Quantile

We propose a new regression method to estimate the impact of explanatory variables on quantiles of the unconditional (marginal) distribution of an outcome variable. The proposed method consists of running a regression of the (recentered) influence function (RIF) of the unconditional quantile on the explanatory variables. The influence function is a widely used tool in robust estimation that can easily be computed for each quantile of interest. We show how standard partial effects, as well as policy effects, can be estimated using our regression approach. We propose three different regression estimators based on a standard OLS regression (RIF-OLS), a logit regression (RIF-Logit), and a nonparametric logit regression (RIF-OLS). We also discuss how our approach can be generalized to other distributional statistics besides quantiles.

We compare capital requirements derived by tail conditional expectation (TCE) with those derived by tail conditional median (TCM) and find that there is no clear-cut relationship between these two measures in empirical data. Our results highlight the relevance of TCM as a robust alternative to TCE, especially for regulatory control.

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.

Quantile regression is applied in two retail credit risk assessment exercises exemplifying the power of the technique to account for the diverse distributions that arise in the financial service industry. The first application is to predict loss given default for secured loans, in particular retail mortgages. This is an asymmetric process since where the security (such as a property) value exceeds the loan balance the banks cannot retain the profit, whereas when the security does not cover the value of the defaulting loan then the bank realises a loss. In the light of this asymmetry it becomes apparent that estimating the low tail of the house value is much more relevant for estimating likely losses than estimates of the average value where in most cases no loss is realised. In our application quantile regression is used to estimate the distribution of property values realised on repossession that is then used to calculate loss given default estimates. An illustration is given for a mortgage portfolio from a European mortgage lender. A second application is to revenue modelling. While credit issuing organisations have access to large databases, they also build models to assess the likely effects of new strategies for which, by definition, there is no existing data. Certain strategies are aimed at increasing the revenue stream or decreasing the risk in specific market segments. Using a simple artificial revenue model, quantile regression is applied to elucidate the details of subsets of accounts, such as the least profitable, as predicted from their covariates. The application uses standard linear and kernel smoothed quantile regression.

In this paper we study both market risks and non-market risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties "coherent". We examine the measures of risk provided and the related actions required by SPAN, by the SEC/NASD rules and by quantile based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.

Using panel data from a developing country on individuals aged 16 to 59 who reported their monthly wages, we estimated a relationship between health (nutrition) measures (i.e. height and BMI) and wages (which proxies productivity/growth). We controlled for endogeneity of BMI and found heterogeneous returns to different human capital indicators. Our findings indicate that productivity is positively and significantly affected

A re lone inventors more or less likely to invent breakthroughs? Recent research has attempted to resolve this question by considering the variance of creative outcome distributions. It has implicitly assumed a symmetric thickening or thinning of both tails, i.e., that a greater probability of breakthroughs comes at the cost of a greater probability of failures. In contrast, we propose that collaboration can have opposite effects at the two extremes: it reduces the probability of very poor outcomes-because of more rigorous selection processes-while simultaneously increasing the probability of extremely successful outcomes-because of greater recombinant opportunity in creative search. Analysis of over half a million patented inventions supports these arguments: Individuals working alone, especially those without affiliation to organizations, are less likely to achieve breakthroughs and more likely to invent particularly poor outcomes. Quantile regressions demonstrate that the effect is more than an upward mean shift. We find partial mediation of the effect of collaboration on extreme outcomes by the diversity of technical experience of team members and by the size of team members' external collaboration networks. Supporting our meta-argument for the importance of examining each tail of the distribution separately, experience diversity helps trim poor outcomes significantly more than it helps create breakthroughs, relative to the effect of external networks.

We show how to use Bayesian modeling to analyze data from an accelerated life test where the test units come from different groups (such as batches) and the group effect is random and significant. Our approach can handle multiple random effects and several accelerating factors. However, we present our approach on the basis on an important application concerning pressure vessels wrapped in Kevlar 49 fibers where the fibers of each vessel comes from a single spool and the spool effect is random. We show how Bayesian modeling using Markov chain Monte Carlo (MCMC) methods can be used to easily answer questions of interest in accelerated life tests with random effects that are not easily answered with more traditional methods. For example, we can predict the lifetime of a pressure vessel wound with a Kevlar 49 fiber either from a spool used in the accelerated life test or from another random spool from the population of spools. We comment on the implications that this analysis has on the estimates of reliability (and safety) for the Space Shuttle, which has a system of 22 such pressure vessels. Our 2 approach is implemented in the freely available WinBUGS software so that readers can easily apply the method to their own data.

Using panel data from a developing country on individuals aged 16 to 59 who reported their monthly wages, we estimated a relationship between health (nutrition) measures (i.e. height and BMI) and wages (which proxies productivity/growth). We controlled for endogeneity of BMI and found heterogeneous returns to different human capital indicators. Our findings indicate that productivity is positively and significantly affected by education, height and BMI. The return to BMI is important both at the lower and upper end of the wage distribution for men while women at the upper end of the distribution suffer a wage penalty due to BMI. Height has been a significant factor affecting men's productivity but not women. The results in general support the high-nutrition and high-productivity equilibrium story. Returns to schooling showed a declining trend as we move from lower to higher quantiles for both sub-samples. This might suggest that schooling is more beneficial for the less able. In addition, the returns to schooling of women are higher than men. The results have important implications for policy making in the form of nutrition interventions and targeted education on women.

Expected Shortfall (ES) in several variants has been proposed as remedy for the deficiencies of Value-at-Risk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss distributions have discontinuities. In this case even the coherence property of ES can get lost unless one took care of the details in its definition. We compare some of the definitions of Expected Shortfall, pointing out that there is one which is robust in the sense of yielding a coherent risk measure regardless of the underlying distributions. Moreover, this Expected Shortfall can be estimated effectively even in cases where the usual estimators for VaR fail.

A statistical analysis of a bank's credit card database is presented. The database is a snapshot of accounts whose holders have missed a payment on a given month but who do not subsequently default. The variables on which there is information are observable measures on the account (such as profit and activity), and whether actions that are available to the bank (such as letters and telephone calls) have been taken. A primary objective for the bank is to gain insight into the effect that collections activity has on ongoing account usage. A neglog transformation that highlights features that are hidden on the original scale and improves the joint distribution of the covariates is introduced. Quantile regression, a novel methodology to the credit scoring industry, is used as it is relatively assumption free, and it is suspected that different relationships may be manifest in different parts of the response distribution. The large size is handled by selecting relatively small subsamples for training and then building empirical distributions from repeated samples for validation. In the application to the database of clients who have missed a single payment a substantive finding is that the predictor of the median of the target variable contains different variables from those of the predictor of the 30% quantile. This suggests that different mechanisms may be at play in different parts of the distribution.

Analysis of massive datasets is challenging owing to limitations of computer primary memory. Composite quantile regression (CQR) is a robust and efficient estimation method. In this paper, we extend CQR to massive datasets and propose a divide-and-conquer CQR method. The basic idea is to split the entire dataset into several blocks, applying the CQR method for data in each block, and finally combining these regression results via weighted average. The proposed approach significantly reduces the required amount of primary memory, and the resulting estimate will be as efficient as if the entire data set was analyzed simultaneously. Moreover, to improve the efficiency of CQR, we propose a weighted CQR estimation approach. To achieve sparsity with high-dimensional covariates, we develop a variable selection procedure to select significant parametric components and prove the method possessing the oracle property. Both simulations and data analysis are conducted to illustrate the finite sample performance of the proposed methods.

We introduce a general approach to nonlinear quantile regression modelling that is based on the specification of the copula function that defines the dependency structure between the variables of interest. Hence we extend Koenker and Bassett's [1978] original statement of the quantile regression problem by determining a distribution for the dependent variable Y conditional on the regressors X and hence the specification of all the quantile regression functions. We use the fact that this multivariate distribution can be split into two parts: the marginals and the dependence function (or copula). We then deduce the form of the non linear conditional quantile relationship implied by the copula. Notice that this can be done with arbitrary distributions assumed for the marginals. Some properties of the copula based quantiles or c-quantiles are then derived. Finally, we develop an empirical application which examines conditional quantile dependency in the foreign exchange market and compare this approach with the standard tail area dependency measures.

This article studies the problem of optimally dividing individuals into peer groups to maximize social gains from heterogeneous peer effects. The specific setting analyzed here concerns efficient ways of allocating roommates in college dormitories. Using confidential data on a sample ...

In order to estimate the effective dose such as the 0.5 quantile ED 50 in a bioassay problem various parametric and semiparametric models have been used in the literature. If the true dose-response curve deviates significantly from the model, the estimates will generally be inconsistent. One strategy is to analyze the data making only a minimal assumption on the model, namely, that the dose-response curve is non-decreasing. In the present paper we first define an empirical dose-response curve based on the estimated response probabilities by using the "pool-adjacent-violators" (PAV) algorithm, then estimate effective doses ED 100p for a large range of p by taking inverse of this empirical dose-response curve. The consistency and asymptotic distribution of these estimated effective doses are obtained. The asymptotic results can be extended to the estimated effective doses proposed by Glasbey [1987. Tolerance-distribution-free analyses of quantal dose-response data. Appl. Statist. 36 (3), 251-259] and Schmoyer [1984. Sigmoidally constrained maximum likelihood estimation in quantal bioassay. J. Amer. Statist. Assoc. 79, 448-453] under the additional assumption that the dose-response curve is symmetric or sigmoidal. We give some simulations on constructing confidence intervals using different methods.

In this paper, we consider the kernel-type estimator of the quantile function based on the kernel smoother under a censored dependent model. The Bahadur-type representation of the kernel smooth estimator is established, and from the Bahadur representation we can show that this estimator is strongly consistent.

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.

Are lone inventors more or less likely to invent breakthroughs? Recent research has attempted to resolve this question by considering the variance of creative outcome distributions. It has implicitly assumed a symmetric thickening or thinning of both tails, that a greater probability of breakthroughs comes at the cost of a greater probability of failures. In contrast, we propose that collaboration can have opposite effects at the two extremes: it reduces the probability of very poor outcomes -due to more rigorous selection processes -while simultaneously increasing the probability of extremely successful outcomes -due to greater recombinant opportunity in creative search. Analysis of over half a million patented inventions supports these arguments: individuals working alone, especially those without affiliation to organizations, are less likely to achieve breakthroughs and more likely to invent particularly poor outcomes. Quantile regressions demonstrate that the effect is more than an upward mean shift. We find partial mediation of the effect of collaboration on extreme outcomes by the diversity of technical experience of team members and by the size of team members' external collaboration networks. Supporting our meta-argument for the importance of examining each tail of the distribution separately, experience diversity helps trim poor outcomes significantly more than it helps create breakthroughs, relative to the effect of external networks. [212 words]

A quantile regression model for counts of breeding Cape Sable seaside sparrows Ammodramus maritimus mirabilis (L.) as a function of water depth and previous year abundance was developed based on extensive surveys, 1992-2005, in the Florida Everglades. The quantile count model extends linear quantile regression methods to discrete response variables, providing a flexible alternative to discrete parametric distributional models, e.g. Poisson, negative binomial and their zero-inflated counterparts.

Given a stationary multidimensional spatial process (Z i = (X i , Y i ) ∈ ℝ d × ℝ, i ∈ ℤ N ), we investigate a kernel estimate of the spatial conditional quantile function of the response variable Y i given the explicative variable X i . Almost complete convergence and consistency in L 2r norm (r ∈

Quantiles of univariate data are frequently used in the construction of popular descriptive statistics like the median, the interquartile range, and various measures of skewness and kurtosis based on percentiles. They are also potentially useful in robust estimation of location (eg, in the ...

We propose a new regression method to estimate the impact of explanatory variables on quantiles of the unconditional (marginal) distribution of an outcome variable. The proposed method consists of running a regression of the (recentered) influence function (RIF) of the unconditional quantile on the explanatory variables. The influence function is a widely used tool in robust estimation that can easily be computed for each quantile of interest. We show how standard partial effects, as well as policy effects, can be estimated using our regression approach. We propose three different regression estimators based on a standard OLS regression (RIF-OLS), a logit regression (RIF-Logit), and a nonparametric logit regression (RIF-OLS). We also discuss how our approach can be generalized to other distributional statistics besides quantiles.

Exact nonparametric inference based on ordinary Type-II right censored samples has been extended here to the situation when there are multiple samples with Type-II censoring from a common continuous distribution. It is shown that marginally, the order statistics from the pooled sample are mixtures of the usual order statistics with multivariate hypergeometric weights. Relevant formulas are then derived for the construction of nonparametric confidence intervals for population quantiles, prediction intervals, and tolerance intervals in terms of these pooled order statistics. It is also shown that this pooled-sample approach assists in achieving higher confidence levels when estimating large quantiles as compared to a single Type-II censored sample with same number of observations from a sample of comparable size. We also present some examples to illustrate all the methods of inference developed here.

The problem of missing values commonly arises in data sets, and imputation is usually employed to compensate for non-response. We propose a novel imputation method based on quantiles, which can be implemented with or without the presence of auxiliary information. The proposed method is extended to unequal sampling designs and nonuniform response mechanisms. Iterative algorithms to compute the proposed imputation methods are presented. Monte Carlo simulations are conducted to assess the performance of the proposed imputation methods with respect to alternative imputation methods. Simulation results indicate that the proposed methods perform competitively in terms of relative bias and relative root mean square error.

We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non-parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non-parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.

This paper uses microeconometric simulations to characterize the distributional changes occurred in the Bolivian economy in the period 1993-2002, and to assess the potential distributional impact of various alternative economic scenarios for the next decade. Wage ...

The purpose of this paper is to provide a strategic collaborative approach to risk and quality control in a cooperative supply chain by using a Neyman-Pearson quantile risk framework for the statistical control of risks. The paper is focused on the statistical quality control of a supplier and a producer, applying the traditional Neyman-Pearson theory to the control of quality in a supply chain environment. In our framework, the risks assumed by the parties in the supply chain depend on the organizational structure, the motivations and the power relationships that exist between members of the supply chain.

Investigating the factors affecting CO2 emissions has always been a challenge. One problem with existing studies is that these studies have been relied on mean-based regression approaches, such as ordinary least squares (OLS) or instrumental variables, which implicitly assumes that the impact of variables along the distribution of CO2 emissions is the same. Unlike previous studies, the present study will use the quantile regression developed by Koenker & Bassett, which is not limited to the assumption. So that, the purpose of this study is to investigate the impacts of technological innovation and renewable energy on CO2 emissions in selected countries of the International Renewable Energy Agency (IRENA) using quantile regression over the period 1990-2016. The results of this study exhibited that the impact of renewable energy on CO2 emissions was negative and statistically significant. This impact is also enhanced in high quantiles (countries with high pollution). In all the studied quantiles, the impact of technological innovations on CO2 emissions was positive, significant and initially decreasing, while increasing again over time. The results of the symmetry test also indicated that by increasing in the volume of CO2 emissions, the variable impact of renewable energy upraised. However, no incremental trend was observed in innovation.

Note to users: The section "Articles in Press" contains peer reviewed accepted articles to be published in this journal. When the final article is assigned to an issue of the journal, the "Article in Press" version will be removed from this section and will appear in the associated ...

In this paper, a technique is presented for assessing the predictive uncertainty of rainfall-runoff and hydraulic forecasts. The technique conditions forecast uncertainty on the forecasted value itself, based on retrospective Quantile Regression of hindcasted water level forecasts and forecast errors. To test the robustness of the method, a number of retrospective forecasts for different catchments across England and Wales having different size and hydrological characteristics have been used to derive in a probabilistic sense the relation between simulated values of water levels and matching errors. From this study, we can conclude that using Quantile Regression for estimating forecast errors conditional on the forecasted water levels provides a relatively simple, efficient and robust means for estimation of predictive uncertainty.

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

In many applications of hydrology, quantiles provide important insights in the statistical problems considered. In this paper, we focus on the estimation of multivariate quantiles based on copulas. We provide a nonparametric estimation procedure for a notion of multivariate quantiles, which has been used in a series of papers. These quantiles are based on particular level sets of copulas and admit the usual probabilistic interpretation that a p-quantile comprises a probability mass p. We also explore the usefulness of a smoothed bootstrap in the estimation process. Our simulation results show that the nonparametric estimation procedure yields excellent results and that the smoothed bootstrap can be beneficially applied. The main purpose of our paper is to provide an easily applicable method for practitioners and applied researchers in domains such as hydrology and coastal engineering.

Given a stationary multidimensional spatial process (Z i = (X i , Y i ) ∈ ℝ d × ℝ, i ∈ ℤ N ), we investigate a kernel estimate of the spatial conditional quantile function of the response variable Y i given the explicative variable X i . Almost complete convergence and consistency in L 2r norm (r ∈

Countries encounter conflicting policy options in reaching fast development goals due to high resource use, rapid economic expansion, and environmental degradation. Thus, the present research examined the connection between CO2 emissions and urbanization, globalization, hydroelectricity, and economic expansion in China utilizing data spanning the period between 1985 and 2018. The novel quantile-on-quantile (QQ) and quantile regression (QR) approaches were applied to assess this interconnection. The QQ approach is characterized by its ability to incorporate quantile regression fundamentals and non-parametric estimation research. As a result, the method appears to transform the quantile of one parameter into another. The QQ outcomes revealed that in all quantiles (0.1–0.95), gross domestic product (GDP), urbanization, and globalization trigger CO2 emissions in China, while in each quantile (0.1–0.985), hydroelectricity consumption mitigates CO2 emissions. The QR outcomes also affirmed...

This paper considers the estimation of extreme quantile autoregression function by using a parametric model. We combine direct estimation of quantiles in the middle region with that of extreme parts using the model and results from extreme value theory (EVT). The volatility used to scale the residuals is estimated indirectly, without estimating conditional mean, using the conditional quantile (CQ) range. The estimators are found to be consistent. A small simulation study carried out shows that the estimator of the volatility function converges to the true function over a range of distributional errors. Finally, the T-periods ahead extreme quantile autoregression function is given.

Over the last decade there has been growing demand for estimates of population characteristics at small area level. Unfortunately, cost constraints in the design of sample surveys lead to small sample sizes within these areas and as a result direct estimation, using only the survey data, is inappropriate since it yields estimates with unacceptable levels of precision. Small area models are designed to tackle the small sample size problem. The most popular class of models for small area estimation is random effects models that include random area effects to account for between area variations. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. An alternative approach to small area estimation that is based on the use of M-quantile models was recently proposed by Chambers and Tzavidis (2006) and Tzavidis and Chambers (2007). Unlike traditional random effects models, M-quantile models do not depend on strong distributional assumption and automatically provide outlier robust inference.

Various regional flood frequency analysis procedures are used in hydrology to estimate hydrological variables at ungauged or partially gauged sites. Relatively few studies have been conducted to evaluate the accuracy of these procedures and estimate the error induced in regional flood frequency estimation models. The objective of this paper is to assess the overall error induced in the residual kriging (RK) regional flood frequency estimation model. The two main error sources in specific flood quantile estimation using RK are the error induced in the quantiles local estimation procedure and the error resulting from the regional quantile estimation process. Therefore, for an overall error assessment, the corresponding errors associated with these two steps must be quantified. Results show that the main source of error in RK is the error induced into the regional quantile estimation method. Results also indicate that the accuracy of the regional estimates increases with decreasing return periods.

We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of Feldman and Tucker [Ann. Math. Statist. 37 (1996) 451-457]. We illustrate the algorithm with an example from credit portfolio risk analysis.