B. L. S. Prakasa Rao

Testing for Second-Order Stochastic Dominance of Two Distributions

A distribution function F is said to stochastically dominate another distribution function G in the second-order sense if null, for all x . Second-order stochastic dominance plays an important role in economics, finance, and accounting. Here a statistical test has been constructed to test null, for some x null [ a , b ], against the hypothesis null, for all x null [ a , b ], where a and b are any two real numbers. The test has been shown to be consistent and has an upper bound α on the asymptotic size. The test is expected to have usefulness for comparison of random prospects for risk averters.

Conditional independence, conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.

Estimation of the survival function for stationary associated processes

Let {Xn, n [greater-or-equal, slanted] 1} be a stationary sequence of associated random variables with survival function (x) = P[X1 > x]. The empirical survival function n(x) based on X1, X2,..., Xn is proposed as an estimator for (x). Strong consistency, pointwise as well as uniform, and asymptotic normality of n(x) are discussed.

A general method of density estimation for associated random variables

Let {X n ;n ≥1} be a sequence of stationary associated random variables having a common marginal density function f (x). Let , be a sequence of Borel-measurable functions defined on R 2. Let be the empirical density function. Here we study a set of sufficient conditions under which the probability at an exponential rate as n → ∞ where the rate possibly depends on ϵ, δ and f and [a, b] is a finite or an infinite interval.

Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion

We investigate the asymptotic properties of the maximum likelihood estimator and Bayes estimator of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by fractional Brownian motion. We obtain a Bernstein-von Mises type theorem also for such a class of processes.

Hajek–Renyi-type inequality for associated sequences

Let be a probability space and {Xn, n[greater-or-equal, slanted]1} be a sequence of random variables defined on it. A finite sequence {X1,...,Xn} is said to be associated if for any two component wise non-decreasing functions f and g on Rn, Cov(f(X1,...,Xn),g(X1,...,Xn))[greater-or-equal, slanted]0. A Hajek-Renyi-type inequality for associated sequences is proved. Some applications are given.

Another characterization of multivariate normal distribution

We establish a characterization of the multivariate normal based on a maximal property relating Var[g([zeta])] and the gradient of g(·).

Kernel-type density and failure rate estimation for associated sequences

Let {Xn,n≥1} be a strictly stationary sequence of associated random variables defined on a probability space (Ω,B, P) with probability density functionf(x) and failure rate functionr(x) forX1. Letfn(x) be a kerneltype estimator off(x) based onX1,...,Xn. Properties offn(x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimatorrn(x) ofr(x) based onfn(x) andFn(x), the empirical survival function, is proposed. The estimatorrn(x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets.

Associated sequences and related inference problems

The concept of association of random variables was introduced by Esary et al. (1967). In several situations, for example, in reliability and survival analysis, the random variables of lifetimes involved are not independent but are associated. Here we review recent results, both probabilistic and statistical inferential, for associated random variables.

Sequential Estimation for Fractional Ornstein–Uhlenbeck Type Process

Abstract We investigate the asymptotic properties of the sequential maximum likelihood estimator of the drift parameter for fractional Ornstein–Uhlenbeck type process satisfying a linear stochastic differential equation driven by a fractional Brownian motion.Abstract We investigate the asymptotic properties of the sequential maximum likelihood estimator of the drift parameter for fractional Ornstein–Uhlenbeck type process satisfying a linear stochastic differential equation driven by a fractional Brownian motion.