The correlated bivariate noncentral F distribution and its application

This paper proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pretest test for the multivariate simple regression model (MSRM)

Local dependence in bivariate copulae with beta marginals

The local dependence function (LDF) describes changes in the correlation structure of continuous bivariate random variables along their range. Bivariate density functions with Beta marginals can be used to model jointly a wide variety of data with bounded outcomes in the (0,1) range, e.g. proportions. In this paper we obtain expressions for the LDF of bivariate densities constructed using three different copula models (Frank, Gumbel and Joe) with Beta marginal distributions, present examples for each, and discuss an application of these models to analyse data collected in a study of marks obtained on a statistics exam by postgraduate students

Copula models for epidemiological research and practice

Investigating associations between random variables (rvs) is one of many topics in the heart of statistical science. Graphical displays show emerging patterns between rvs, and the strength of their association is conventionally quantified via correlation coefficients. When two or more of these rvs are thought of as outcomes, their association is governed by a joint probability distribution function (pdf). When the joint pdf is bivariate normal, scalar correlation coefficients will produce a satisfactory summary of the association, otherwise alternative measures are needed. Local dependence functions, together with their corresponding graphical displays, quantify and show how the strength of the association varies across the span of the data. Additionally, the multivariate distribution function can be explicitly formulated and explored. Copulas model joint distributions of varying shapes by combining the separate (univariate) marginal cumulative distribution functions of each rv under a specified correlation structure. Copula models can be used to analyse complex relationships and incorporate covariates into their parameters. Therefore, they offer increased flexibility in modelling dependence between rvs. Copula models may also be used to construct bivariate analogues of centiles, an application for which few references are available in the literature though it is of particular interest for many paediatric applications. Population centiles are widely used to highlight children or adults who have unusual univariate outcomes. Whilst the methodology for the construction of univariate centiles is well established there has been very little work in the area of bivariate analogues of centiles where two outcomes are jointly considered. Conditional models can increase the efficiency of centile analogues in detection of individuals who require some form of intervention. Such adjustments can be readily incorporated into the modelling of the marginal distributions and of the dependence parameter within the copula model

Advances in Semi-Nonparametric Density Estimation and Shrinkage Regression

This thesis advocates the use of shrinkage and penalty techniques for estimating the parameters of a regression model that comprises both parametric and nonparametric components and develops semi-nonparametric density estimation methodologies that are applicable in a regression context. First, a moment-based approach whereby a univariate or bivariate density function is approximated by means of a suitable initial density function that is adjusted by a linear combination of orthogonal polynomials is introduced. Such adjustments are shown to be mathematically equivalent to making use of standard polynomials in one or two variables. Once extended to apply to density estimation, in which case the sample moments are being utilized, the proposed technique readily lends itself to the modeling of massive univariate or bivariate data sets. As well, the resulting density functions are shown to be expressible as kernel density estimates via the Christoffel-Darboux formula. Additionally, it is established that a set of n observations is entirely specified by its first n moments. It is also explained that a univariate bona fide density approximation can be obtained by assuming that the derivative of the logarithm of the density function under consideration is expressible as a rational function or a polynomial. An explicit representation of the density function so obtained is derived and jointly sufficient statistics for its parameters are identified. Then, extensions of the proposed methodology to density estimation and multivariate settings are discussed. As a matter of fact, this approach constitutes a generalization of Pearson\u27s system of frequency curves. Several illustrative examples are presented including regression applications. Finally, an iterative algorithm involving shrinkage and pretest techniques is introduced for estimating the parameters of a certain semi-nonparametric model. It is theoretically established and numerically verified that the proposed estimators are more accurate than the unrestricted ones. This methodology is successfully applied to a mass spectrometry data set