Empirical Quantile Function Research Articles

INFERENCE ON A DISTRIBUTION FROM NOISY DRAWS

We consider a situation where the distribution of a random variable is being estimated by the empirical distribution of noisy measurements of that variable. This is common practice in, for example, teacher value-added models and other fixed-effect models for panel data. We use an asymptotic embedding where the noise shrinks with the sample size to calculate the leading bias in the empirical distribution arising from the presence of noise. The leading bias in the empirical quantile function is equally obtained. These calculations are new in the literature, where only results on smooth functionals such as the mean and variance have been derived. We provide both analytical and jackknife corrections that recenter the limit distribution and yield confidence intervals with correct coverage in large samples. Our approach can be connected to corrections for selection bias and shrinkage estimation and is to be contrasted with deconvolution. Simulation results confirm the much-improved sampling behavior of the corrected estimators. An empirical illustration on heterogeneity in deviations from the law of one price is equally provided.

Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing

In this paper, we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al. (Ann. Statist. 45 (2017) 223–256) and Hallin et al. (Ann. Statist. 49 (2021) 1139–1165). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. We derive the uniform consistency of these empirical estimates to their population versions, under certain assumptions. In fact, we prove a Glivenko–Cantelli type theorem that shows the asymptotic stability of the empirical rank map in any direction. Under mild structural assumptions, we provide global and local rates of convergence of the empirical quantile and rank maps. We also provide a sub-Gaussian tail bound for the global L2-loss of the empirical quantile function. Further, we propose tuning parameter-free multivariate nonparametric tests—a two-sample test and a test for mutual independence—based on our notion of multivariate quantiles/ranks. Asymptotic consistency of these tests are shown and the rates of convergence of the associated test statistics are derived, both under the null and alternative hypotheses.

Structural health monitoring by a novel probabilistic machine learning method based on extreme value theory and mixture quantile modeling

This article proposes a novel probabilistic machine learning method based on unsupervised novelty detection for health monitoring of civil structures. The core of this method is based on extreme value theory (EVT) and mixture quantile modeling. Accordingly, a mixture quantile value by combining non-parametric and parametric quantile estimators is proposed as a new decision-making or novelty score. The non-parametric estimator relies on an empirical quantile function, while the parametric estimator stems from modeling a generalized extreme value distribution. Generally, the proposed method is composed of some main steps; that is, calculating distances between feature samples, sorting the distances in ascending order, changing their signs for providing negated quantities, selecting some negated maximum distances or extreme values in an iterative algorithm by using a goodness-of-fit test based on Kullback-Leibler information, and computing a mixture quantile. Furthermore, an EVT-based approach is proposed to estimate an alarming threshold. The major innovation in this article is to develop a novel probabilistic novelty detector with a new score for decision-making. The advantages of the proposed method contain preparing discriminative novelty scores for damage detection, dealing with the major challenge of environmental and/or operational variability, estimating a reliable threshold, and implementing both the procedures of decision-making and threshold estimation within a single framework. Dynamic and statistical features of two full-scale bridge structures are utilized to validate the proposed method along with comparative studies. Results demonstrate that the method is a reliable and influential tool for health monitoring of civil structures under varying various environmental and/or operational conditions.

Recurrent Neural Network-Based Joint Chance Constrained Stochastic Model Predictive Control*

A novel recurrent neural network (RNN)-based approach is proposed in this work to handle joint chance-constrained stochastic model predictive control (SMPC) problem. In the proposed approach, the joint chance constraint (JCC) is first reformulated as a quantile-based inequality to reduce the complexity in approximation. Then, the quantile function (QF) in the quantile-based inequality is replaced by the empirical QF using sample average approximation (SAA). Afterwards, the empirical QF is approximated via an RNN-based surrogate model, which is embedded into the SMPC problem formulation to predict quantile values at different sampling instants. By employing the RNN-based approximation, the resulting deterministic optimization problem is finally solved through a nonlinear optimization solver. The proposed approach is applied to a hydrodesulphurisation process to demonstrate its efficiency in handling the SMPC problem.

On Some Smooth Estimators of the Quantile Function for a Stationary Associated Process

Let {Xn, n ≥ 1} be a sequence of stationary non-negative associated random variables with common marginal distribution function F(x) and quantile function Q(u), where Q(u) is defined as F(Q(u)) = u. Here we consider the smooth estimation of Q(u), adapted from generalized kernel smoothing (Cheng and Parzen J. Stat. Plann. Infer. 59, 291–307, 1997) of the empirical quantile function. Some asymptotic properties of the kernel quantile estimator, for associated sequences, are also established parallel to those in the i.i.d. case. Various estimators in this class of estimators are contrasted, through a simulation study, among themselves and with an indirect smooth quantile estimator obtained by inverting the Poisson weights based estimator of the distribution function studied in Chaubey et al. (Statist. Probab. Lett. 81, 267–276, 2011). The indirect smoothing estimator seems to be the best estimator on account of smaller MSE, however, a quantile estimator based on the Bernstein polynomials and that using the corrected Poisson weights turn out to be almost as good as the inverse distribution function estimator using Poisson weights.

On the approximation of a membership function by empirical quantile functions

The Average Cumulative representation of fuzzy intervals is connected with the possibility theory in the sense that the possibility and necessity functions are substituted by a pair of non decreasing functions defined as the positive and negative variations in the Jordan decomposition of a membership function. In this paper we motivate the crucial role of ACF in determining the membership function from experimental data; some examples and simulations are shown to state the robustness of the proposed construction.

Применение многомерной квантильной функции в задаче пептид-белок докинга

The paper presents an exploration of using evolutionary optimization algorithms in protein-peptide docking. The main assumptions that reduce docking to the continuous global optimization problem are described. Some special features of the given problem and the difficulties of using evolutionary algorithms are discussed. The paper provides a way of using evolutionary optimization algorithms based on using empirical quantile function. The multivariate quantile function structure is defined recursively using univariate quantile transform. The grid-based approach of using quantile function is presented. The disadvantages of this approach are indicated. The deterministic sampling algorithm is proposed. The used scheme of parallel sampling and the resulting speed-up are described. The GPU-accelerated approach for quantile function evaluation is presented. This paper provides multiple GPU-based ways which use a sample in explicit form. Their speed-up depending on sample size is shown. The paper presents the results of docking using an evolutionary algorithm and its quantile-function-based modification. The comparison with the relevant docking method within a particular force-field is made. The results of the experiments are analyzed.

Nonparametric estimation of mean residual quantile function under right censoring

ABSTRACTIn 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.

On the tail index estimation of an autoregressive Pareto process

In this paper we consider an autoregressive Pareto process which can be used as an alternative to heavy tailed MARMA. We focus on the tail behavior and prove that the tail empirical quantile function can be approximated by a Gaussian process. This result allows to derive a class of consistent and asymptotically normal estimators for the shape parameter. We will see through simulation that the usual estimation procedure based on an i.i.d. setting may fall short of the desired precision.

Semiparametrically weighted robust estimation of regression models

A class of two-step robust regression estimators that achieve a high relative efficiency for data from light-tailed, heavy-tailed, and contaminated distributions irrespective of the sample size is proposed and studied. In particular, the least weighted squares (LWS) estimator is combined with data-adaptive weights, which are determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the LWS estimator with the proposed weights preserves robust properties of the initial robust estimate. However, contrary to the existing methods and despite the data-dependent weights, the first-order asymptotic behavior of LWS is fully independent of the initial estimate under mild conditions. Moreover, the proposed estimation method is asymptotically efficient if errors are normally distributed. A simulation study documents these theoretical properties in finite samples; in particular, the relative efficiency of LWS with the proposed weighting schemes can reach 85%–100% in samples of several tens of observations under various distributional models.

Nonparametric estimation of hazard quantile function

In this paper, we study the estimation of the hazard quantile function based on right censored data. Two nonparametric estimators, one based on the empirical quantile density function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the kernel-based estimator are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated for a real data set.

A Markov Chain Monte Carlo approach to joint simulation of regional areas burned annually in Canadian forest fires

A simulation of regional and national forest carbon balance in Canada requires, due to regional correlations, a joint simulation of areas burned ( BA) in regional fires. Regional correlations of BA are largely determined by concurrent years of relatively large ( LF) and small fires ( SF). A binary Markov Chain Monte Carlo procedure (MCMC) is constructed for forecasting regional LF( SF) status on an annual basis. For each forecast year the regional BA-value is obtained by a random draw from region-specific empirical quantile functions for LF and SF years. In the MCMC the conditional likelihood of a regional allocation of nLF* LF-years is maximized; whereby nLF* is drawn from a distribution fitted to LF( SF) classified data of area burned in 29 Canadian forest fire regions between 1959 and 1999. Regional allocation is governed by region-specific autologistic functions. MCMC results confirmed regional and national means and variances while regional correlations were generally somewhat lower than in the data.

On estimating the conditional expected shortfall

AbstractUnlike the value at risk, the expected shortfall is a coherent measure of risk. In this paper, we discuss estimation of the expected shortfall of a random variable Yt with special reference to the case when auxiliary information is available in the form of a set of predictors Xt. We consider three classes of estimators of the conditional expected shortfall of Yt given Xt: a class of fully non‐parametric estimators and two classes of analog estimators based, respectively, on the empirical conditional quantile function and the empirical conditional distribution function. We study their sampling properties by means of a set of Monte Carlo experiments and analyze their performance in an empirical application to financial data. Copyright © 2008 John Wiley & Sons, Ltd.

ASYMPTOTIC PROPERTIES OF NONPARAMETRIC FRONTIER ESTIMATORS

Aragon, Daouia, and Thomas-Agnan (2005, Econometric Theory 21, 358–389) introduced a new nonparametric frontier estimation. We prove the weak convergence of the empirical conditional quantile function. The distribution of the limit depends on the unknown conditional quantile density function. We provide a method to construct uniform confidence bands without estimating the conditional quantile density.

Efficient robust estimation of time-series regression models

The paper studies a new class of robust regression estimators based on the two-step least weighted squares (2S-LWS) estimator which employs data-adaptive weights determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However, contrary to the existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions. We propose data-adaptive weighting schemes that perform well both in the cross-section and time-series data and prove the asymptotic normality and efficiency of the resulting procedure. A simulation study documents these theoretical properties in finite samples.

On Estimating the Conditional Expected Shortfall

Unlike the value at risk, the expected shortfall is a coherent measure of risk. In this paper, we discuss estimation of the expected shortfall of a random variable Yt with special reference to the case when auxiliary information is available in the form of a set of predictors Xt. We consider three classes of estimators of the conditional expected shortfall of Yt given Xt: a class of fully non-parametric estimators and two classes of analog estimators based, respectively, on the empirical conditional quantile function and the empirical conditional distribution function.We study their sampling properties by means of a set of Monte Carlo experiments and analyze their performance in an empirical application to financial data.

Tail and dependence behavior of levels that persist for a fixed period of time

This work emerges from a study of the extremal behavior of a daily maximum sea water levels series, {X i } , presented in Draisma (Duration of extremes at sea. In: Parametric and semi-parametric methods in E. V. T., pp. 137–143. PhD thesis, Erasmus, University, 2001). In its approach, a new series, {Y i }, is defined, consisting of water levels that persist for a fixed period of time. In this paper, we study the tail behavior of {Y i } , in case {X i } is independent and identically distributed (i.i.d.) and in case {X i } is a max-autoregressive sequence (we will consider two different max-autoregressive processes), whose distribution function is in the Frechet domain of attraction. We also determine Ledford and Tawn tail dependence index (Ledford and Tawn, Biometrika 83:169–187, 1996, J. R. Stat. Soc. B 59:475–499, 1997) and we analyze the asymptotic tail dependence of the random pair (Y i , Y i + m ), in all considered cases. According to Drees (Bernoulli 9:617–657, 2003), we obtain the limit behavior of the tail empirical quantile function associated with a random sample (Y 1, Y 2,...Y n ) and hence the asymptotic normality of a class of estimators of the tail index that includes Hill estimator.

Efficient Robust Estimation of Regression Models

This paper introduces a new class of robust regression estimators. The proposed twostep least weighted squares (2S-LWS) estimator employs data-adaptive weights determined from the empirical distribution, quantile, or density functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However contrary to existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions; most importantly, the initial estimator does not need to be square root of n consistent. Moreover, we prove that 2S-LWS is asymptotically normal under beta-mixing conditions and asymptotically efficient if errors are normally distributed. A simulation study documents these theoretical properties in finite samples; in particular, the relative efficiency of 2S-LWS can reach 85-90% in samples of several tens of observations under various distributional models.

Efficient Robust Estimation of Time-Series Regression Models

This paper studies a new class of robust regression estimators based on the two-step least weighted squares (2S-LWS) estimator which employs data-adaptive weights determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However contrary to existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions. We propose data-adaptive weighting schemes that perform well both in the cross-section and time-series data and prove the asymptotic normality and efficiency of the resulting procedure. A simulation study documents these theoretical properties in finite samples.

FUNCTIONAL ESTIMATION BY ASYMPTOTIC REGRESSION

Through an appeal to asymptotic Gaussian representations of certain empirical stochastic processes, the techniques of continuous regression are applied to derive estimates for underlying parametric probability laws. This asymptotic regression approach yields estimates for a wide range of statistical problems, including estimation based on the empirical quantile function, Poisson process intensity estimation, and parametric density estimation.