Comparing two independent groups via the lower and upper quantiles

The most common strategy for comparing two independent groups is in terms of some measure of location intended to reflect the typical observation. However, it can be informative and important to compare the lower and upper quantiles as well, but when there are tied values, extant techniques suffer from practical concerns reviewed in the paper. For the special case where the goal is to compare the medians, a slight generalization of the percentile bootstrap method performs well in terms of controlling Type I errors when there are tied values [Wilcox RR. Comparing medians. Comput. Statist. Data Anal. 2006;51:1934–1943]. But our results indicate that when the goal is to compare the quartiles, or quantiles close to zero or one, this approach is highly unsatisfactory when the quantiles are estimated using a single order statistic or a weighted average of two order statistics. The main result in this paper is that when using the Harrell–Davis estimator, which uses all of the order statistics to estimate a quantile, control over the Type I error probability can be achieved in simulations, even when there are tied values, provided the sample sizes are not too small. It is demonstrated that this method can also have substantially higher power than the distribution free method derived by Doksum and Sievers [Plotting with confidence: graphical comparisons of two populations. Biometrika 1976;63:421–434]. Data from two studies are used to illustrate the practical advantages of the method studied here.     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

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.     

Pitman Closeness of Order Statistics to Population Quantiles

In this article, Pitman closeness of sample order statistics to population quantiles of a location-scale family of distributions is discussed. Explicit expressions are derived for some specific families such as uniform, exponential, and power function. Numerical results are then presented for these families for sample sizes n = 10,15, and for the choices of p = 0.10, 0.25, 0.75, 0.90. The Pitman-closest order statistic is also determined in these cases and presented.     

Small sample approximations for spacing statistics

Edgeworth expansions as well as saddle-point methods are used to approximate the distributions of some spacing statistics for small to moderate sample sizes. By comparing with the exact values when available, it is shown that a particular form of Edgeworth expansion produces extremely good results even for fairly small sample sizes. However, this expansion suffers from negative tail probabilities and an accurate approximation without this disadvantage, is shown to be the one based on saddle-point method. Finally, quantiles of some spacing statistics whose exact distributions are not known, are tabulated, making them available in a variety of testing contexts.     

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.     

Exact nonparametric confidence,prediction and tolerance intervals based on multi-sample Type-II right censored data

Exact nonparametric inference based on ordinary Type-II right censored samples has been extended here to the situation when there are multiple samples with Type-II censoring from a common continuous distribution. It is shown that marginally, the order statistics from the pooled sample are mixtures of the usual order statistics with multivariate hypergeometric weights. Relevant formulas are then derived for the construction of nonparametric confidence intervals for population quantiles, prediction intervals, and tolerance intervals in terms of these pooled order statistics. It is also shown that this pooled-sample approach assists in achieving higher confidence levels when estimating large quantiles as compared to a single Type-II censored sample with same number of observations from a sample of comparable size. We also present some examples to illustrate all the methods of inference developed here.     

Empirical Likelihood Inference for Population Quantiles with Unbalanced Ranked Set Samples

We consider nonparametric interval estimation for the population quantiles based on unbalanced ranked set samples. We derived the large sample distribution of the empirical log likelihood ratio statistic for the quantiles. Approximate intervals for quantiles are obtained by inverting the likelihood ratio statistic. The performance of the empirical likelihood interval is investigated and compared with the performance of the intervals based on the ranked set sample order statistics.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

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.     

Comparing two dependent groups via quantiles

This paper considers two general ways dependent groups might be compared based on quantiles. The first compares the quantiles of the marginal distributions. The second focuses on the lower and upper quantiles of the usual difference scores. Methods for comparing quantiles have been derived that typically assume that sampling is from a continuous distribution. There are exceptions, but generally, when sampling from a discrete distribution where tied values are likely, extant methods can perform poorly, even with a large sample size. One reason is that extant methods for estimating the standard error can perform poorly. Another is that quantile estimators based on a single-order statistic, or a weighted average of two-order statistics, are not necessarily asymptotically normal. Our main result is that when using the Harrell–Davis estimator, good control over the Type I error probability can be achieved in simulations via a standard percentile bootstrap method, even when there are tied values, provided the sample sizes are not too small. In addition, the two methods considered here can have substantially higher power than alternative procedures. Using real data, we illustrate how quantile comparisons can be used to gain a deeper understanding of how groups differ.     

A note on the nonparametric test based on theL 1-version of the Cramér-von Mises statistic

We consider the test based on theL ₁-version of the Cramér-von Mises statistic for the nonparametric two-sample problem. Some quantiles of the exact distribution under H₀ of the test statistic are computed for small sample sizes. We compare the test in terms of power against general alternatives to other two-sample tests, namely the Wilcoxon rank sum test, the Smirnov test and the Cramér-von Mises test in the case of unbalanced small sample sizes. The computation of the power is rather complicated when the sample sizes are unequal. Using Monte Carlo power estimates it turns out that the Smirnov test is more sensitive to non stochastically ordered alternatives than the new test. And under location-contamination alternatives the power estimates of the new test and of the competing tests are equal.     

QUANTILE ESTIMATION AND PROBABILITY COVERAGES

When estimating population quantiles via a random sample from an unknown continuous distribution function it is well known that a pair of order statistics may be used to set a confidence interval for any single desired, population quantile. In this paper the technique is generalized so that more than one pair of order statistics may be used to obtain simultaneous confidence intervals for the various quantiles that might be required. The generalization immediately extends to the problem of obtaining interval estimates for quantile intervals. Distributions of the ordered and unordered probability coverages of these confidence intervals are discussed as are the associated distributions of linear combinations of the coverages.     

A low-end quantile estimator from a right-skewed distribution

ABSTRACT

In many statistical applications estimation of population quantiles is desired. In this study, a log–flip–robust (LFR) approach is proposed to estimate, specifically, lower-end quantiles (those below the median) from a continuous, positive, right-skewed distribution. Characteristics of common right-skewed distributions suggest that a logarithm transformation (L) followed by flipping the lower half of the sample (F) allows for the estimation of the lower-end quantile using robust methods (R) based on symmetric populations. Simulations show that this approach is superior in many cases to current methods, while not suffering from the sample size restrictions of other approaches.     

Sample quantiles and additive statistics: Information,sufficiency, estimation

The order of the increase in the Fisher information measure contained in a finite number k of additive statistics or sample quantiles, constructed from a sample of size n, as n → ∞, is investigated. It is shown that the Fisher information in additive statistics increases asymptotically in a manner linear with respect to n, if 2 + δ moments of additive statistics exist for some δ > 0. If this condition does not hold, the order of increase in this information is non-linear and the information may even decrease. The problem of asymptotic sufficiency of sample quantiles is investigated and some linear analogues of maximum likelihood equations are constructed.     

Noninformative nonparametric quantile estimation for simple random samples

For noninformative nonparametric estimation of finite population quantiles under simple random sampling, estimation based on the Polya posterior is similar to estimation based on the Bayesian approach developed by Ericson (J. Roy. Statist. Soc. Ser. B 31 (1969) 195) in that the Polya posterior distribution is the limit of Ericson's posterior distributions as the weight placed on the prior distribution diminishes. Furthermore, Polya posterior quantile estimates can be shown to be admissible under certain conditions. We demonstrate the admissibility of the sample median as an estimate of the population median under such a set of conditions. As with Ericson's Bayesian approach, Polya posterior-based interval estimates for population quantiles are asymptotically equivalent to the interval estimates obtained from standard frequentist approaches. In addition, for small to moderate sized populations, Polya posterior-based interval estimates for quantiles of a continuous characteristic of interest tend to agree with the standard frequentist interval estimates.     

Estimating ordered quantiles of two exponential populations with a common minimum guarantee time

The problem of estimating ordered quantiles of two exponential populations is considered, assuming equality of location parameters (minimum guarantee times), using the quadratic loss function. Under order restrictions, we propose new estimators which are the isotonized version of the MLEs, call it, restricted MLE. A sufficient condition for improving equivariant estimators is derived under order restrictions on the quantiles. Consequently, estimators improving upon the old estimators have been derived. A detailed numerical study has been done to evaluate the performance of proposed estimators using the Monte-Carlo simulation method and recommendations have been made for the use of the estimators.     

Approximations for a fixed sample size selection procedure for negative binomial populations

A large sample approximation of the least favorable configuration for a fixed sample size selection procedure for negative binomial populations is proposed. A normal approximation of the selection procedure is also presented. Optimal sample sizes required to be drawn from each population and the bounds for the sample sizes are tabulated. Sample sizes obtained using the approximate least favorable configuration are compared with those obtained using the exact least favorable configuration. Alternate form of the normal approximation to the probability of correct selection is also presented. The relation between the required sample size and the number of populations involved is studied.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.     

Quasi-nonparametric upper tolerance regions based on the bootstrap

In this paper the upper tolerance problem for random samples will be investigated. Here we will be concerned with populations that are skewed to the right and possess a finite minimum (e.g. the Exponential distribution). The method developed here takes the form of a multiplicative factor times a quantile estimate. The multiplicative factor is simple in form and is based on bootstrapped quantiles of order statistics, though no resampling is required. The quantile estimate itself could be of any desired form; for example, it could be nonparametric, and, therefore based on order statistics as well. The proposed tolerance limit has the desirable property of allowing for the possibility of exceeding the sample maximum, and therefore capturing more probability content, while avoiding, in general, use of the extreme order statistics.