Arjun K. Gupta

Elliptically contoured models in statistics

Series Editors Preface. Preface. 1. Preliminaries. 2. Basic Properties. 3. Probability Density Function and Expected Values. 4. Mixtures of Normal Distributions. 5. Quadratic Forms and other Functions of Elliptically Contoured Matrices. 6. Characterization Results. 7. Estimation. 8. Hypothesis Testing. 9. Linear Models. References. Author Index. Subject Index.

Handbook of beta distribution and its applications

Foundations. Limit Theorems. Characterization. Interacting particles. Arithmetical functions. Miscellaneous results. Author Index. Subject Index.

Testing and Locating Variance Changepoints with Application to Stock Prices

Abstract This article explores testing and locating multiple variance changepoints in a sequence of independent Gaussian random variables (assuming known and common mean). This type of problem is very common in applied economics and finance. A binary procedure combined with the Schwarz information criterion (SIC) is used to search all of the possible variance changepoints existing in the sequence. The simulated power of the proposed procedure is compared to that of the CUSUM procedure used by Inclan and Tiao to cope with variance changepoints. The SIC and unbiased SIC for this problem are derived. To obtain the percentage points of the SIC criterion, the asymptotic null distribution of a function of the SIC is obtained, and then the approximate percentage points of the SIC are tabulated. Finally, the results are applied to the weekly stock prices. The unknown but common mean case is also outlined at the end.

Multivariate skew t-distribution

In this paper, we define multivariate skew t-distribution which has some of the properties of multivariate t-distribution and has a shape parameter to represent skewness. Some of its properties are also studied including the moments. Multivariate skew-Cauchy distribution is given as a special case.

Some skew-symmetric models

The skew-normal distributions have been introduced by many authors, e.g. Azzalini (1985), Arnold et al. (1993), Aigner et al. (1977), Andel et al. (1984). This class of distributions includes the normal distribution and possesses several properties which coincide or are close to the properties of the normal family. However, this class has a skewness parameter which makes it possible to have a reasonable model for a skewed population distribution thus providing a more flexible model which represents the data as adequately as possible. Besides being useful in modeling, they are helpful in studying the robustness, and in Bayesian analysis as priors. The construction of such models is based on the following lemma (see Azzalini, 1985).

A multivariate skew normal distribution

In this paper, we define a new class of multivariate skew-normal distributions. Its properties are studied. In particular we derive its density, moment generating function, the first two moments and marginal and conditional distributions. We illustrate the contours of a bivariate density as well as conditional expectations. We also give an extension to construct a general multivariate skew normal distribution.

ON CHANGE POINT DETECTION AND ESTIMATION

In this paper, a survey of the change point detection and estimation will be given. Change point problem primarily arose from the process of quality control in which one is concerned about the output of a production line and wants to find any departure from an acceptable standard of the products. The problem of abrupt changes is often encountered in various experimental and mathematical sciences. From the statistical point of view, we want to infer (detect) whether there is a statistically significant change point in a sequence of chronologically ordered data. In the case there is a COMMUN. STATIST.—SIMULA., 30(3), 665–697 (2001)

GOODNESS-OF-FIT TESTS FOR THE SKEW-NORMAL DISTRIBUTION

Given a set of data, one of the statistical issues is to see how well the data fit into a postulated model. This technique necessitates the corresponding table of the probability distribution for the proposed model. In this paper, we present the table for skew normal distributions with different values of skew factor λ. The table presented makes it workable to examine whether skew normal distribution is an appropriate model for a set of data. Two conventional testing methods are discussed, one is Kolmogorov-Smirnov test and the other one is Pearsons χ2 test. An example in medical study and a simulation example are included to illustrate the use of the table.

On the distribution of the product of independent beta random variables

In this paper, exact distribution of the product of independent beta random variables has been derived and its structural form is given together with recurrence relations for the coefficients of this representation. These recurrence relations yield a direct computational algorithm for computing the percentage points of many test criteria in multivariate statistical analysis.

On the Moments of the Beta Normal Distribution

Abstract Eugene et al. [Eugene, N., Lee, C., Famoye, F. (2002). Beta-normal distribution and its applications. Commun. Stat. Theory Meth. 31:497–512] introduced the beta normal distribution, generated from the logit of a beta random variable. The only properties of the distribution studied by them are its first moments for some particular values of the parameters. In this article, we derive a more general expression for the nth moment of the beta normal distribution. We also consider several particular cases (including those cases considered by Eugene et al.). Apart from being more general, the proof of our results are simpler and avoid the differential equations approach taken by Eugene et al.