Let X be a unit vector random variable taking values on a k-dimensional sphere [Omega] with probability density function f(x). The problem considered is one of estimating f(x) based on n independent observation X1,...,Xn on X. The proposed estimator is of the form fn(x) = (nhk-1)-1C(h)...

The paper is devoted to metrization of probability spaces through the introduction of a quadratic differential metric in the parameter space of the probability distributions. For this purpose, a [phi]-entropy functional is defined on the probability space and its Hessian along a direction of the...

The paper consists of two parts. The first part deals with solutions to some optimization problems. The general problem is one of minimssing Tr AVA'U, where V and U are positive definite matrices when the elements of the matrix are subject to linear restrictions of the type AX = O or X'AX = O...

The Choquet-Deny theorem on an integral equation is extended using an elementary technique based on a certain inequality for exchangeable random variables. Previous proofs for partial results have involved amongst other things the Hewitt-Savage zero-one law and the martingale convergence...

General functional equations of the type [summation operator][phi]i(Ai't+Bi'u)=Ca(ut)+Db(tu)+Pk(t,u) and [Sigma][phi]i(Ci't) = Pk(t) have been solved, where Pk represents a polynomial of degree k in all the arguments, Cn(u [short parallel] t), a polynomial of degree a in u given t, and Db(t...

In the general Gauss-Markoff model (Y, X[beta], [sigma]2V), when V is singular, there exist linear functions of Y which vanish with probability 1 imposing some restrictions on Y as well as on the unknown [beta]. In all earlier work on linear estimation, representations of best-linear unbiased...

Likelihood ratio tests are derived for testing the structure of mean values in a two-way classification. The most general hypothesis considered is when the mean values are subject to row and column effects and interaction has a given complexity. The observations corresponding to a row or a...

The variance of a quadratic function of the random variables in a linear model is minimized to obtain locally best unbiased estimators (MIVQUE) of variance components. Condition for such estimators to be independent of the kurtosis of the variables is given. When the variables are normally...

Kantorovich gave an upper bound to the product of two quadratic forms, (X'AX) (X'A-1X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants X'AX X'A-1X where X is n - k matrix such that X'X = Ik....