A composite quantile function estimator with applications in bootstrapping

In this note we define a composite quantile function estimator in order to improve the accuracy of the classical bootstrap procedure in small sample setting. The composite quantile function estimator employs a parametric model for modelling the tails of the distribution and uses the simple linear interpolation quantile function estimator to estimate quantiles lying between 1/(n+1) and n/(n+1). The method is easily programmed using standard software packages and has general applicability. It is shown that the composite quantile function estimator improves the bootstrap percentile interval coverage for a variety of statistics and is robust to misspecification of the parametric component. Moreover, it is also shown that the composite quantile function based approach surprisingly outperforms the parametric bootstrap for a variety of small sample situations.     

Nonparametric estimation of quantile functions for randomly right censored data

In this paper we compare four nonparametric quantile function estimators for randomly right censored data: the Kaplan–Meier estimator, the linearly interpolated Kaplan–Meier estimator, the kernel-type survival function estimator, and the Bézier curve smoothing estimator. Also, we compare several kinds of confidence intervals of quantiles for four nonparametric quantile function estimators.     

Estimation of monotone bivariate quantile residual life

The bivariate quantile residual life function can play an important role in statistical reliability and survival analysis. In many situations assuming a decreasing form for it is recommended. Here, we propose a new non-parametric estimator of this measure under such restriction. It has been shown that the new estimator is consistent and, with proper normalization, weakly converges to a bivariate Gaussian process. A simulation study shows that the proposed estimator is an alternative to the unrestricted estimator when the bivariate quantile residual life is decreasing. Finally, the new estimators are applied to two real data sets.     

An Investigation of Quantile Function Estimators Relative to Quantile Confidence Interval Coverage

In this article, we investigate the limitations of traditional quantile function estimators and introduce a new class of quantile function estimators, namely, the semi-parametric tail-extrapolated quantile estimators, which has excellent performance for estimating the extreme tails with finite sample sizes. The smoothed bootstrap and direct density estimation via the characteristic function methods are developed for the estimation of confidence intervals. Through a comprehensive simulation study to compare the confidence interval estimations of various quantile estimators, we discuss the preferred quantile estimator in conjunction with the confidence interval estimation method to use under different circumstances. Data examples are given to illustrate the superiority of the semi-parametric tail-extrapolated quantile estimators. The new class of quantile estimators is obtained by slight modification of traditional quantile estimators, and therefore, should be specifically appealing to researchers in estimating the extreme tails.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

Quantile Estimation Using Ranked Set Samples from a Population with Known Mean

Ranked set sampling (RSS) is a cost-efficient technique for data collection when the units in a population can be easily judgment ranked by any cheap method other than actual measurements. Using auxiliary information in developing statistical procedures for inference about different population characteristics is a well-known approach. In this work, we deal with quantile estimation from a population with known mean when data are obtained according to RSS scheme. Through the simple device of mean-correction (subtract off the sample mean and add on the known population mean), a modified estimator is constructed from the standard quantile estimator. Asymptotic normality of the new estimator and its asymptotic efficiency relative to the original estimator are derived. Simulation results for several underlying distributions show that the proposed estimator is more efficient than the traditional one.     

A weighted Harrell–Davis distance test with applications to censored data

Consider the standard treatment-control model with a time-to-event endpoint. We propose a novel interpretable test statistic from a quantile function point of view. The large sample consistency of our estimator is proven for fixed bandwidth values theoretically and validated empirically. A Monte Carlo simulation study also shows that given small sample sizes, utilization of a tuning parameter through the application of a smooth quantile function estimator shows an improvement in efficiency in terms of the MSE when compared to direct application of classic Kaplan–Meier survival function estimator. The procedure is finally illustrated via an application to epithelial ovarian cancer data.     

Jackknifed kernel quantile estimators

A new quantile estimator is obtained by jackknifing the kernel quantile estimator. The asymptotic relative deficiency of the kernel quantile estimator relative to the jackknifed quantile estimator is investigated.     

Maintaining the exchangeability assumption for a two-group permutation test in the non-randomized setting

In this note, we develop a new two-group bootstrap-permutation test that utilizes the tail-extrapolated quantile function estimator for the bootstrap component. This test is an extension of the standard two-group permutation test, that through its construction is defined to meet the exchangeability assumption, and thus it guarantees that the type I error is appropriately bounded by definition. This methodology is particularly useful in the non-randomized two-group setting for which the exchangeability assumption for the traditional two-group permutation test is untestable. We develop some theoretical results for the new test, followed by a simulation study and an example.     

Jackknife method for intermediate quantiles

Quantile function plays an important role in statistical inference, and intermediate quantile is useful in risk management. It is known that Jackknife method fails for estimating the variance of a sample quantile. By assuming that the underlying distribution satisfies some extreme value conditions, we show that Jackknife variance estimator is inconsistent for an intermediate order statistic. Further we derive the asymptotic limit of the Jackknife-Studentized intermediate order statistic so that a confidence interval for an intermediate quantile can be obtained. A simulation study is conducted to compare this new confidence interval with other existing ones in terms of coverage accuracy.     

Bootstrapping quantiles in a fixed design regression model with censored data

We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation.     

Asymptotic normality for a non parametric estimator of conditional quantile with left-truncated data

In this paper, we construct a non parametric estimator of conditional distribution function by the double-kernel local linear approach for left-truncated data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators is also established. Finite-sample performance of the estimator is investigated via simulation.     

Composite Estimation for Single‐Index Models with Responses Subject to Detection Limits

We propose a semiparametric estimator for single‐index models with censored responses due to detection limits. In the presence of left censoring, the mean function cannot be identified without any parametric distributional assumptions, but the quantile function is still identifiable at upper quantile levels. To avoid parametric distributional assumption, we propose to fit censored quantile regression and combine information across quantile levels to estimate the unknown smooth link function and the index parameter. Under some regularity conditions, we show that the estimated link function achieves the non‐parametric optimal convergence rate, and the estimated index parameter is asymptotically normal. The simulation study shows that the proposed estimator is competitive with the omniscient least squares estimator based on the latent uncensored responses for data with normal errors but much more efficient for heavy‐tailed data under light and moderate censoring. The practical value of the proposed method is demonstrated through the analysis of a human immunodeficiency virus antibody data set.     

Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

On the mean squared error of nonparametric quantile estimators under random right-censorship

For randomly right-censored data, new asymptotic expressions for the mean squared errors of the product-limit quantile estimator and a kernel-type quantile estimator are presented in this paper. From these results a comparison of the two quantile estimators with respect to their mean squared errors is given.     

Consistent model identification of varying coefficient quantile regression with BIC tuning parameter selection

Quantile regression provides a flexible platform for evaluating covariate effects on different segments of the conditional distribution of response. As the effects of covariates may change with quantile level, contemporaneously examining a spectrum of quantiles is expected to have a better capacity to identify variables with either partial or full effects on the response distribution, as compared to focusing on a single quantile. Under this motivation, we study a general adaptively weighted LASSO penalization strategy in the quantile regression setting, where a continuum of quantile index is considered and coefficients are allowed to vary with quantile index. We establish the oracle properties of the resulting estimator of coefficient function. Furthermore, we formally investigate a Bayesian information criterion (BIC)-type uniform tuning parameter selector and show that it can ensure consistent model selection. Our numerical studies confirm the theoretical findings and illustrate an application of the new variable selection procedure.     

Non Parametric Regression Quantile Estimation for Dependent Functional Data under Random Censorship: Asymptotic Normality

In this paper we study a smooth estimator of the regression quantile function in the censorship model when the covariates take values in some abstract function space. The main goal of this paper is to establish the asymptotic normality of the kernel estimator of the regression quantile, under α-mixing condition and, on the concentration properties on small balls probability measure of the functional regressors. Some applications and particular cases are studied. This study can be applied in time series analysis to the prediction and building confidence bands. Some simulations are drawn to lend further support to our theoretical results and to compare the quality of behavior of the estimator for finite samples with different rates of censoring and sizes.     

Quantile estimation in the proportional hazards model of random censorship

The asymptotic theory is given for quantile estimation in the proportional hazards model of random censorship. In this model, the tail of the censoring distribution function is some power of the tail of the survival distribution function. The quantile estimator is based on the maximum likelihood estimator for the survival time distribution, due to Abdushukurov, Cheng and Lin.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.