Thus far, likelihood-based interval estimates for quantiles have not been studied in the literature on interval censored case 2 data and partly interval censored data, and, in this context, the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval censored data. The fourth order expansion of the weighted empirical log-likelihood ratio is derived and the theoretical coverage accuracy equation for the proposed WELRCI is established, which generally guarantees at least `first order' accuracy. In particular, for right censored data, we show that the coverage accuracy is at least $O(n^{-1/2})$ and our simulation studies show that in comparison with empirical likelihood-based methods, the smoothing used in WELRCI generally provides a shorter confidence interval with comparable coverage accuracy. For interval censored data, it is interesting to find that with an adjusted rate $n^{-1/3}$, the weighted empirical log-likelihood ratio has an asymptotic distribution completely different from that obtained by the empirical likelihood approach and the resulting WELRCI perform favorably in the available comparison simulation studies.
In this paper, we consider binary response models with linear quantile restrictions. Considerably generalizing previous research on this topic, our analysis focuses on an infinite collection of quantile estimators. We derive a uniform linearisation for the properly standardized empirical quantile process and discover some surprising differences with the setting of continuously observed responses. Moreover, we show that considering quantile processes provides an effective way of estimating binary choice probabilities without restrictive assumptions on the form of the link function, heteroskedas…
Censored quantile regression (CQR) has become a valuable tool to study the heterogeneous association between a possibly censored outcome and a set of covariates, yet computation and statistical inference for CQR have remained a challenge for large-scale data with many covariates. In this paper, we focus on a smoothed martingale-based sequential estimating equations approach, to which scalable gradient-based algorithms can be applied. Theoretically, we provide a unified analysis of the smoothed sequential estimator and its penalized counterpart in increasing dimensions. When the covariate dimen…
This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on …
Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. In this paper, we consider statistical inference for quantile regression with large-scale data in the "increasing dimension" regime. We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression. This method, which we refer to as {\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differ…
This paper presents two results concerning uniform confidence intervals for the tail index and the extreme quantile. First, we show that it is impossible to construct a length-optimal confidence interval satisfying the correct uniform coverage over a local non-parametric family of tail distributions. Second, in light of the impossibility result, we construct honest confidence intervals that are uniformly valid by incorporating the worst-case bias in the local non-parametric family. The proposed method is applied to simulated data and a real data set of National Vital Statistics from National C…
$\ell_1$-penalized quantile regression is widely used for analyzing high-dimensional data with heterogeneity. It is now recognized that the $\ell_1$-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized $M$-estimation with strongly convex loss functions have been well studied, the extant literature on quantile regression is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non…
Penalized quantile regression (QR) is widely used for studying the relationship between a response variable and a set of predictors under data heterogeneity in high-dimensional settings. Compared to penalized least squares, scalable algorithms for fitting penalized QR are lacking due to the non-differentiable piecewise linear loss function. To overcome the lack of smoothness, a recently proposed convolution-type smoothed method brings an interesting tradeoff between statistical accuracy and computational efficiency for both standard and penalized quantile regressions. In this paper, we propose…
The moment conditions or estimating equations for instrumental variables quantile regression involve the discontinuous indicator function. We instead use smoothed estimating equations (SEE), with bandwidth $h$. We show that the mean squared error (MSE) of the vector of the SEE is minimized for some $h>0$, leading to smaller asymptotic MSE of the estimating equations and associated parameter estimators. The same MSE-optimal $h$ also minimizes the higher-order type I error of a SEE-based $χ^2$ test and increases size-adjusted power in large samples. Computation of the SEE estimator also becomes …
We consider a regression modeling of the quantiles of residual life, remaining lifetime at a specific time. We propose a smoothed induced version of the existing non-smooth estimating equations approaches for estimating regression parameters. The proposed estimating equations are smooth in regression parameters, so solutions can be readily obtained via standard numerical algorithms. Moreover, the smoothness in the proposed estimating equations enables one to obtain a robust sandwich-type covariance estimator of regression estimators aided by an efficient resampling method. To handle data subje…
Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their most general form, provide a full characterization of the copulas associated with the pairs $(X_t,X_{t-k})$ in a process $(X_t)_{t\in\mathbb{Z}}$, and account for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes that their traditional counterparts cannot capture. Despite various proposals for estimation strategies, only quite incomplete asymptotic distributional results are availabl…
We propose to smooth the entire objective function, rather than only the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also all…
Quasi-Monte Carlo confidence intervals using quantiles of randomized nets
Johann Radon Institute for Computational and Applied Mathematics, ÖAW（约翰·拉多纳计算与应用数学研究所，奥地利科学院）；Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学基础与交叉研究学院）
Bootstrap inference is a powerful tool for obtaining robust inference for quantiles and difference-in-quantiles estimators. The computationally intensive nature of bootstrap inference has made it infeasible in large-scale experiments. In this paper, the theoretical properties of the Poisson bootstrap algorithm and quantile estimators are used to derive alternative resampling-free algorithms for Poisson bootstrap inference that reduce the computational complexity substantially without additional assumptions. These findings are connected to existing literature on analytical confidence intervals …
We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate $λ_0$ in the Cox model. We analyze their asymptotic behavior and show that they are asymptotically normal at rate $n^{m/(2m+1)}$, when~$λ_0$ is $m\geq 2$ times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.
In this paper, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the …
Jędrzejczak et al. (2018) constructed a confidence interval for a ratio of quantiles coming from the Dagum distribution, which is frequently applied as a theoretical model in numerous income distribution analyses. The proposed interval is symmetric with respect to the ratio of sample quantiles, which result may be unsatisfactory in many practical applications. The search for a confidence interval with a smaller length led to the derivation of the shortest interval with the ends being asymmetric relative to the ratio of sample quantiles. In the paper, the existence of the shortest confidence in…
Ratio of medians or other suitable quantiles of two distributions is widely used in medical research to compare treatment and control groups or in economics to compare various economic variables when repeated cross-sectional data are available. Inspired by the so-called growth incidence curves introduced in poverty research, we argue that the ratio of quantile functions is a more appropriate and informative tool to compare two distributions. We present an estimator for the ratio of quantile functions and develop corresponding simultaneous confidence bands, which allow to assess significance of…
In economic research inequality measures based on ratios of quantiles are frequently applied to the analysis of income distributions. In the paper, we construct a confidence interval for such measures under the Dagum distribution which has been widely assumed as a~model for income distributions in empirical analyses Its properties are investigated on the basis of computer simulations. The constructed confidence interval is applied to the analysis of inequality income in Poland in 2015.
We propose inferential tools for functional linear quantile regression where the conditional quantile of a scalar response is assumed to be a linear functional of a functional covariate. In contrast to conventional approaches, we employ kernel convolution to smooth the original loss function. The coefficient function is estimated under a reproducing kernel Hilbert space framework. A gradient descent algorithm is designed to minimize the smoothed loss function with a roughness penalty. With the aid of the Banach fixed-point theorem, we show the existence and uniqueness of our proposed estimator…
We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator (SMLE) and constructed using bootstrap methods. Other methods to construct confidence intervals, using the non-standard limit distribution of the (restricted) MLE, are compared to our approach via simulations and real data applications.
In this paper, we consider estimation of the conditional mode of an outcome variable given regressors. To this end, we propose and analyze a computationally scalable estimator derived from a linear quantile regression model and develop asymptotic distributional theory for the estimator. Specifically, we find that the pointwise limiting distribution is a scale transformation of Chernoff's distribution despite the presence of regressors. In addition, we consider analytical and subsampling-based confidence intervals for the proposed estimator. We also conduct Monte Carlo simulations to assess the…
Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. The key result of Efron (2014) is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron (2014) proposed a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard d…
In the literature, quite a few measures have been proposed for quantifying the deviation of a probability distribution from symmetry. The most popular of these skewness measures are based on the third centralized moment and on quantiles. However, there are major drawbacks in using these quantities. These include a strong emphasis on the distributional tails and a poor asymptotic behaviour for the (empirical) moment based measure as well as difficult statistical inference and strange behaviour for discrete distributions for quantile based measures. Therefore, in this paper, we introduce skewnes…
Population quantiles are important parameters in many applications. Enthusiasm for the development of effective statistical inference procedures for quantiles and their functions has been high for the past decade. In this article, we study inference methods for quantiles when multiple samples from linked populations are available. The research problems we consider have a wide range of applications. For example, to study the evolution of the economic status of a country, economists monitor changes in the quantiles of annual household incomes, based on multiple survey datasets collected annually…
Modern problems in statistics tend to include estimators of high computational complexity and with complicated distributions. Statistical inference on such estimators usually relies on asymptotic normality assumptions, however, such assumptions are often not applicable for available sample sizes, due to dependencies in the data and other causes. A common alternative is the use of re-sampling procedures, such as the bootstrap, but these may be computationally intensive to an extent that renders them impractical for modern problems. In this paper we develop a method for fast construction of test…
Bootstrap smoothed (bagged) estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. Efron, 2014, derived a widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. He also considered a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this standard deviation. Kabaila and Wijethunga, 2019, assessed the performance of this confidence interval in the scenario of two nested linear regression models, the full model and …
Interval estimation of quantiles has been treated by many in the literature. However, to the best of our knowledge there has been no consideration for interval estimation when the data are available in grouped format. Motivated by this, we introduce several methods to obtain confidence intervals for quantiles when only grouped data is available. Our preferred method for interval estimation is to approximate the underlying density using the Generalized Lambda Distribution (GLD) to both estimate the quantiles and variance of the quantile estimators. We compare the GLD method with some other meth…
This work proposes new inference methods for a regression coefficient of interest in a (heterogeneous) quantile regression model. We consider a high-dimensional model where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable approximation to the conditional quantile function. The proposed methods are (explicitly or implicitly) based on orthogonal score functions that protect against moderate model selection mistakes, which are often inevitable in the approximately sparse model considered in the present paper. We establish the unif…
In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap met…
