Topp–Leone normal distribution with application to increasing failure rate data

In this article, we proposed a new three-parameter probability distribution, called Topp–Leone normal, for modelling increasing failure rate data. The distribution is obtained by using Topp–Leone-X family of distributions with normal as a baseline model. The basic properties including moments, quantile function, stochastic ordering and order statistics are derived here. The estimation of unknown parameters is approached by the method of maximum likelihood, least squares, weighted least squares and maximum product spacings. An extensive simulation study is carried out to compare the long-run performance of the estimators. Applicability of the distribution is illustrated by means of three real data analyses over existing distributions.     

The generalized Gudermannian distribution: inference and volatility modelling

In this paper, we introduce a new distribution, called generalized Gudermannian (GG) distribution, and its skew extension for GARCH models in modelling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are obtained including probability density and cumulative distribution functions, moments, and stochastic representation. The maximum likelihood method is used to estimate unknown parameters of the proposed model and finite sample performance of maximum likelihood estimates are evaluated by means of Monte-Carlo simulation study. The real data application on Nikkei 225 index is given to demonstrate the performance of GARCH model specified under skew extension of GG innovation distribution against normal, Student's-t, skew normal and generalized error and skew generalized error distributions in terms of the accuracy of VaR forecasts. The empirical results show that the GARCH model with GG innovation distribution produces the most accurate VaR forecasts for all confidence levels.     

The McDonald arcsine distribution: a new model to proportional data

We propose a new three-parameter continuous model called the McDonald arcsine distribution, which is a very competitive model to the beta, beta type I and Kumaraswamy distributions for modelling rates and proportions. We provide a mathematical treatment of the new distribution including explicit expressions for the density function, moments, generating and quantile functions, mean deviations, two probability measures based on the Bonferroni and Lorenz curves, Shannon entropy, Rényi entropy and cumulative residual entropy. Maximum likelihood is used to estimate the model parameters and the expected information matrix is determined. An application of the proposed model to real data shows that it can give consistently a better fit than other important statistical models.     

A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.     

The exponentiated exponential–geometric distribution: a distribution with decreasing,increasing and unimodal failure rate

In this paper, we proposed a new family of distributions namely exponentiated exponential–geometric (E2G) distribution. The E2G distribution is a straightforwardly generalization of the exponential–geometric (EG) distribution proposed by Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], which accommodates increasing, decreasing and unimodal hazard functions. It arises on a latent competing risk scenarios, where the lifetime associated with a particular risk is not observable but only the minimum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime and modal value. Maximum-likelihood inference is implemented straightforwardly. From a mis-specification simulation study performed in order to assess the extent of the mis-specification errors when testing the EG distribution against the E2G, and we observed that it is usually possible to discriminate between both distributions even for moderate samples with presence of censoring. The practical importance of the new distribution was demonstrated in three applications where we compare the E2G distribution with several lifetime distributions.     

Weighted composite quantile regression analysis for nonignorable missing data using nonresponse instrument

Efficient statistical inference on nonignorable missing data is a challenging problem. This paper proposes a new estimation procedure based on composite quantile regression (CQR) for linear regression models with nonignorable missing data, that is applicable even with high-dimensional covariates. A parametric model is assumed for modelling response probability, which is estimated by the empirical likelihood approach. Local identifiability of the proposed strategy is guaranteed on the basis of an instrumental variable approach. A set of data-based adaptive weights constructed via an empirical likelihood method is used to weight CQR functions. The proposed method is resistant to heavy-tailed errors or outliers in the response. An adaptive penalisation method for variable selection is proposed to achieve sparsity with high-dimensional covariates. Limiting distributions of the proposed estimators are derived. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies. An application to the ACTG 175 data is analysed.     

Inverse Weibull power series distributions: properties and applications

Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.     

Survival and lifetime data analysis with a flexible class of distributions

We introduce a general class of continuous univariate distributions with positive support obtained by transforming the class of two-piece distributions. We show that this class of distributions is very flexible, easy to implement, and contains members that can capture different tail behaviours and shapes, producing also a variety of hazard functions. The proposed distributions represent a flexible alternative to the classical choices such as the log-normal, Gamma, and Weibull distributions. We investigate empirically the inferential properties of the proposed models through an extensive simulation study. We present some applications using real data in the contexts of time-to-event and accelerated failure time models. In the second kind of applications, we explore the use of these models in the estimation of the distribution of the individual remaining life.     

Computational Examples of a New Method for Distribution Selection in the Pearson System

A considerable problem in statistics and risk management is finding distributions that capture the complex behaviour exhibited by financial data. The importance of higher order moments in decision making has been well recognized and there is increasing interest in modelling with distributions that are able to account for these effects. The Pearson system can be used to model a wide scale of distributions with various skewness and kurtosis. This paper provides computational examples of a new easily implemented method for selecting probability density functions from the Pearson family of distributions. We apply this method to daily, monthly, and annual series using a range of data from commodity markets to macroeconomic variables.     

A test procedure for detecting super-heavy tails

The aim of this work is to develop a test to distinguish between heavy and super-heavy tailed probability distributions. These classes of distributions are relevant in areas such as telecommunications and insurance risk, among others. By heavy tailed distributions we mean probability distribution functions with polynomially decreasing upper tails (regularly varying tails). The term super-heavy is reserved for right tails decreasing to zero at a slower rate, such as logarithmic, or worse (slowly varying tails). Simulations are presented for several models and an application with telecommunications data is provided.     

Gaussian Markov random field spatial models in GAMLSS

This paper describes the modelling and fitting of Gaussian Markov random field spatial components within a Generalized AdditiveModel for Location, Scale and Shape (GAMLSS) model. This allows modelling of any or all the parameters of the distribution for the response variable using explanatory variables and spatial effects. The response variable distribution is allowed to be a non-exponential family distribution. A new package developed in R to achieve this is presented. We use Gaussian Markov random fields to model the spatial effect in Munich rent data and explore some features and characteristics of the data. The potential of using spatial analysis within GAMLSS is discussed. We argue that the flexibility of parametric distributions, ability to model all the parameters of the distribution and diagnostic tools of GAMLSS provide an ideal environment for modelling spatial features of data.     

Discrete distributions based on inter arrival times with application to football data

Motivated by McShane, Adrian, Bradlow and Fader [Journal of Business and Economic Statistics, 26, 2008, 369–378], we introduce thirteen discrete distributions. We give explicit expressions for their probability mass functions. We analyze two football data sets and show that some of the proposed distributions provide better fits than the distribution due to McShane, Adrian, Bradlow, and Fader.     

A new class of probability distributions via cosine and sine functions with applications

ABSTRACT

In this paper, we introduce a new class of (probability) distributions, based on a cosine-sine transformation, obtained by compounding a baseline distribution with cosine and sine functions. Some of its properties are explored. A special focus is given to a particular cosine-sine transformation using the exponential distribution as baseline. Estimations of parameters of a particular cosine-sine exponential distribution are performed via the maximum likelihood estimation method. A simulation study investigates the performances of these estimates. Applications are given for four real data sets, showing a better fit in comparison to some existing distributions based on some goodness-of-fit tests.     

Bayesian estimation of quantile distributions

Use of Bayesian modelling and analysis has become commonplace in many disciplines (finance, genetics and image analysis, for example). Many complex data sets are collected which do not readily admit standard distributions, and often comprise skew and kurtotic data. Such data is well-modelled by the very flexibly-shaped distributions of the quantile distribution family, whose members are defined by the inverse of their cumulative distribution functions and rarely have analytical likelihood functions defined. Without explicit likelihood functions, Bayesian methodologies such as Gibbs sampling cannot be applied to parameter estimation for this valuable class of distributions without resorting to numerical inversion. Approximate Bayesian computation provides an alternative approach requiring only a sampling scheme for the distribution of interest, enabling easier use of quantile distributions under the Bayesian framework. Parameter estimates for simulated and experimental data are presented.     

Rapid penalized likelihood-based outlier detection via heteroskedasticity test

Outlier detection is fundamental to statistical modelling. When there are multiple outliers, many traditional approaches in use are stepwise detection procedures, which can be computationally expensive and ignore stochastic error in the outlier detection process. Outlier detection can be performed by a heteroskedasticity test. In this article, a rapid outlier detection method via multiple heteroskedasticity test based on penalized likelihood approaches is proposed to handle these kinds of problems. The proposed method detects the heteroskedasticity of all data only by one step and estimate coefficients simultaneously. The proposed approach is distinguished from others in that a rapid modelling approach uses a weighted least squares formulation coupled with nonconvex sparsity-including penalization. Furthermore, the proposed approach does not need to construct test statistics and calculate their distributions. A new algorithm is proposed for optimizing penalized likelihood functions. Favourable theoretical properties of the proposed approach are obtained. Our simulation studies and real data analysis show that the newly proposed methods compare favourably with other traditional outlier detection techniques.     

On tail-ordering and comparison of failure rates

In this paper, we have studied some implications between tail-ordering (also known as dispersive ordering) and failure rate ordering (also called TP₂ ordering) of two probability distribution functions. Based on independent random samples from these distributions, a class of distribution-free tests has been proposed for testing the null hypothesis that the two life distributions are identical against the alternative that one failure rate is uniformly smaller than the other. The tests have good efficiencies as compared to their competitors.     

A folded model for compositional data analysis

A folded type model is developed for analysing compositional data. The proposed model involves an extension of the α‐transformation for compositional data and provides a new and flexible class of distributions for modelling data defined on the simplex sample space. Despite its rather seemingly complex structure, employment of the EM algorithm guarantees efficient parameter estimation. The model is validated through simulation studies and examples which illustrate that the proposed model performs better in terms of capturing the data structure, when compared to the popular logistic normal distribution, and can be advantageous over a similar model without folding.     

On zero-distorted generalized geometric distribution

ABSTRACT

We propose a new generalized geometric distribution which permits inflation/deflation of the zero count probability and study some of its properties. We also present an actuarial application of this distribution and fit it to three datasets used by other researchers. It is observed that the proposed distribution fits reasonably well to these data. Further, in a regression setup, the performance of this distribution is studied vis–a–vis other competing distributions used for explaining variability in a response variable.     

A General Quantile Function Model for Economic and Financial Time Series

This article proposed a general quantile function model that covers both one- and multiple-dimensional models and that takes several existing models in the literature as its special cases. This article also developed a new uniform Bayesian framework for quantile function modelling and illustrated the developed approach through different quantile function models. Many distributions are defined explicitly only via their quanitle functions as the corresponding distribution or density functions do not have an explicit mathematical expression. Such distributions are rarely used in economic and financial modelling in practice. The developed methodology makes it more convenient to use these distributions in analyzing economic and financial data. Empirical applications to economic and financial time series and comparisons with other types of models and methods show that the developed method can be very useful in practice.     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. 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.