proposed. The proposed estimator is essentially a kernel type of estimator and has the same asymptotic distribution as the conventional sample quantile. The mean squared convergence rate of the proposed estimator is also estimated. Monte Carlo studies are conducted to compare the proposed estimator with the sample quantile and the estimator proposed by Kaigh and Lachenbruch. The feasibility of using bootstrap techniques to estimate the optimal window width for the proposed estimator is also considered.

Logistic growth curve of chickens: heritability of parameters.

Parameters of the logistic function of growth, fit to individual body weight curves of two randombred control populations of each sex of chickens from hatching to 45 weeks of age, were evaluated. Growth-rate constant and age at the infection point in the curve were estimated by the method of sample quantiles from individual weekly body weights of 225 males and 281 females of the Rhode Island Red (RIR) line, and 164 males and 239 females of the White Leghorn (WL) line. Heritability estimates, based on correlation among full-sibs, of growth rate constant were 0.18 +/- 0.32 in males and 0.29 +/- 0.29 in females of the RIR line, and 0.41 +/- 0.40 in males and 0.46 +/- 0.32 in females of the WL line. Estimates of heritability of age at the inflection point were 0.36 +/- 0.44 in males and 0.42 +/- 0.32 in females of the RIR line, and 0.46 +/- 0.41 in males and 0.50 +/- 0.28 in females of the WL line. Observed variation for each trait probably does not provide evidence for heritable differences. No genetic correlations were evident among growth-rate constant, age at the point of inflection, and initial or maximum weight. According to these results, it does not appear that selection for growth-rate constant or age at the point of inflection will change the shape of the growth curve of these populations genetically. Moreover, correlated genetic change in initial or maximum weight would not be expected.

Estimating a Quantile of a Symmetric Distribution

The problem is to estimate a quantile of a symmetric distribution. The cases of known and unknown center are studied for small and large samples. The estimators for known center are the sample quantile, the symmetrized sample quantile, the sample quantile from flipped over data, the Rao-Blackwellized sample quantile, and a Bayes estimator using a Dirichlet prior. For center unknown, we study the analogues of the first four estimators listed above. For small samples and center known, the Rao-Blackwellized sample quantile performs very well for normal and double exponential distributions while for the Cauchy distribution the flipped over estimator did well. In the center known case, the latter four estimators are asymptotically equivalent, asymptotically optimal in the sense of Hajek's convolution, and asymptotically minimax in the Hajek-LeCam sense. For center unknown, those properties remain true if one uses an adaptive estimator of the center for the symmetrized sample quantile, the flipped over estimator, and the Rao-Blackwell estimator.

Logistic growth curve of chickens: a comparison of techniques to estimate parameters

Parameters of a mathematical function of growth, fit to the body weight curve of two randombred control populations of each sex of chickens from hatching through 45 weeks of age, were estimated. The logistic function was chosen from among growth formulae that express rate of gain as a function of weight at a given time and gain to be made. Two logistic parameters, growth-rate constant and age at the point of inflection, were estimated by the methods of sample quantiles and nonlinear regression from weekly mean body weights of 225 males and 281 females of the Rhode Island Red (RIR) line, and 164 males and 239 females of the White Leghorn (WL) line. Males had a larger growth-rate constant than females of the same line. The RIR line had a larger rate constant than the WL line, for each sex. Age at the point of inflection was similar for males and females in the RIR line, but smaller for males than females in the WL line. Sample quantiles yielded larger, less precise estimates of the growth-rate constant than nonlinear regression. Estimates of age at the point of inflection were usually smaller using sample quantiles.

Asymptotic Properties of a Family of Minimum Quantile Distance Estimators

Abstract A family of estimators based upon M specified quantiles is defined. These procedures take as the estimate the vector that minimizes a quadratic distance measure between M sample quantiles and a parametric family of quantile functions. Under regularity conditions these estimators are consistent, asymptotically normal, and robust. For a specific quadratic form the estimator is optimal among a class of asymptotically normal estimators, and it approaches full efficiency as M approaches infinity. The asymptotic relative efficiency is computed for various sets of quantiles and various parameter values of the three-parameter lognormal distribution. The small-sample properties and robustness of the optimal M-quantile estimator for the three-parameter lognormal distribution are investigated in Monte Carlo studies.

Asymptotic Properties of a Family of Minimum Quantile Distance Estimators

Abstract A family of estimators based upon M specified quantiles is defined. These procedures take as the estimate the vector that minimizes a quadratic distance measure between M sample quantiles and a parametric family of quantile functions. Under regularity conditions these estimators are consistent, asymptotically normal, and robust. For a specific quadratic form the estimator is optimal among a class of asymptotically normal estimators, and it approaches full efficiency as M approaches infinity. The asymptotic relative efficiency is computed for various sets of quantiles and various parameter values of the three-parameter lognormal distribution. The small-sample properties and robustness of the optimal M-quantile estimator for the three-parameter lognormal distribution are investigated in Monte Carlo studies.

Distribution-free confidence intervals for quantiles in small samples

Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.

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.

Weak Convergence of the Weighted Empirical Quantile Process in $L^2(0, 1)$

Sufficient conditions are developed for various versions of the weighted empirical quantile process to converge weakly in $L^2(0, 1)$ to a weighted Brownian bridge. The results are directly applicable to the derivation of the asymptotic distribution of goodness of fit tests based on the sample quantiles that can be written as a functional defined on $L^2(0, 1)$ continuous in the norm topology. In the process, tight bounds for the moments of transformed uniform order statistics are derived that are likely to have applications elsewhere.

Inner tolerance limits controlling tail proportions for the normal distribution

Inner tolerance limits are defined to be statistics based on a random sample which lie between certain quantiles of the distribution sampled with a certain confidence. Assuming an underlying normal distribution, tables are presented which allow inner limits to be implemented over a range of sample sizes, quantiles, and confidences.

A Space-Efficient Recursive Procedure for Estimating a Quantile of an Unknown Distribution

Consider the problem of computing an estimate of a percentile or quantile of an unknown population based on a random sample of n observations. By viewing this problem as a problem in stochastic approximation, we obtain an estimator that requires only a small amount of direct access storage space that does not increase with the sample size. We show that a modified version of the simple stochastic approximation estimator has the same large-sample behavior as the sample quantile, which has the smallest asymptotic variance among all reasonable estimators. The modified procedure also yields an estimate of the asymptotic variance of the estimator. Some simulation results are presented to show that the proposed estimator performs well in samples of moderate size.