Isha Dewan

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.

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.

Nonparametric estimation of quantile density function

In the present article, a new nonparametric estimator of quantile density function is defined and its asymptotic properties are studied. The comparison of the proposed estimator has been made with estimators given by Jones (1992), graphically and in terms of mean square errors for the uncensored and censored cases.

Asymptotic normality for U-statistics of associated random variables

Abstract Let {X n , n⩾1} be a stationary sequence of associated random variables and Un be a U-statistic based on this sample. We establish a central limit theorem for Un when the U-statistic is degenerate or non-degenerate using an orthogonal expansion for the kernel associated with Un. We extend the results to U-statistics of kernels of degree 3 and to V-statistics of arbitrary degree. We also establish a central limit theorem for the two sample U-statistic based on observations of two independent stationary associated sequences.

Wavelet linear density estimation for associated stratified size-biased sample

Ramirez and Vidakovic [(2010), ‘Wavelet Density Estimation for Stratified Size-Biased Sample’, Journal of Statistical Planning and Inference, 140, 419–432] considered an estimator of the density function based on wavelets with independent stratified random variables from weighted distributions. They proved that it is L 2-consistent. In this paper, we complete this result by determining the rate of convergence attained by a slightly modified version of their estimator (including an estimator of the normalisation parameters). Then, we explore the case when the random variables are negatively and positively associated within strata. The theory is illustrated with a simulation study.

ON CONDITIONAL MARGINAL AND CONDITIONAL JOINT RELIABILITY IMPORTANCE

The reliability importance of one or more components when another component is assumed to be working/non-working is measured by Conditional Marginal Reliability Importance (CMRI) and Conditional Joint Reliability Importance (CJRI) respectively. We consider two systems viz the series-in-parallel and series-parallel. The expressions for CMRI and CJRI are derived for both the systems when the components are independent but not identically distributed. It is shown that the sign of the joint importance of three components and Conditional Joint Importance (CJI) can be determined using Schur-convexity (concavity) of the reliability function. The difference in the reliability functions of two coherent systems with n ≥ 3 statistically independent and with dependent components is derived. It is shown to be measured by their covariance, the JRI and the CJRIs. CMRIs and CJRIs of a phased type electronic system and a bridge structure are worked out.

Ageing concepts for discrete data — A relook

Many a times a product lifetime can be described through a nonnegative integer valued random variable. For example, a piece of equipment operates in cycles and the experimenter observes the number of cycles successfully completed prior to failure. A frequently referred example is a Xerox machine whose life length would be the total number of copies it produces before the failure. Another example is the length of the hospital stay of patients who were hospitalized due to an accident. We review ageing concepts like ILR, IHR, IHRA, NBU, NBUE, MRL, etc for discrete random variables and look at stochastic orderings between two discrete random variables.

Mann-Whitney test for associated sequences

Let {X1, ...,Xm} and {Y1, ...,Yn} be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functionsF andG, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.

Central limit theorem for U-statistics of associated random variables

Let {Xn, n[greater-or-equal, slanted]1} be a sequence of stationary associated random variables. Let Un be a U-statistic based on this sample. We establish the Central Limit Theorem for Un using the Hoeffdings decomposition.

On testing dependence between time to failure and cause of failure when causes of failure are missing.

The hypothesis of independence between the failure time and the cause of failure is studied by using the conditional probabilities of failure due to a specific cause given that there is no failure up to certain fixed time. In practice, there are situations when the failure times are available for all units but the causes of failures might be missing for some units. We propose tests based on U-statistics to test for independence of the failure time and the cause of failure in the competing risks model when all the causes of failure cannot be observed. The asymptotic distribution is normal in each case. Simulation studies look at power comparisons for the proposed tests for two families of distributions. The one-sided and the two-sided tests based on Kendall type statistic perform exceedingly well in detecting departures from independence.