Simultaneous estimation for non-crossing multiple quantile regression with right censored data

In this paper, we consider the estimation problem of multiple conditional quantile functions with right censored survival data. To account for censoring in estimating a quantile function, weighted quantile regression (WQR) has been developed by using inverse-censoring-probability weights. However, the estimated quantile functions from the WQR often cross each other and consequently violate the basic properties of quantiles. To avoid quantile crossing, we propose non-crossing weighted multiple quantile regression (NWQR), which estimates multiple conditional quantile functions simultaneously. We further propose the adaptive sup-norm regularized NWQR (ANWQR) to perform simultaneous estimation and variable selection. The large sample properties of the NWQR and ANWQR estimators are established under certain regularity conditions. The proposed methods are evaluated through simulation studies and analysis of a real data set.     

Improved distribution quantile estimation

A technique for estimating the quantiles or percentiles of a distribution is developed. The parametric form of the distribution is assumed unknown. The estimation procedure is based on a kernel estimator of a probability density function and on aquantile estimator suggested by Harrell and Davis (1982). Simulation studies show that estimation of quantiles in moderately heavyto heavy tails of a distribution is substantially improved by use of the technique.     

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.     

Random weighting-based quantile estimation via importance resampling

Abstract

This paper presents a new method to estimate the quantiles of generic statistics by combining the concept of random weighting with importance resampling. This method converts the problem of quantile estimation to a dual problem of tail probabilities estimation. Random weighting theories are established to calculate the optimal resampling weights for estimation of tail probabilities via sequential variance minimization. Subsequently, the quantile estimation is constructed by using the obtained optimal resampling weights. Experimental results on real and simulated data sets demonstrate that the proposed random weighting method can effectively estimate the quantiles of generic statistics.     

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.     

Single-pass low-storage arbitrary quantile estimation for massive datasets

We present a single-pass, low-storage, sequential method for estimating an arbitrary quantile of an unknown distribution. The proposed method performs very well when compared to existing methods for estimating the median as well as arbitrary quantiles for a wide range of densities. In addition to explaining the method and presenting the results of the simulation study, we discuss intuition behind the method and demonstrate empirically, for certain densities, that the proposed estimator converges to the sample quantile.     

Improving extreme quantile estimation via a folding procedure

In many applications (geosciences, insurance, etc.), the peaks-over-thresholds (POT) approach is one of the most widely used methodology for extreme quantile inference. It mainly consists of approximating the distribution of exceedances above a high threshold by a generalized Pareto distribution (GPD). The number of exceedances which is used in the POT inference is often quite small and this leads typically to a high volatility of the estimates. Inspired by perfect sampling techniques used in simulation studies, we define a folding procedure that connects the lower and upper parts of a distribution. A new extreme quantile estimator motivated by this theoretical folding scheme is proposed and studied. Although the asymptotic behaviour of our new estimate is the same as the classical (non-folded) one, our folding procedure reduces significantly the mean squared error of the extreme quantile estimates for small and moderate samples. This is illustrated in the simulation study. We also apply our method to an insurance dataset.     

Kernel type smoothed quantile estimation under long memory

This paper studies nonparametric kernel type (smoothed) estimation of quantiles for long memory stationary sequences. The uniform strong consistency and asymptotic normality of the estimates with rates are established. Finite sample behaviors are investigated in a small Monte Carlo simulation study.     

Extreme quantile estimation based on financial time series

Estimation of market risk is an important problem in finance. Two well-known risk measures, viz., value at risk and median shortfall, turn out to be extreme quantiles of the marginal distribution of asset return. Time series on asset returns are known to exhibit certain stylized facts, such as heavy tails, skewness, volatility clustering, etc. Therefore, estimation of extreme quantiles in the presence of such features in the data seems to be of natural interest. It is difficult to capture most of these stylized facts using one specific time series model. This motivates nonparametric and extreme value theory-based estimation of extreme quantiles that do not require exact specification of the asset return model. We review these quantile estimators and compare their known properties. Their finite sample performance are compared using Monte Carlo simulation. We propose a new estimator that exhibits encouraging finite sample performance while estimating extreme quantile in the right tail region.     

A finite population quantile estimation by unequal probability sampling

Abstract

Using a model-assisted approach, this paper studies asymptotically design-unbiased (ADU) estimation of a population “distribution function” and extends to deriving an asymptotic and approximate unbiased estimator for a population quantile from a sample chosen with varying probabilities. The respective asymptotic standard errors and confidence intervals are then worked out. Numerical findings based on an actual data support the theory with efficient results.     

Randomized quantile regression estimation for heteroskedastic non parametric model

In this paper, we propose robust randomized quantile regression estimators for the mean and (condition) variance functions of the popular heteroskedastic non parametric regression model. Unlike classical approaches which consider quantile as a fixed quantity, our method treats quantile as a uniformly distributed random variable. Our proposed method can be employed to estimate the error distribution, which could significantly improve prediction results. An automatic bandwidth selection scheme will be discussed. Asymptotic properties and relative efficiencies of the proposed estimators are investigated. Our empirical results show that the proposed estimators work well even for random errors with infinite variances. Various numerical simulations and two real data examples are used to demonstrate our methodologies.     

Simultaneous estimation of gamma means using a hierarchical generalized linear model

The problem of simultaneously estimating p Gamma means is investigated when the means are believed a priori to satisfy an r-dimensional generalized linear model. Using a Bayesian hierarchical model to reflect the uncertainty in the linear model, approximate methods are proposed to compute the posterior densities. The resulting estimator shrinks the usual estimator toward a prior estimator where the size of the shrinkage depends upon the agreement of the observed data with the proposed generalized linear model.     

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.     

Nonparametric estimation of mean residual quantile function under right censoring

In this paper, we develop non-parametric estimation of the mean residual quantile function based on right-censored data. Two non-parametric estimators, one based on the empirical quantile function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the estimators are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated with the aid of two real data sets.     

Shrinkage estimation of varying covariate effects based on quantile regression

Varying covariate effects often manifest meaningful heterogeneity in covariate-response associations. In this paper, we adopt a quantile regression model that assumes linearity at a continuous range of quantile levels as a tool to explore such data dynamics. The consideration of potential non-constancy of covariate effects necessitates a new perspective for variable selection, which, under the assumed quantile regression model, is to retain variables that have effects on all quantiles of interest as well as those that influence only part of quantiles considered. Current work on l ₁-penalized quantile regression either does not concern varying covariate effects or may not produce consistent variable selection in the presence of covariates with partial effects, a practical scenario of interest. In this work, we propose a shrinkage approach by adopting a novel uniform adaptive LASSO penalty. The new approach enjoys easy implementation without requiring smoothing. Moreover, it can consistently identify the true model (uniformly across quantiles) and achieve the oracle estimation efficiency. We further extend the proposed shrinkage method to the case where responses are subject to random right censoring. Numerical studies confirm the theoretical results and support the utility of our proposals.     

Bayesian variable selection and estimation in maximum entropy quantile regression

Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis.     

Simultaneous estimation of ordered parameters

The problem of estimating ordered parameters is encountered in biological, agricultural, reliability and various other experiments. Consider two populations with densities f₁(x₁-ω₁) and f₂(x₂-ω₂) where ω₁#ω₂. The estimation of ω₁,ω₂) with the loss function, the sum of squared errors, is studied. when f_i is the f_i(,_i,,_i ²) density with ,_i known, i=1,2; we obtain a class of minimax estimators. When ω₁ #ω₂ we show some of these estimators are improved by the maximum likelihood estimator. For a general f_i we give sufficient conditions for the minimaxity of the analogue of the Pitman estimator.