Y. S. Chow

On optimal stopping rules

Let y 1,y 2, … be a sequence of random variables with a given joint distribution. Assume that we can observe the y’s sequentially but that we must stop some time, and that if we stop with yn we will receive a payoff x n = f n(y 1, …, y n). What stopping rule will maximize the expected value of the payoff?

Iterated Logarithm laws for weighted averages

It Let {if,, n > 1 } be independent, identically distributed (i.i.d.) random variables with zero means and unit variances. The Law of the Iterated Logarithm is shown to hold for sequences {an Xn} provided that the constants {an} satisfy (i) n a, 2 < C ~ a~, (ii) ~ a~ --* oo. j=t j=l

A renewal theorem and its applications to some sequential procedures

SummaryA renewal theorem of the elementary type for some stopping times which arise from some statistical estimation problems has been established. It is applied to prove the asymptotic risk efficiency for the problem of estimating the mean when the loss function is a weighted sum of the squared error and the sample size, and the variance is unknown. It is also applied to verify a conjecture of Robbins and Siegmund (1974) on the evaluation of the variance of the estimator of the logarithm of the odds ratio for a sequential procedure.