Centile estimation for a proportion response variable

This paper introduces two general models for computing centiles when the response variable Y can take values between 0 and 1, inclusive of 0 or 1. The models developed are more flexible alternatives to the beta inflated distribution. The first proposed model employs a flexible four parameter logit skew Student t (logitSST) distribution to model the response variable Y on the unit interval (0, 1), excluding 0 and 1. This model is then extended to the inflated logitSST distribution for Y on the unit interval, including 1. The second model developed in this paper is a generalised Tobit model for Y on the unit interval, including 1. Applying these two models to (1-Y) rather than Y enables modelling of Y on the unit interval including 0 rather than 1. An application of the new models to real data shows that they can provide superior fits

GAMLSS: a distributional regression approach

A tutorial of the generalized additive models for location, scale and shape (GAMLSS) is given here using two examples. GAMLSS is a general framework for performing regression analysis where not only the location (e.g., the mean) of the distribution but also the scale and shape of the distribution can be modelled by explanatory variables

Flexible statistical models: Methods for the ordering and comparison of theoretical distributions

Statistical models usually rely on the assumption that the shape of the distribution is fixed and that it is only the mean and volatility that varies. Although the fitting of heavy tail distributions has become easier due to computational advances, the fitting of the appropriate heavy tail distribution requires knowledge of the properties of the different theoretical distributions. The selection of the appropriate theoretical distribution is not trivial. Therefore, this paper provides methods for the ordering and comparison of continuous distributions by making a threefold contribution. Firstly, it provides an ordering of the heaviness of distribution tails of continuous distributions. The resulting classification of over 30 important distributions is given. Secondly it provides guidance on choosing the appropriate tail for a given variable. As an example, we use the USA box-office revenues, an industry characterised by extreme events affecting the supply schedule of the films, to illustrate how the theoretical distribution could be selected. Finally, since moment based measures may not exist or may be unreliable, the paper uses centile based measures of skewness and kurtosis to compare distributions. The paper therefore makes a substantial methodological contribution towards the development of conditional densities for statistical model in the presence of heavy tails

Beyond location and dispersion models: The Generalized Structural Time Series Model with Applications

In many settings of empirical interest, time variation in the distribution parameters is important for capturing the dynamic behaviour of time series processes. Although the fitting of heavy tail distributions has become easier due to computational advances, the joint and explicit modelling of time-varying conditional skewness and kurtosis is a challenging task. We propose a class of parameter-driven time series models referred to as the generalized structural time series (GEST) model. The GEST model extends Gaussian structural time series models by a) allowing the distribution of the dependent variable to come from any parametric distribution, including highly skewed and kurtotic distributions (and mixed distributions) and b) expanding the systematic part of parameter-driven time series models to allow the joint and explicit modelling of all the distribution parameters as structural terms and (smoothed) functions of independent variables. The paper makes an applied contribution in the development of a fast local estimation algorithm for the evaluation of a penalised likelihood function to update the distribution parameters over time \textit{without} the need for evaluation of a high-dimensional integral based on simulation methods

Beyond location and dispersion models: The Generalized Structural Time Series Model with Applications

In many settings of empirical interest, time variation in the distribution parameters is important for capturing the dynamic behaviour of time series processes. Although the fitting of heavy tail distributions has become easier due to computational advances, the joint and explicit modelling of time-varying conditional skewness and kurtosis is a challenging task. We propose a class of parameter-driven time series models referred to as the generalized structural time series (GEST) model. The GEST model extends Gaussian structural time series models by a) allowing the distribution of the dependent variable to come from any parametric distribution, including highly skewed and kurtotic distributions (and mixed distributions) and b) expanding the systematic part of parameter-driven time series models to allow the joint and explicit modelling of all the distribution parameters as structural terms and (smoothed) functions of independent variables. The paper makes an applied contribution in the development of a fast local estimation algorithm for the evaluation of a penalised likelihood function to update the distribution parameters over time \textit{without} the need for evaluation of a high-dimensional integral based on simulation methods

Modelling location, scale and shape parameters of the birnbaumsaunders generalized t distribution

The Birnbaum-Saunders generalized t (BSGT) distribution is a very flflexible family of distributions that admits different degrees of skewness and kurtosis and includes some important special or limiting cases available in the literature, such as the Birnbaum-Saunders and Birnbaum-Saunders t distributions. In this paper we provide a regression type model to the BSGT distribution based on the generalized additive models for location, scale and shape (GAMLSS) framework. The resulting model has high flflexibility and therefore a great potential to model the distribution parameters of response variables that present light or heavy tails, i.e. platykurtic or leptokurtic shapes, as functions of explanatory variables. For different parameter settings, some simulations are performed to investigate the behavior of the estimators. The potentiality of the new regression model is illustrated by means of a real motor vehicle insurance data set

Principal component regression in GAMLSS applied to Greek-German government bond yield spreads

A solution to the problem of having to deal with a large number of interrelated explanatory variables within a generalized additive model for location, scale, and shape (GAMLSS) is given here using as an example the Greek-German government bond yield spreads from the 25th of April 2005 to the 31th of March 2010. Those were turbulent financial years, and in order to capture the spreads behaviour, a model has to be able to deal with the complex nature of the financial indicators used to predict the spreads. Fitting a model, using principal components regression of both main and first order interaction terms, for all the parameters of the assumed distribution of the response variable seems to produce promising results

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 Additive-Model 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

Generalized Additive Models for Location, Scale and Shape: A Distributional Regression Approach, with Applications

An emerging field in statistics, distributional regression facilitates the modelling of the complete conditional distribution, rather than just the mean. This book introduces generalized additive models for location, scale and shape (GAMLSS) – one of the most important classes of distributional regression. Taking a broad perspective, the authors consider penalized likelihood inference, Bayesian inference, and boosting as potential ways of estimating models and illustrate their usage in complex applications. Written by the international team who developed GAMLSS, the text's focus on practical questions and problems sets it apart. Case studies demonstrate how researchers in statistics and other data-rich disciplines can use the model in their work, exploring examples ranging from fetal ultrasounds to social media performance metrics. The R code and data sets for the case studies are available on the book's companion website, allowing for replication and further study