SKEW NORMAL REGRESSION MODEL

Skew Normal Regression Model

Table of Contents

Chapter 1: Introduction3

The Problem of Regression4

Chapter 2: Normal Distribution14

The Standard Normal Distribution15

Linear Transformation16

Multivariate Normal Distribution17

Log-Normal Distribution17

Characterization of the Normal Distribution18

Cumulative Distribution Function20

Impact on Psychological Measurement26

Caveats26

Implications for Analysis30

Testing for Goodness of Fit31

Chapter 3: Skew-Normal Distribution35

A generalized skew-normal distribution39

The skew-normal nonlinear regression model41

Geometric Distribution55

t-Distribution56

Chapter 5: Examples of Skew-Normal and Skew-t Distributions59

Distributions of order statistics from bivariate skew-normal distribution59

A generalized skew-t? distribution65

Distributions of conditional minima and maxima66

Distributions of order statistics from bivariate skew-t? distribution67

Bivariate Skew-Normal And Skew-T? Distributions69

Score test for homogeneity of skewness parameter70

Simulation studies73

An Illustrative example77

Chapter 6: Summary80

Summary of Project80

Final Remarks81

Refrences82

End Notes85

Chapter 1: Introduction

Statistical analysis on the treatment of continuous observations within a parametric approach is usually proceeded by assuming:

(i) simplicity of the structure for the mean of the data,

(ii) constancy of error variability, and

(iii) normality of error distributions.

The requirement of assumptions (i) and (ii) aims both to allow an efficient analysis and to achieve easiness of understanding. A typical example of (i) is the assumption of additivity. Assumption (iii) is mainly driven by the formal properties of the normal distribution, in particular its analytical beauty and also the simplicity when dealing with fundamental operations like marginalization, conditioning and linear combinations. The other reason of imposing the normality assumption is that the outcomes of the experiment are usually expected to obey the central limit theorem, thereby resulting in an approximately normally distributed observations (Weiss, 1996).

In general terms, there are two ways of dealing with data which do not satisfy the above assumptions. The first one is to develop new methods of analysis with assumptions which fit the data in original scare. The second most commonly adopted approach , however, is to bend the data in order that assumptions (i), ii) and (iii) are approximately satisfied b) taking a monotonic non-linear transformation.

Thereby the customary purposes of transformation are of threefold, but the primary motivation of transformation has tended to be on obtaining normality so as to exploit the unrivalled mathematical tractability of the normal distribution. Nevertheless, there have often been doubts, reservations and criticisms about the use of transformation for normality for two major reasons. Firstly, in multivariate setting, transformations are usually carried out on each component separately (Sahu, 2003). Thus the appropriateness of joint normality assumption is highly questionable. Secondly, the requirement for variance stability or simplicity in the mean surface often demands a transformation which is different from that for achieving normality. Therefore it seems too demanding to accomplish three goals simultaneously by means of transformation alone.

Should there be a conflict between the requirements for normality and for model simplicity (e.g. improving additivity and homoscedasticity), it is best to pay most attention to the latter to allow for ease of description and interpretation (Piessens, deDoncker Kapenga, Uberhuber, Kahaner, 1983). Hence less restrictive families of distributions that can accommodate asymmetry and non-normal peakedness and allow a continuous departure from normality to non-normality can be valuable in analyzing non-Gaussian data. The aim of this dissertation is to present an ...