Theophilos Cacoullos

On upper bounds for the variance of functions of random variables

The upper bounds for the variance of a function g of a random variable X obtained in Cacoullos (1982) (for short CP) are improved in the case [mu] = E(X) [not equal to] 0. A main feature of these bounds is that they involve the second moment of the derivative or the difference of g. A multivariate extension for functions of independent random variables is also given.

Characterizations of discrete distributions by a conditional distribution and a regression function

SummaryThe bivariate distribution of (X, Y), whereX andY are non-negative integer-valued random variables, is characterized by the conditional distribution ofY givenX=x and a consistent regression function ofX onY. This is achieved when the conditional distribution is one of the distributions: a) binomial, Poisson, Pascal or b) a right translation of these. In a) the conditional distribution ofY is anx-fold convolution of another random variable independent ofX so thatY is a generalized distribution. A main feature of these characterizations is that their proof does not depent on the specific form of the regression function. It is also indicated how these results can be used for good-ness-of-fit purposes.

Lower variance bounds and a new proof of the central limit theorem

Lower variance bounds are derived for functions of a random vector X, thus extending previous results. Moreover, the w-function associated X is shown to characterize its distribution, and a special application shows the multivariate central limit theorem.

Variance inequalities for covariance kernels and applications to central limit theorems

A simple estimate for the error in the CLT, valid for a wide class of absolutely continuous r.v.s, is derived without Fourier techniques. This is achieved by using a simple convolution inequality for the variance of covariance kernels or w-functions in conjunction with bounds for the total variation distance. The results are extended to the multivariate case. Finally, a simple proof of the classical Darmois--Skitovich characterization of normality is obtained.

Bounds for the variance of functions of random variables by orthogonal polynomials and Bhattacharya bounds

Upper and lower bounds for the variance of a function g of a random variable X are obtained by expanding g in a series of orthogonal polynomials associated with the distribution of X or by using the convergence of Bhattacharya bounds for exponential families of distribution.

Characterizations of mixtures of continuous distributions by their posterior means

Abstract Let X be a continuous r. v. with probability density f(x|θ) where θ is a continuous or discrete parameter. Mixtures of normal or gamma densities f(x|θ) with respect to the parameter θ are characterized by the posterior mean of θ given X. The (prior) mixing distribution of θ is also characterized at the same time, by using the identifiability of such mixtures. Some applications in collective risk theory are indicated. It is pointed out that the characterizations hold more generally for the exponential family.

On Bivariate Discrete Distributions Generated by Compounding

A discrete r.v. X is generalized (compounded) by another discrete r.v. Zi to yield the compound distribution of Z = Z1+ … + ZX. Distributional properties are given concerning the bivariate structure of X and Z. The joint, marginal, and conditional distributions arising out of (X, Z) are derived via probability generating function techniques. Special attention is given to power series distributions (PSD), in particular when Z is a compound Poisson. Recurrences for joint probabilities and cumulants are indicated. Several ad hoc estimation techniques are discussed.

Characterizing priors by posterior expectations in multiparameter exponential families

SummaryExploiting the notion of identifiability of mixtures of exponential families with respect to a vector parameter θ, it is shown that the posterior expectation of θ characterizes the prior distribution of θ. The result is applied to normal and negative multinomial distributions.

The F-test of homoscedasticity for correlated normal variables

Using the F-representation of t, the Pitman-Morgan t-test for homoscedasticity under a bivariate normal setup is shown to be equivalent to an F-test on n-2 and n-2 degrees of freedom. This yields an F-test of independence under normality.

Two LDF characterizations of the normal as a spherical distribution

Two optimal characteristic properties of the normal distribution are shown: (a) Of all the SNM (spherical scale normal mixtures) the normal with the same Mahalanobis distances between [Pi]i:SNM([mu]i) and [Pi]j:SNM([mu]j), i [not equal to] j, maximizes the probabilities of correct classification determined by a certain subclass of the LDF classification rules; (b) The class of LDF (linear discriminant function) rules is the admissible class for the discrimination problem with spherical population alternatives iff the spherical distribution is normal.