Quantile Density Function Research Articles

Consumer heterogeneity and inefficiency in oligopoly markets

Motivated by several recent studies on oligopolistic markets with consumer heterogeneity, we characterize the geometric properties of a class of inequalities involving the product of two integrals of the probability density quantile function and its reciprocal. Through three applications, we illustrate how our characterization can help enrich our understanding of the existing findings and shed some new insights, regarding the extent to which consumer welfare and social efficiency depend on the properties of the distributions of consumer valuation heterogeneity.

Estimating Income Distributions From Grouped Data: A Minimum Quantile Distance Approach

This paper focuses on the estimation of income distribution from grouped data in the form of quantiles. We propose a novel application of the minimum quantile distance (MQD) approach and compare its performance with the maximum likelihood (ML) technique. The estimation methods are applied using three parametric distributions: the generalized beta distribution of the second kind (GB2), the Dagum distribution, and the Singh–Maddala distribution. We provide the density-quantile functions for these distributions, along with reproducible R code. A simulation study is conducted to evaluate the performance of the MQD and ML methods. The proposed methods are then applied to data from 30 European countries, utilizing the aforementioned parametric distributions. To validate the accuracy of the estimates, we compare them with estimates obtained from more detailed and informative microdata sets. The findings confirm the excellent performance of the considered parametric distributions in estimating income distribution. Additionally, the MQD approach is identified as a straightforward and reliable method for this purpose. Notably, the MQD method displays superior robustness in comparison to the ML technique when it comes to selecting suitable starting values for the underlying computation algorithm, specifically when dealing with the GB2 distribution.

On quantile-based dynamic survival extropy and its applications

The cumulative residual extropy is an uncertainty measure that parallels extropy in an absolutely continuous cumulative distribution function. The dynamic version of this measure is known as dynamic survival extropy. In this paper, we study some properties of the dynamic survival extropy using quantile function approach. Unlike the dynamic survival extropy, the quantile-based dynamic survival extropy determines the quantile density function uniquely through a simple relationship. We also extend the definition of quantile-based dynamic survival extropy into order statistics. Finally, an application of new quantile-based uncertainty measure as a risk measure is derived.

Evaluation of threshold selection methods for adaptive wavelet quantile density estimation in the presence of bias

In this paper, the estimation of the quantile density function based on i.i.d biased observations is investigated. The bias function is assumed to be positive and bounded. Of the various smoothing methods for selecting the model parameters, hard and block thresholding methods are proposed and two adaptive estimators based on them are constructed. We evaluate these theoretical performances via the minimax approach over Besov balls. We show that these estimators obtain near-optimal and optimal convergence rates under some mild assumptions. Finally, with a simulation study and application on a real set of data, the performance quality of these estimators will be compared to other wavelet methods.

On some non parametric estimators of the quantile density function for a stationary associated process

In this article, we consider smooth estimators for the quantile density function (qdf) for a sequence { X n , n ≥ 1 } of stationary non negative associated random variables with a common marginal distribution function. The qdf is given by q ( u ) = Q ′ ( u ) , u ∈ ( 0 , 1 ) , Q ( u ) representing the corresponding quantile function. The smooth estimators of q ( u ) considered here are adapted from those of Q ( u ) considered in Chaubey, Dewan, and Li (2021). A few asymptotic properties of these estimators are established parallel to those in the i.i.d. case. A numerical study comparing the mean squared errors of various estimators indicates the advantages and a few limitations of various estimators. The smoothing parameter is selected based on the BCV and RLCV (a variation of likelihood cross-validation) criteria. It is concluded, based on the numerical studies, that the RLCV criterion may produce over-smoothing, hence BCV criterion may be preferable. The numerical studies also suggest that, overall, the estimator proposed by Soni, Dewan, and Jain (2012) seems to have some advantage over the other estimators considered in this article.

On weighted extropy of ranked set sampling and its comparison with simple random sampling counterpart

The extropy measure proposed by Lad, Sanfilippo, and Agro (Statistical Science 30 (1):40–58, 2015) as a dual function of Shannon entropy has received a considerable attention in the last five years. In this paper, we first show a representation for the weighted extropy of ranked set sampling in terms of quantile and density-quantile functions. Then we provide some related results including monotone properties, stochastic orders, characterizations and sharp bounds. Moreover, the so obtained results show how the weighted extropy of ranked set sampling compares with its counterpart of simple random sampling.

Nonparametric estimation of first price auctions via density–quantile function

We propose a nonparametric estimator of bidders’ value function based on a kernel estimator of the density–quantile function of bids in first price auctions. This estimator provides certain advantage over the conventional approach that relies on the distribution/density ratio of the bids. We use a boundary-adaptive kernel for boundary bias correction and propose a practical method of bandwidth selection. Our numerical experiments demonstrate good performance of the proposed method in the estimation of first price auctions with risk neutral and risk averse bidders.

An empirical estimate of quantile density function in presence of censoring

In survival or reliability studies, the quantile density function is an important characteristic in understanding the survival or aging process. In this various types of studies, a typical problem encountered in the data collection stage is that the samples may be censored from the right. In this article, we consider the problem of nonparametric estimation of a quantile density with right-censored data. A new adaptive wavelet estimator of this function is defined and its asymptotic properties are studied. In order to evaluate the performance of our estimator, small Monte Carlo simulations are carried out. Results show that the proposed estimators work well especially when the sample size is small and their calculations are simple. Finally, a real data example is provided.

Consumer Heterogeneity and Inefficiency in Oligopoly Markets

Motivated by several recent papers on oligopoly theory, we characterize geometric properties of a class of inequalities involving the product of two integrals of the probability density quantile function and its reciprocal. Through three applications, we illustrate how our characterization can help enrich our understanding of the existing findings and shed some new insight, regarding the extent to which market inefficiency depends on the properties of the distributions of consumer heterogeneity.

Inference for conditional value-at-risk of a predictive regression

Conditional value-at-risk is a popular risk measure in risk management. We study the inference problem of conditional value-at-risk under a linear predictive regression model. We derive the asymptotic distribution of the least squares estimator for the conditional value-at-risk. Our results relax the model assumptions made in (Oper. Res. 60 (2012) 739–756) and correct their mistake in the asymptotic variance expression. We show that the asymptotic variance depends on the quantile density function of the unobserved error and whether the model has a predictor with infinite variance, which makes it challenging to actually quantify the uncertainty of the conditional risk measure. To make the inference feasible, we then propose a smooth empirical likelihood based method for constructing a confidence interval for the conditional value-at-risk based on either independent errors or GARCH errors. Our approach not only bypasses the challenge of directly estimating the asymptotic variance but also does not need to know whether there exists an infinite variance predictor in the predictive model. Furthermore, we apply the same idea to the quantile regression method, which allows infinite variance predictors and generalizes the parameter estimation in (Econometric Theory 22 (2006) 173–205) to conditional value-at-risk in the Supplementary Material. We demonstrate the finite sample performance of the derived confidence intervals through numerical studies before applying them to real data.

Wavelet-Based Quantile Density Estimation By Block Thresholding Method Under L2 Loss Function

In this paper, we consider an adaptive wavelet estimation for quantile density function based on block thresholding method and obtain it's convergence rate under L2 loss function over Besove function spaces. This work is an extension of results in Chesneau et. al. (2016) and shows that the block threshold estimator gets better convergence rate (Optimal) than the estimators proposed by Chesneau et. al. (2016). The performance of the proposed estimator is investigated with a simulation study.

Nonparametric estimation of a quantile density function under Lp risk via block thresholding method

Here we propose a new quantile density function estimator via block thresholding methods and investigate its asymptotic convergence rates under Lp risk with over Besov balls. We show that the considered estimator achieves optimal or near optimal rates of convergence according to the values of the parameter ν of the Besov classes . We show that this estimator attain optimal and nearly optimal rates of convergence over a wide range of Besov function classes, and in particular enjoys those faster rates without the extraneous logarithmic penalties that given in Chesneau et al. A simulation study shows new proposed estimator performs better at the tails than existing competitors.

The Odd Gamma Weibull-Geometric Model: Theory and Applications

In this paper, we study a new four-parameter distribution called the odd gamma Weibull-geometric distribution. Having the qualities suggested by its name, the new distribution is a special member of the odd-gamma-G family of distributions, defined with the Weibull-geometric distribution as baseline, benefiting of their respective merits. Firstly, we present a comprehensive account of its mathematical properties, including shapes, asymptotes, quantile function, quantile density function, skewness, kurtosis, moments, moment generating function and stochastic ordering. Then, we focus our attention on the statistical inference of the corresponding model. The maximum likelihood estimation method is used to estimate the model parameters. The performance of this method is assessed by a Monte Carlo simulation study. An empirical illustration of the new distribution is presented by the analyses two real-life data sets. The results of the proposed model reveal to be better as compared to those of the useful beta-Weibull, gamma-Weibull and Weibull-geometric models.

On the estimation of the quantile density function by orthogonal series

The classical estimator of a quantile density function by orthogonal series depends on the empirical distribution function estimator Hn. The fact that Hn is a step function even when the underlying cumulative distribution function is continuous, has called for the need (in certain areas of application like estimating the quantile density function ) for smooth estimators of Hn. The present work has two goals. The first one is to introduce a new technique for estimating by orthogonal series for any orthonormal system in , a smooth nonparametric estimators of and are proposed. Asymptotic properties of the proposed estimators are studied. The second is to introduce a new method for selection of a smoothing parameter. A simulation study is done to compare the performances of the new approach with the (Chesneau et al 2016) one, when comparing mean integrated square error of the two estimators.

Nonparametric estimators for quantile density function under length-biased sampling

In this article, the strong uniform consistency of two nonparametric estimators for the quantile density function is established under length-biased sampling. The rate of the strong approximation of the resulting processes of these estimators will be presented as well. A Monte Carlo simulation study is carried out to compare the proposed estimators with each other in terms of mean squared errors.

Recovery of quantile and quantile density function using the frequency moments

The problem of recovering quantiles and quantile density functions of a positive random variable via the values of frequency moments is studied. The uniform upper bounds of the proposed approximations are derived. Several simple examples and corresponding plots illustrate the behavior of the recovered approximations. Some applications of the constructions are discussed as well. Namely, using the empirical counterparts of the constructions yield the estimates of the quantiles, and the quantile density functions. By means of simulations, the average errors in terms of L2-norm are evaluated to justify the consistency of the estimate of the quantile density function. As an application of the constructions, the question of estimating the so-called expected shortfall measure in risk models is also studied.

Nonparametric Confidence Interval for Quantiles

Quantiles, which are also known as values-at-risk in finance, frequently arise in practice as measures of risk. Confidence interval for quantiles are typically constructed via large sample theory or the sectioning. One of the ways for achieving the confidence interval for quantiles is direct use of a central limit theorem. In this approach, we require a good estimator of the quantile density function. In this paper, we consider the nonparametric estimator of the quantile density function from Soni et al. (2012) and we obtain confidence interval for quantiles. In the following, by using simulation, the coverage probability and mean square error of this confidence interval is calculated. Also, we compare our proposed approach with alternative approaches such as sectioning and jackknife.

Analytical expressions of risk quantities for composite models

Composite models have received a lot of attention in the recent actuarial literature. In this paper, we obtain analytic expressions of different actuarial and statistical quantities for a general class of composite models derived from the McDonald’s family of probability distributions. These quantities include probability density functions, cumulative distribution functions, quantile functions, raw moments, value-at-risk, tail moments, moments of loss variables and the quantile density function, which is very useful for the computation of expectations of order statistics. Finally, applications of these models to two sets of unimodal and positively skewed insurance claim size data are provided.

Duplex output software effort estimation model with self-guided interpretation

ContextSoftware effort estimation (SEE) plays a key role in predicting the effort needed to complete software development task. However, the conclusion instability across learners has affected the implementation of SEE models. This instability can be attributed to the lack of an effort classification benchmark that software researchers and practitioners can use to facilitate and interpret prediction results. ObjectiveTo ameliorate the conclusion instability challenge by introducing a classification and self-guided interpretation scheme for SEE. MethodWe first used the density quantile function to discretise the effort recorded in 14 datasets into three classes (high, low and moderate) and built regression models for these datasets. The results of the regression models were an effort estimate, termed output 1, which was then classified into an effort class, termed output 2. We refer to the models generated in this study as duplex output models as they return two outputs. The introduced duplex output models trained with the leave-one-out cross validation and evaluated with MAE, BMMRE and adjusted R2, can be used to predict both the software effort and the class of software effort estimate. Robust statistical tests (Welch's t-test and Kruskal-Wallis H-test) were used to examine the statistical significant differences in the models’ prediction performances. ResultsWe observed the following: (1) the duplex output models not only predicted the effort estimates, they also offered a guide to interpreting the effort expended; (2) incorporating the genetic search algorithm into the duplex output model allowed the sampling of relevant features for improved prediction accuracy; and (3) ElasticNet, a hybrid regression, provided superior prediction accuracy over the ATLM, the state-of-the-art baseline regression. ConclusionThe results show that the duplex output model provides a self-guided benchmark for interpreting estimated software effort. ElasticNet can also serve as a baseline model for SEE.

Nonparametric Estimation of Quantile and Quantile Density Function

In this article, we derive a new and unique method of estimating quantile and quantile density function, which is based on moments of fractional order statistics. A comparison of the proposed estimators is made with existing popular nonparametric quantile and quantile density estimators, in terms of mean squared error (MSE) for censored and uncensored data. Recommendations for the choice of quantile and/or quantile density estimators are given.