A fractional order statistic towards defining a smooth quantile function for discrete data

This work is motivated in part by a recent publication by Ma et al. (2011) who resolved the asymptotic non-normality problem of the classical sample quantiles for discrete data through defining a new mid-distribution based quantile function. This work is the motivation for defining a new and improved smooth population quantile function given discrete data. Our definition is based on the theory of fractional order statistics. The main advantage of our definition as compared to its competitors is the capability to distinguish the uth quantile across different discrete distributions over the whole interval, u∈(0,1). In addition, we define the corresponding estimator of the smooth population quantiles and demonstrate the convergence and asymptotic normal distribution of the corresponding sample quantiles. We verify our theoretical results through a Monte Carlo simulation, and illustrate the utilization of our quantile function in a Q-Q plot for discrete data.     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

Bayesian composite Tobit quantile regression

Composite quantile regression models have been shown to be effective techniques in improving the prediction accuracy [H. Zou and M. Yuan, Composite quantile regression and the oracle model selection theory, Ann. Statist. 36 (2008), pp. 1108–1126; J. Bradic, J. Fan, and W. Wang, Penalized composite quasi-likelihood for ultrahighdimensional variable selection, J. R. Stat. Soc. Ser. B 73 (2011), pp. 325–349; Z. Zhao and Z. Xiao, Efficient regressions via optimally combining quantile information, Econometric Theory 30(06) (2014), pp. 1272–1314]. This paper studies composite Tobit quantile regression (TQReg) from a Bayesian perspective. A simple and efficient MCMC-based computation method is derived for posterior inference using a mixture of an exponential and a scaled normal distribution of the skewed Laplace distribution. The approach is illustrated via simulation studies and a real data set. Results show that combine information across different quantiles can provide a useful method in efficient statistical estimation. This is the first work to discuss composite TQReg from a Bayesian perspective.     

Composite quantile regression for varying-coefficient single-index models

ABSTRACT

The varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example.     

Nonparametric confidence intervals for quantiles and quantile intervals from inversely sampled record breaking data in a multi-sample plan

There are a number of situations in which an observation is retained only if it is a record value, which include studies in industrial quality control experiments, destructive stress testing, meteorology, hydrology, seismology, athletic events and mining. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. In this paper, we study the problems of constructing the nonparametric confidence intervals for quantiles and quantile intervals of the parent distribution based on record data. For a single record-breaking sample, the confidence coefficients of the confidence intervals for the pth quantile cannot exceed p and 1?p, on the basis of upper and lower records, respectively; hence, replication is required. So, we develop the procedure based on k independent record-breaking samples. Various cases have been studied and in each case, the optimal k and the exact nonparametric confidence intervals are obtained, and exact expressions for the confidence coefficients of these confidence intervals are derived. Finally, the results are illustrated by numerical computations.     

Goodness-of-fit tests for location–scale families based on random distance

For location–scale families, we consider a random distance between the sample order statistics and the quasi sample order statistics derived from the null distribution as a measure of discrepancy. The conditional qth quantile and expectation of the random discrepancy on the given sample are chosen as test statistics. Simulation results of powers against various alternatives are illustrated under the normal and exponential hypotheses for moderate sample size. The proposed tests, especially the qth quantile tests with a small or large q, are shown to be more powerful than other prominent goodness-of-fit tests in most cases.     

Competing risks quantile regression at work: in-depth exploration of the role of public child support for the duration of maternity leave

Despite its emergence as a frequently used method for the empirical analysis of multivariate data, quantile regression is yet to become a mainstream tool for the analysis of duration data. We present a pioneering empirical study on the grounds of a competing risks quantile regression model. We use large-scale maternity duration data with multiple competing risks derived from German linked social security records to analyse how public policies are related to the length of economic inactivity of young mothers after giving birth. Our results show that the model delivers detailed insights into the distribution of transitions out of maternity leave. It is found that cumulative incidences implied by the quantile regression model differ from those implied by a proportional hazards model. To foster the use of the model, we make an R-package (cmprskQR) available.     

Nonparametric Testing of an Exclusion Restriction in Quantile Regression

Using the framework proposed by Bickel et al. (2006 Bickel , P. J. , Ritov , Y. , Stoker , T. ( 2006 ). Tailor-made tests for goodness-of-fit to semiparametric hypotheses . Ann. Stat. 34 ( 2 ): 721 – 741 . [Google Scholar]), we provide a score-based testing method to check the exclusion restriction in quantile regression, i.e., H: ν_α(Y|U, V) = ν_α(Y|U) w.p.1, where ν_α denotes the αth (0 < α < 1) quantile. A subsampling method is suggested to acquire the critical values and justified. The tests are all found to be consistent against fixed alternatives and have discriminating power against local alternatives at root-n scale. We address this particular problem as a representative among a wide family of semiparametric model checking problems. The methodology can be carried over to other goodness-of-fit testing of semiparametric models, possibly involve non smooth functions.     

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.     

Bayesian Elastic Net Tobit Quantile Regression

In this article, the problem of parameter estimation and variable selection in the Tobit quantile regression model is considered. A Tobit quantile regression with the elastic net penalty from a Bayesian perspective is proposed. Independent gamma priors are put on the l₁ norm penalty parameters. A novel aspect of the Bayesian elastic net Tobit quantile regression is to treat the hyperparameters of the gamma priors as unknowns and let the data estimate them along with other parameters. A Bayesian Tobit quantile regression with the adaptive elastic net penalty is also proposed. The Gibbs sampling computational technique is adapted to simulate the parameters from the posterior distributions. The proposed methods are demonstrated by both simulated and real data examples.     

Limiting distributions of MLE and UMVUE in the biparametric uniform distribution

In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ₁, θ₂) when sampling from a biparametric uniform distributionU(θ₁, θ₂). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution of UMVUE is a shift of the limiting distribution of MLE.     

Adaptive penalized quantile regression for high dimensional data

We propose a new adaptive L₁ penalized quantile regression estimator for high-dimensional sparse regression models with heterogeneous error sequences. We show that under weaker conditions compared with alternative procedures, the adaptive L₁ quantile regression selects the true underlying model with probability converging to one, and the unique estimates of nonzero coefficients it provides have the same asymptotic normal distribution as the quantile estimator which uses only the covariates with non-zero impact on the response. Thus, the adaptive L₁ quantile regression enjoys oracle properties. We propose a completely data driven choice of the penalty level _λn${\lambda }_n$, which ensures good performance of the adaptive L₁ quantile regression. Extensive Monte Carlo simulation studies have been conducted to demonstrate the finite sample performance of the proposed method.     

Local linear double and asymmetric kernel estimation of conditional quantiles

Abstract

In this work, we propose and investigate a family of non parametric quantile regression estimates. The proposed estimates combine local linear fitting and double kernel approaches. More precisely, we use a Beta kernel when covariate’s support is compact and Gamma kernel for left-bounded supports. Finite sample properties together with asymptotic behavior of the proposed estimators are presented. It is also shown that these estimates enjoy the property of having finite variance and resistance to sparse design.     

Quantile inference for heteroscedastic regression models

Consider the nonparametric heteroscedastic regression model Y=m(X)+σ(X)?, where m(·) is an unknown conditional mean function and σ(·) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ(X) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.     

Box–Cox power transformation unconditional quantile regressions with an application on wage inequality

This study proposes a semi-parametric estimation method, Box–Cox power transformation unconditional quantile regression, to estimate the impact of changes in the distribution of the explanatory variables on the unconditional quantile of the outcome variable. The proposed method consists of running a nonlinear regression of the recentered influence function (RIF) of the outcome variable on the explanatory variables. We also show the asymptotic properties of the proposed estimator and apply the estimation method to address an existing puzzle in labor economics–why the 50th/10th percentile wage gap has been falling in the USA since the late 1980s. Our results show that declining unionization can explain approximately 10% of the decline in the 50/10 wage gap in 1990–2000 and 23% in 2000–2010.     

Comparing Conditional Quantile Curves

Abstract. We consider the problem of testing the equality of J quantile curves from independent samples. A test statistic based on an L²‐distance between non‐crossing non‐parametric estimates of the quantile curves from the individual samples is proposed. Asymptotic normality of this statistic is established under the null hypothesis, local and fixed alternatives, and the finite sample properties of a bootstrap‐based version of this test statistic are investigated by means of a simulation study.     

NONPARAMETRIC QUANTILE ESTIMATION FROM RECORD-BREAKING DATA

Sometimes, in industrial quality control experiments and destructive stress testing, only values smaller than all previous ones are observed. Here we consider nonparametric quantile estimation, both the ‘sample quantile function’ and kernel-type estimators, from such record-breaking data. For a single record-breaking sample, consistent estimation is not possible except in the extreme tails of the distribution. Hence replication is required, and for m. such independent record-breaking samples the quantile estimators are shown to be strongly consistent and asymptotically normal as m-→∞. Also, for small m, the mean-squared errors, biases and smoothing parameters (for the smoothed estimators) are investigated through computer simulations.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

Distribution estimation with auxiliary information for missing data

There is much literature on statistical inference for distribution under missing data, but surprisingly very little previous attention has been paid to missing data in the context of estimating distribution with auxiliary information. In this article, the auxiliary information with missing data is proposed. We use Zhou, Wan and Wang's method (2008) to mitigate the effects of missing data through a reformulation of the estimating equations, imputed through a semi-parametric procedure. Whence we can estimate distribution and the τth quantile of the distribution by taking auxiliary information into account. Asymptotic properties of the distribution estimator and corresponding sample quantile are derived and analyzed. The distribution estimators based on our method are found to significantly outperform the corresponding estimators without auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

A class of distributions with the quadratic mean residual quantile function

Abstract

The present paper introduces a new family of distributions with quadratic mean residual quantile function. Various distributional properties as well as reliability characteristics are discussed. Some characterizations of the class of distributions are presented. The estimation of parameters of the model using method of L-moments is studied. The practical application of the class of models is illustrated with a real life data set.