 Open Access Publications from the University of California

## Inference for the Bivariate and Multivariate Hidden Truncated Pareto(type II) and Pareto(type IV) Distribution and Some Measures Of Divergence Related To Incompatibility of Probability Distribution.

• Author(s): Ghosh, Indranil
• et al.
Abstract

Consider a discrete bivariate random variable (X; Y ) with possible values x1,x2, ..., xI for X

and y1, y2, ...., yJ for Y . Further suppose that the corresponding families of conditional distributions,

for X given values of Y and of Y for given values of X are available. We specifically

consider those situations where the above mentioned conditional distributions are not compatible.

In such a case we seek a joint probability matrix (P), say, that is minimally incompatible

with the given conditional distributions. The Kullback-Leibler Information function provides

a convenient measure of (pseudo) distance between two distributions. However we will use

a more general measure which is called the \power divergence criterion" which includes the

Kullback-Leibler Function as a special case. We will, along with this measure, also consider

some other measures of diversity which are widely used in the field of Information Theory.

Using all these measures we have developed algorithms for nding the joint probability matrix

which is minimally incompatible with the given conditionals. We will also propose here

some alternative measures of compatibility related to discrepancy. Our main objective here

is to find among the various measures of discrepancy (for example the power divergence test

statistic, modified Renyi's measure etc.), along with the proposed measures, which one give

us the minimally incompatible distribution with a faster convergence rate. This topic will be

discussed in detail in chapter 1.

Next we consider an alternative approach to determine whether or not any two given

matrices (say, A and B) with non negative elements (where for the matrix A, the column

sums add up to one and for the matrix B, the row sums add up to one) are compatible in

the sense that there exists a joint probability matrix for which has the columns and rows,

respectively, of A and B as its conditional distributions. We formulate the above problem as

a homogeneous and consistent set of equations and consider the LPP (Linear Programming

Problem) approach to solve for the unknown quantities. Furthermore we will also discuss

briefly,

under the condition of compatibility, how can we find some of the elements of the two

given conditional matrices A and B; in case they are unknown. This is the subject matter

for chapter 2.

Next we consider the hidden truncation paradigm for the bivariate Pareto (type II) distribution

when one variable is subject to hidden truncation from above. We consider the

classical method of estimation and a reasonable testing procedure for the truncation parameter

and also the other parameters involved in the model along with an application of the

above mentioned model to a real life data. We will also focus on the estimation procedure

under the Bayesian paradigm. This will be the subject matter in chapter 3.

In chapter 4 we will consider the hidden truncation paradigm for a bivariate Pareto (type IV)

distribution where both the marginals as well as the conditionals are again members of

the Pareto (type IV) family. Here also we will consider inference for such a distribution under

the classical approach as well as the Bayesian approach along with an application to a real

life data.

Next we will consider a possible extension of hidden truncation concept to the multivariate

case for the Pareto (type II) family. In particular we will focus on single variable truncation as

well as more than one variable truncation. We will discuss about the tractability of such type

of models in the context of estimation and testing for the parameters involved in the model.

In particular we focus on a trivariate Pareto (type II) set-up and we will consider estimation

procedures under both the classical and Bayesian paradigm. Two specific situations are dealt

with: (1) when one of the concomitant variables is truncated from above and (2) when more

than one variable is truncated from above. This material is discussed in detail in chapter

5. In chapter 6 we provide a general discussion of the topics which are covered in chapter 1

through 5, together with a brief discussion of potential future work.