Topic
Quantile function
About: Quantile function is a research topic. Over the lifetime, 1595 publications have been published within this topic receiving 23148 citations. The topic is also known as: percent point function & inverse cumulative distribution function.
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TL;DR: An approach to statistical data analysis which is simultaneously parametric and nonparametric is described, and density-quantile functions, autoregressive density estimation, estimation of location and scale parameters by regression analysis of the sample quantile function, and quantile-box plots are introduced.
Abstract: This article attempts to describe an approach to statistical data analysis which is simultaneously parametric and nonparametric. Given a random sample X 1, …, X n of a random variable X, one would like (1) to test the parametric goodness-of-fit hypothesis H 0 that the true distribution function F is of the form F(x) = F0[(x − μ)/σ)], where F 0 is specified, and (2) when H 0 is not accepted, to estimate nonparametrically the true density-quantile function fQ(u) and score function J(u) = − (fQ)'(u). The article also introduces density-quantile functions, autoregressive density estimation, estimation of location and scale parameters by regression analysis of the sample quantile function, and quantile-box plots.
719 citations
TL;DR: In this paper, the authors proposed a method to address the problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem.
Abstract: This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem (Bassett and Koenker (1982)). The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural distribution and quantile functions using data on Vietnam veteran status and earnings.
487 citations
TL;DR: In this paper, the authors examined the complete pattern of 256 model variants that can be constructed from twenty functions, and their explanatory power was assessed, and the best model has a power value function, a risky weighting function due to Prelec (1998), and a Logit function.
Abstract: Many different functional forms have been suggested for both the value function and probability weighting function of Cumulative Prospect Theory (Tversky and Kahneman, 1992). There are also many stochastic choice functions available. Since these three components only make predictions when considered in combination, this paper examines the complete pattern of 256 model variants that can be constructed from twenty functions. All these variants are fit to experimental data and their explanatory power assessed. Significant interaction effects are observed. The best model has a power value function, a risky weighting function due to Prelec (1998), and a Logit function.
439 citations
TL;DR: A simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution is developed and it is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution.
Abstract: This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.
392 citations
TL;DR: The Weibull distribution is the most important distribution for problems in reliability as discussed by the authors, and it has been studied extensively in the literature, including in the context of the wider Weibbull-G family of distributions.
Abstract: The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.
391 citations