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.     

Global comparisons of medians and other quantiles in a one-way design when there are tied values

For J ? 2 independent groups, the article deals with testing the global hypothesis that all J groups have a common population median or identical quantiles, with an emphasis on the quartiles. Classic rank-based methods are sometimes suggested for comparing medians, but it is well known that under general conditions they do not adequately address this goal. Extant methods based on the usual sample median are unsatisfactory when there are tied values except for the special case J = 2. A variation of the percentile bootstrap used in conjunction with the Harrell–Davis quantile estimator performs well in simulations. The method is illustrated with data from the Well Elderly 2 study.     

Modeling of maximum precipitation using maximal generalized extreme value distribution

Distribution of maximum or minimum values (extreme values) of a dataset is especially used in natural phenomena including sea waves, flow discharge, wind speeds, and precipitation and it is also used in many other applied sciences such as reliability studies and analysis of environmental extreme events. So if we can explain the extremal behavior via statistical formulas, we can estimate how their behavior would be in the future. In this paper, we study extreme values of maximum precipitation in Zahedan using maximal generalized extreme value distribution, which all maxima of a data set are modeled using it. Also, we apply four methods to estimate distribution parameters including maximum likelihood estimation, probability weighted moments, elemental percentile and quantile least squares then compare estimates by average scaled absolute error criterion and obtain quantiles estimates and confidence intervals. In addition, goodness-of-fit tests are described. As a part of result, the return period of maximum precipitation is computed.     

Optimal sign test for quantiles in ranked set samples

This paper considers the one-sample sign test for population quantiles in general ranked set sampling, and proposes a weighted sign test because observations with different ranks are not identically distributed. It is shown analytically that optimal weight always improves the Pitman efficiency for all distributions. For each quantile, the sampling allocation that maximizes the sign test efficacy is identified and shown to not depend on the population distribution. Moreover, distribution-free confidence intervals for quantiles based on ordered values of optimal ranked set samples are discussed.     

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.     

Bayesian non-crossing quantile regression for regularly varying distributions

Quantile regression is a very important statistical tool for predictive modelling and risk assessment. For many applications, conditional quantile at different levels are estimated separately. Consequently the monotonicity of conditional quantiles can be violated when quantile regression curves cross each other. In this paper, we propose a new Bayesian multiple quantile regression based on heavy tailed distribution for non-crossing. We consider a linear quantile regression model for simultaneous Bayesian estimation of multiple quantiles based on a regularly varying assumptions. The numerical and competitive performance of the proposed method is illustrated by simulation.     

Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.     

Comparison of bootstrap and generalized bootstrap methods for estimating high quantiles

The generalized bootstrap is a parametric bootstrap method in which the underlying distribution function is estimated by fitting a generalized lambda distribution to the observed data. In this study, the generalized bootstrap is compared with the traditional parametric and non-parametric bootstrap methods in estimating the quantiles at different levels, especially for high quantiles. The performances of the three methods are evaluated in terms of cover rate, average interval width and standard deviation of width of the 95% bootstrap confidence intervals. Simulation results showed that the generalized bootstrap has overall better performance than the non-parametric bootstrap in high quantile estimation.     

Semi-parametric expected shortfall forecasting in financial markets

Bayesian methods have proved effective for quantile estimation, including for financial Value-at-Risk forecasting. Expected shortfall (ES) is a competing tail risk measure, favoured by the Basel Committee, that can be semi-parametrically estimated via asymmetric least squares. An asymmetric Gaussian density is proposed, allowing a likelihood to be developed, that facilitates both pseudo-maximum likelihood and Bayesian semi-parametric estimation, and leads to forecasts of quantiles, expectiles and ES. Further, the conditional autoregressive expectile class of model is generalised to two fully nonlinear families. Adaptive Markov chain Monte Carlo sampling schemes are developed for the Bayesian estimation. The proposed models are favoured in an empirical study forecasting eight financial return series: evidence of more accurate ES forecasting, compared to a range of competing methods, is found, while Bayesian estimated models tend to be more accurate. However, during a financial crisis period most models perform badly, while two existing models perform best.     

Fishery and Forestry Management

There are a large number of different definitions used for sample quantiles in statistical computer packages. Often within the same package one definition will be used to compute a quantile explicitly, while other definitions may be used when producing a boxplot, a probability plot, or a QQ plot. We compare the most commonly implemented sample quantile definitions by writing them in a common notation and investigating their motivation and some of their properties. We argue that there is a need to adopt a standard definition for sample quantiles so that the same answers are produced by different packages and within each package. We conclude by recommending that the median-unbiased estimator be used because it has most of the desirable properties of a quantile estimator and can be defined independently of the underlying distribution.     

Parametric inference for quantile event times with adjustment for covariates on competing risks data

ABSTRACT

We propose parametric inferences for quantile event times with adjustment for covariates on competing risks data. We develop parametric quantile inferences using parametric regression modeling of the cumulative incidence function from the cause-specific hazard and direct approaches. Maximum likelihood inferences are developed for estimation of the cumulative incidence function and quantiles. We develop the construction of parametric confidence intervals for quantiles. Simulation studies show that the proposed methods perform well. We illustrate the methods using early stage breast cancer data.     

Combining least-squares and quantile regressions

Least-squares and quantile regressions are method of moments techniques that are typically used in isolation. A leading example where efficiency may be gained by combining least-squares and quantile regressions is one where some information on the error quantiles is available but the error distribution cannot be fully specified. This estimation problem may be cast in terms of solving an over-determined estimating equation (EE) system for which the generalized method of moments (GMM) and empirical likelihood (EL) are approaches of recognized importance. The major difficulty with implementing these techniques here is that the EEs associated with the quantiles are non-differentiable. In this paper, we develop a kernel-based smoothing technique for non-smooth EEs, and derive the asymptotic properties of the GMM and maximum smoothed EL (MSEL) estimators based on the smoothed EEs. Via a simulation study, we investigate the finite sample properties of the GMM and MSEL estimators that combine least-squares and quantile moment relationships. Applications to real datasets are also considered.     

Do maternal health problems influence child's worrying status? Evidence from the British Cohort Study

Conventional methods apply symmetric prior distributions such as a normal distribution or a Laplace distribution for regression coefficients, which may be suitable for median regression and exhibit no robustness to outliers. This work develops a quantile regression on linear panel data model without heterogeneity from a Bayesian point of view, i.e. upon a location-scale mixture representation of the asymmetric Laplace error distribution, and provides how the posterior distribution is summarized using Markov chain Monte Carlo methods. Applying this approach to the 1970 British Cohort Study (BCS) data, it finds that a different maternal health problem has different influence on child's worrying status at different quantiles. In addition, applying stochastic search variable selection for maternal health problems to the 1970 BCS data, it finds that maternal nervous breakdown, among the 25 maternal health problems, contributes most to influence the child's worrying status.     

A revised simplex algorithm for the absolute deviation curve fitting problem

The statistical properties of quasi-ranges in small samples from a gamma density are the objects of study in this paper.The methods of computing the "Coefficient MatrixnA(j,k), which plays a major role for computing moments and quantiles from such a density using high speed digital computers, are presented. Limited tables of central and non-central moments as well as tables of quantile values, are given. More extensive tables can be easily constructed by using the methods described here when the need arises. Also, a table of values for kurtosis and skewness is presented.     

Rank statistics expressible as integrals under P–P-plots and receiver operating characteristic curves

Given a probability measure on the unit square, the measure of the region under an empirical P – P -plot defines a two-sample rank statistic. Instances include trimmed and censored versions of the Mann–Whitney–Wilcoxon statistic and a class of statistics with applications in the analysis of receiver operating characteristic (ROC) curves. A large sample distribution for such a statistic is obtained, which is valid under sampling from general populations. Explicit results are presented for comparing arbitrary quantile segments of two populations. The results are not restricted to continuous data and incorporate adjustments for tied values in the discrete case. A multivariate version of the large sample distribution extends the class of tractable statistics in ROC analysis and facilitates the use of methods based on partial areas when the data are discrete.     

MULTI-STAGE KERNEL-BASED CONDITIONAL QUANTILE PREDICTION IN TIME SERIES

We present a multi-stage conditional quantile predictor for time series of Markovian structure. It is proved that at any quantile level, p ∈ (0, 1), the asymptotic mean squared error (MSE) of the new predictor is smaller than the single-stage conditional quantile predictor. A simulation study confirms this result in a small sample situation. Because the improvement by the proposed predictor increases for quantiles at the tails of the conditional distribution function, the multi-stage predictor can be used to compute better predictive intervals with smaller variability. Applying this predictor to the changes in the U.S. short-term interest rate, rather smooth out-of-sample predictive intervals are obtained.     

Normal quantile estimation based on minimizing the error in predicted distribution functions

Estimators of the form [Xbar] + kS for estimating the p quantile of a normal distribution are studied when k is chosen to either minimize the mean square error in the predicted distribution function or to make the predicted distribution function unbiased for p. Here, [Xbar] and S are the usual sample mean and standard deviation, respectively, and the predicted distribution function is the true (normal) distribution function evaluated at the estimated quantile.

These k values are presented for various sample sizes and values of p, and application to warranty determination is discussed.     

Estimation of the quantile function using Bernstein–Durrmeyer polynomials

This paper studies quantile estimation using Bernstein–Durrmeyer polynomials in terms of its mean squared error and integrated mean squared error including rates of convergence as well as its asymptotic distribution. Whereas the rates of convergence are achieved for i.i.d. samples, we also show that the consistency more or less directly follows from the consistency of the sample quantiles, such that our proposal can also be applied for risk measurement in finance and insurance. Furthermore, an improved estimator based on an error-correction approach is proposed for which a general consistency result is established. A crucial issue is how to select the degree of Bernstein–Durrmeyer polynomials. We propose a novel data-adaptive approach that controls the number of modes of the corresponding density estimator. Its consistency including an uniform error bound as well as its limiting distribution in the sense of a general invariance principle are established. The finite sample properties are investigated by a Monte Carlo study. Finally, the results are illustrated by an application to photovoltaic energy research.     

Adaptive composite quantile regressions and their asymptotic relative efficiency

The composite quantile regression (CQR for short) provides an efficient and robust estimation for regression coefficients. In this paper we introduce two adaptive CQR methods. We make two contributions to the quantile regression literature. The first is that, both adaptive estimators treat the quantile levels as realizations of a random variable. This is quite different from the classic CQR in which the quantile levels are typically equally spaced, or generally, are treated as fixed values. Because the asymptotic variances of the adaptive estimators depend upon the generic distribution of the quantile levels, it has the potential to enhance estimation efficiency of the classic CQR. We compare the asymptotic variance of the estimator obtained by the CQR with that obtained by quantile regressions at each single quantile level. The second contribution is that, in terms of relative efficiency, the two adaptive estimators can be asymptotically equivalent to the CQR method as long as we choose the generic distribution of the quantile levels properly. This observation is useful in that it allows to perform parallel distributed computing when the computational complexity issue arises for the CQR method. We compare the relative efficiency of the adaptive methods with respect to some existing approaches through comprehensive simulations and an application to a real-world problem.     

分位数回归技术综述

普通最小二乘回归建立了在自变量X=x下因变量Y的条件均值与X的关系的线性模型。而分位数回归（Quantile Regression）则利用自变量X和因变量y的条件分位数进行建模。与普通的均值回归相比，它能充分反映自变量X对于因变量y的分布的位置、刻度和形状的影响，有着十分广泛的应用，尤其是对于一些非常关注尾部特征的情况。文章介绍了分位数回归的概念以及分位数回归的估计、检验和拟合优度，回顾了分位数回归的发展过程以及其在一些经济研究领域中的应用，最后做了总结。