M. Bhaskara Rao

Complete convergence and almost sure convergence of weighted sums of random variables

AbstractLetr>1. For eachn≥1, let {Xnk, −∞<k<∞} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee

Some comments on fuzzy variables

\Sigma _{n \geqslant 1} n^{r - 2} P\{ |\Sigma _{k = - \infty }^\infty X_{nk} | \geqslant \varepsilon \}< \infty

Nonparametric Estimation of Specific Occurrence/Exposure Rate in Risk and Survival Analysis

for every ε>0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.

Large deviations for moving average processes

Let {Yi; −∞<i<∞} be a doubly infinite sequence of i.i.d. random variables,{ai − ∞ < i < ∞} an absolutely summable sequence of real numbers and 1 ⩽ t < 2. In this paper, we prove the complete convergence of {∑nk=1∑∞i=−∞ai+kYi/n1/t;n⩾1}, assuming EY1=0and E|E|Y1|2t∞.

A strong law for

Abstract Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X1, X2,…,Xn are mutually unrelated fuzzy variables with common membership function μ and α1, α2,…,αn are real numbers satisfying αi ⩾ o for every i and Σi=1n αi=1, when does does Z = Σi = 1n αiXi have the same membership function μ?

B

Let Yn, n>=1, be a sequence of integrable random variables with EYn = xn1[beta]1 + xn2[beta]2 + ... + xnp[beta]p, where the xijs are known and [beta]T = ([beta]1, [beta]2,..., [beta]p) unknown. Let bn be the least-squares estimator of [beta] based on Y1, Y2,..., Yn. Weak consistency of bn, n>=1, has been considered in the literature under the assumption that each Yn is square integrable. In this paper, we study weak consistency of bn, n>=1, and associated rates of convergence under the minimal assumption that each Yn is integrable.

-valued arrays

Abstract A cohort of individuals exposed to some risk is followed up to a point of time M, and observations on two random variables (Y, Δ) are recorded for each individual. The variable Δ refers to one of the four possible events that can occur for an individual in the period [0, M]: (i) dies of a specific disease, say cancer, (ii) dies of a natural cause, (iii) withdraws from the study, and (iv) is alive and still under study at time M. The variable Y refers to the time at which an event occurs. Based on such data for n individuals, we consider the problem of estimation of a specific occurrence/exposure rate (SOER), which is a risk ratio defined as the ratio of probability of death due to cancer in the interval [0, M] to the mean lifetime of all individuals up to the time point M. The asymptotic distribution of a nonparametric estimator of SOER is shown to be normal, and the asymptotic variance involves unknown parameters. Various ways of bootstrapping are discussed for construction of confidence interva...

A structure theorem on bivariate positive quadrant dependent distributions and tests for independence in two-way contingency tables

Let Z = {hellip;, - 1, 0, 1, ...}, [xi], [xi]n, n [epsilon] Z a doubly infinite sequence of i.i.d. random variables in a separable Banach space B, and an, n [epsilon] Z, a doubly infinite sequence of real numbers with 0 [not equal to] [summation operator]n [epsilon] zan