Generalized Concept of Relative Risk and Wider Applications of the Proportional Hazards Model and the Kaplan–Meier Estimator

The concepts of relative risk and hazard ratio are generalized for ordinary ordinal and continuous response variables, respectively. Under the generalized concepts, the Cox proportional hazards model with the Breslow's and Efron's methods can be regarded as generalizations of the Mantel–Haenszel estimator for dealing with broader types of covariates and responses. When ordinal responses can be regarded as discretized observations of a hypothetical continuous variable, the estimated relative risks from the Cox model reflect the associations between the responses and covariates. Examples are given to illustrate the generalized concepts and wider applications of the Cox model and the Kaplan–Meier estimator.     

Strong consistency result of a non parametric conditional mode estimator under random censorship for functional regressors

Abstract

Let (T, C, X) be a vector of random variables (rvs) where T, C, and X are the interest variable, a right censoring rv, and a covariate, respectively. In this paper, we study the kernel conditional mode estimation when the covariate takes values in an infinite dimensional space and is α-mixing. Under some regularity conditions, the almost complete convergence of the estimate with rates is established.     

Construction of peculiar diffusion process having Gaussian marginals

ABSTRACT

In this note, we construct a new strange diffusion process whose drift coefficient does not satisfy the usual Lipschitz and linear-growth conditions. Beginning with the necessary characteristic equation between drift and diffusion coefficients established in Mudakkar (2012 Mudakkar, S.R. (2012). A necessary characteristic equation of diffusion processes having gaussian marginals. Abstr. Appl. Anal. 2012, Article ID 598590:9. doi:10.1155/2012/598590.[Crossref] , [Google Scholar]) and using the Foster–Lyapunov criterion and truncation technique, the existence is established.     

Convergence of weighted sums for sequences of pairwise NQD random variables

ABSTRACT

The authors discuss the convergence for weighted sums of pairwise negatively quadrant dependent (NQD) random variables and obtain some new results which extend and improve the result of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar]. In addition, we relax some restrictions of the conditions in their result. Some new methods are used in this article which differ from that of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar].     

On pitman nearness and variance of estimators

In this article, some conditions on variances are presented under which the (Generalized) Pitman Nearness Criterion Would Prefer one estimator over another. Results for univariate as well as multivariate cases are derived. An exact expression for a result of Rao, Keating and Mason (1985) is provided.     

MULTIVARIATE GENERALIZED POLYA DISTRIBUTION OF ORDER k

ABSTRACT

An order k (or cluster) generalized Polya distribution and a multivariate generalized Polya-Eggenberger one where derived in (Sen, K.; Jain, R. Cluster Generalized Negative Binomial Distribution. In Probability Models and Statistics, A. J. Medhi Festschrift on the Occasion of his 70th Birthday; Borthakur, A.C. et al., Eds.; New Age International Publishers: New Delhi, 1996; 227–241 and Sen, K.; Jain, R. A Multivariate Generalized Polya-Eggenberger Probability Model-First Passage Approach. Communications in Statistics—Theory and Methods 1997, 26, 871–884). Presently, both distributions are generalized to a multivariate generalized Polya distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases new multivariate Polya and inverse Polya distributions of order k and the multivariate generalized negative binomial distribution of the same order derived recently in (Tripsiannis, G.A.; Philippou, A.N.; Papathanasiou, A.A. Multivariate Generalized Distributions of Order k. Medical Statistics Technical Report #41: Democritus University of Thrace, Greece, 2001). Limiting cases are considered and applications are indicated.     

Optimal transitive procedures for comparing large numbers of parameters

Independent random samples are selected from each of a set of N independent populations, P₁,…,P_n. Interest centers around comparing N (unknown) scalar parameters θ₁,…,θ_N associated respectively with the N populations P₁,…,P_n. Procedures are constructed for estimating the magnitude of each of the differences δ_t,j = θ_i ? θ_j (1 ≤ i,j ≤ N) between pairs of populations. A loss function which adopts appropriate penalties for magnitude errors in estimation of differences is constructed. Magnitude estimators of differences are called transitive if they give rise to a transitive (i.e., consistent) relationship between pairwise differences of parameters. We show how to construct optimal effcient transitive magnitude–estimation procedures and demonstrate their usefulness through an example involving estimating the magnitude of the differences between disease incidence in paired towns for different pairs. Optimal transitive pairwise–comparison procedures are optimum (i.e., have the smallest posterior Bayes risks) in the class of all transitive pairwise–comparison procedures; as such they replace classical Bayes procedures which are usually not transitive when the number N of parameters compared is large. The posterior Bayes risk of optimal transitive pairwise comparison procedures are compared with that for alternative ‘adapted’ procedures, constructed from optimal simultaneous estimators and adapted for the purpose of pairwise comparisons. It is shown that the optimal transitive pairwise comparison procedures dominated the adapted procedures (in posterior Bayes risk) and typically represent only a small increase in posterior risk over the classical Bayes procedures which generally fail to be consistent. Optimal Bayes procedures are shown, for large numbers of parameters to be reasonably easy to construct using the algorithms outlined in this paper