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Covariance formulas via marginal martingales
Author(s) -
Wellner J. A.
Publication year - 1994
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1994.tb01443.x
Subject(s) - mathematics , covariance , covariance function , marginal distribution , function (biology) , random variable , variance (accounting) , combinatorics , statistics , accounting , evolutionary biology , business , biology
Prenctice and Cai recently introduced and studied the function C defined as the covariance function of the two marginal counting process martingales of a pair of dependent survival times (T 1 , T 2 ). They show that the function C together with the marginal distributions determines the joint survival function F of (T 1 , T 2 ). In this note we show how the key characterizing equation of Prentice and Cai yields a formula for the covariance of T 1 and T 2 in termsof the marginal mean residual life functions and C. The resulting formula generalizes a formula for the variance of a one‐dimensional random variable Tdueto Pyke (1965). We also explore several generalizations of the covariance formula, and obtain a valid k‐dimensional version of the Prentice and Cai formula.