A one-sided testing problem based on an i.i.d. sample of observations is considered. The usual one-sided sequential probability ratio test would be based on a random walk derived from these observations. Here we propose a sequential test where the random walk is replaced by Lindley’s random...

The problem of sequentially testing two simple hypotheses is considered for i.i.d. observations. We give explicit lower bounds for the kth moments of the expected sample size for a sequential test with error probabilities [alpha] and [beta]. These bounds give the optimal asymptotic rate as...

In this paper a method for obtaining a.s. consistency in nonparametric estimation is presented which only requires the handling of covariances. This method is applied to kernel density estimation and kernel and nearest neighbour regression estimation. It leads to conditions for a.s. consistency...

A general result on r-quick convergence for time-averages of regenerative stochastic processes is derived and then applied to Markov processes. The notion of r-quick convergence was used by Lai (1981) to show asymptotic optimality of invariant sequential probability ratio tests. In the last...

For an extremal process (Zt)t the optimal stopping problem for Xt = f(Zt)-g(t) gives the continuous time analogue of the optimal stopping problem for max{Y1,...,Yk}-ck where Y1, Y2,... are i.i.d. For the continuous time problem we derive optimal stopping times in explicit form and also show that...

We treat the problem of finding asymptotic expansions for the variance of stopping times for Wiener processes with positive drift (continuous time case) as well as sums of i.i.d. random variables with positive mean (discrete time case). Carrying over the setting of nonlinear renewal theory to...