On the Lp-quantiles for the Student t distribution

L_p-quantiles represent an important class of generalised quantiles and are defined as the minimisers of an expected asymmetric power function, see Chen (1996). For p=1 and p=2 they correspond respectively to the quantiles and the expectiles. In his paper Koenker (1993) showed that the tau quantile and the tau expectile coincide for every tau in (0,1) for a class of rescaled Student t distributions with two degrees of freedom. Here, we extend this result proving that for the Student t distribution with p degrees of freedom, the tau quantile and the tau L_p-quantile coincide for every tau in (0,1) and the same holds for any affine transformation. Furthermore, we investigate the properties of L_p-quantiles and provide recursive equations for the truncated moments of the Student t distribution

An analysis of life expectancy and economic production using expectile frontier zones

The wealth of a country is assumed to have a strong non-linear influence on the life expectancy of its inhabitants. We follow up on research by Preston and study the relationship with gross domestic product. Smooth curves for the average but also for (upper) frontiers are constructed by a combination of least asymmetrically weighted squares and P-splines. Guidelines are given for optimizing the amount of smoothing and the definition of frontiers. The model is applied to a large set of countries in different years. It is also used to estimate life expectancy performance for individual countries and to show how it changed over time.frontier estimation, gross domestic product, least asymmetrically weighted squares, life expectancy, production frontier, smoothing

Do maternal health problems influence child's worrying status? Evidence from the British Cohort Study

Conventional methods apply symmetric prior distributions such as a normal distribution or a Laplace distribution for regression coefficients, which may be suitable for median regression and exhibit no robustness to outliers. This work develops a quantile regression on linear panel data model without heterogeneity from a Bayesian point of view, i.e. upon a location-scale mixture representation of the asymmetric Laplace error distribution, and provides how the posterior distribution is summarized using Markov chain Monte Carlo methods. Applying this approach to the 1970 British Cohort Study (BCS) data, it finds that a different maternal health problem has different influence on child's worrying status at different quantiles. In addition, applying stochastic search variable selection for maternal health problems to the 1970 BCS data, it finds that maternal nervous breakdown, among the 25 maternal health problems, contributes most to influence the child's worrying status

An analysis of life expectancy and economic production using expectile frontier zones

The wealth of a country is assumed to have a strong non-linear influence on the life expectancy of its inhabitants. We follow up on research by Preston and study the relationship with gross domestic product. Smooth curves for the average but also for upper frontiers are constructed by a combination of least asymmetrically weighted squares and P-splines. Guidelines are given for optimizing the amount of smoothing and the definition of frontiers. The model is applied to a large set of countries in different years. It is also used to estimate life expectancy performance for individual countries and to show how it changed over time

deepregression: A Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression

In this paper we describe the implementation of semi-structured deep distributional regression, a flexible framework to learn conditional distributions based on the combination of additive regression models and deep networks. Our implementation encompasses (1) a modular neural network building system based on the deep learning library TensorFlow for the fusion of various statistical and deep learning approaches, (2) an orthogonalization cell to allow for an interpretable combination of different subnetworks, as well as (3) pre-processing steps necessary to set up such models. The software package allows to define models in a user-friendly manner via a formula interface that is inspired by classical statistical model frameworks such as mgcv. The package's modular design and functionality provides a unique resource for both scalable estimation of complex statistical models and the combination of approaches from deep learning and statistics. This allows for state-of-the-art predictive performance while simultaneously retaining the indispensable interpretability of classical statistical models

A three-step approach to production frontier estimation and the Matsuoka's distribution

In this work, we introduce a three-step semiparametric methodology for the estimation of production frontiers. We consider a model inspired by the well-known Cobb-Douglas production function, wherein input factors operate multiplicatively within the model. Efficiency in the proposed model is assumed to follow a continuous univariate uniparametric distribution in $(0,1)$, referred to as Matsuoka's distribution, which is introduced and explored. Following model linearization, the first step of the procedure is to semiparametrically estimate the regression function through a local linear smoother. The second step focuses on the estimation of the efficiency parameter in which the properties of the Matsuoka's distribution are employed. Finally, we estimate the production frontier through a plug-in methodology. We present a rigorous asymptotic theory related to the proposed three-step estimation, including consistency, and asymptotic normality, and derive rates for the convergences presented. Incidentally, we also introduce and study the Matsuoka's distribution, deriving its main properties, including quantiles, moments, $\alpha$-expectiles, entropies, and stress-strength reliability, among others. The Matsuoka's distribution exhibits a versatile array of shapes capable of effectively encapsulating the typical behavior of efficiency within production frontier models. To complement the large sample results obtained, a Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed three-step methodology. An empirical application using a dataset of Danish milk producers is also presented

Expectile depth: Theory and computation for bivariate datasets

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble the quantiles, and just like them, expectiles are indexed by a level α in the unit interval. In the present paper, we introduce and discuss the main properties of the (multivariate) expectile regions, a nested family of sets, whose instance with level 0 < α ≤ 1/2 is built up by all points whose univariate projections lie between the expectiles of levels α and 1 − α of the projected dataset. Such level is interpreted as the degree of centrality of a point with respect to a multivariate distribution and therefore serves as a depth function. We propose here algorithms for determining all the extreme points of the bivariate expectile regions as well as for computing the depth of a point in the plane. We also study the convergence of the sample expectile regions to the population ones and the uniform consistency of the sample expectile depth. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.This research was partially supported by the Spanish Ministry of Science and Innovation under grant ECO2015-66593-P

Modelling expected shortfall using tail entropy

Given the recent replacement of value-at-risk as the regulatory standard measure of risk with expected shortfall (ES) undertaken by the Basel Committee on Banking Supervision, it is imperative that ES gives correct estimates for the value of expected levels of losses in crisis situations. However, the measurement of ES is affected by a lack of observations in the tail of the distribution. While kernel-based smoothing techniques can be used to partially circumvent this problem, in this paper we propose a simple nonparametric tail measure of risk based on information entropy and compare its backtesting performance with that of other standard ES models