Showing papers by "Herman Chernoff published in 1965"

Sequential tests for the mean of a normal distribution iv (discrete case)

Abstract: The problem of sequentially testing whether the mean of a normal distribution is positive has been approximated by the continuous analogue where one must decide whether the mean drift of a Wiener-Levy process is positive or negative [3]. The asymptotic behavior of the solution of the latter problem has been studied as $t \rightarrow \infty$ and as $t \rightarrow 0$ [1], [2], [4], [5]. The original (discrete) problem, can be regarded as a variation of the continuous problem where one is permitted to stop observation only at the discrete time points $t_0, t_0 + \delta, t_0 + 2\delta, \cdots$. Especially since the numerical computation of the solution of the continuous version can be carried out by solving the discrete version for small $\delta$, it is important to study the relationship between the solutions of the discrete and continuous problems. These solutions are represented by symmetric continuation regions whose upper boundaries are $\tilde x_\delta(t)$ and $\tilde x(t)$ respectively. The main result of this paper is that \begin{equation*}\tag{(1.1)}\tilde x_\delta(t) = \tilde x(t) + \hat z\sqrt\delta + o(\sqrt\delta).\end{equation*} This result involves relating the original problem to an associated problem and studying the limiting behavior of the solution of the associated problem. This solution corresponds to the solution of a Wiener-Hopf equation. Results of Spitzer [6], [7] can be used to characterize the solution of the Wiener-Hopf equation and yield $\hat z$ as an integral, which, as Gordon Latta pointed out to the author, is equal to $\zeta(\frac{1}{2})/(2\pi)^{\frac{1}{2}} = -.5824$. The associated problem referred to above is the following. A Wiener-Levy process $Z_t$ starting at a point $(z, t), t < 0$ is observed at a cost of one per unit time. If the observation is stopped before $t = 0$, there is no payoff. If $t = 0$ is reached, the payoff is $Z^2_0$ if $Z_0 < 0$ and 0 if $Z_0 \geqq 0$. Stopping is permitted at times $t = -1, -2, \cdots$.

SEQUENTIAL TEST FOR THE MEAN OF A NORMAL DISTRIBUTION III (SMALL t)

Abstract: : Asymptotic expansions are derived for the behavior of the optimal sequential test of whether the unknown drift mu of a Wiener-Levy process is positive or negative for the case where the process has been observed for a short time. The test is optimal in the sense that it is the Bayes test for the problem where we have an a priori normal distribution of mu, the regret for coming to the wrong conclusion is proportional to the absolute value of mu and the cost of observation is one per unit time. The Bayes procedure is compared with the best sequential likelihood ratio test and with the procedure which calls for stopping when no fixed additional sampling time is better than stopping. The derivations allow for generalizing to variations of this problem with different cost structure. (Author)

A Bayes Sequential Sampling Inspection Plan

Abstract: 1. Introduction and summary. Given a lot of size N whose items are obtained from a statistically controlled process with an unknown probability p, 0 0 is the loss involved in sending out a defective item instead of replacing it with a good one. The cost of inspecting n items (defective items detected during inspection are replaced by good ones) is cn, c > 0. An inspection plan is to be devised that minimizes (in some suitable sense) the risk (expected loss plus inspection cost). Let Vb denote any sequential inspection plan, which determines sequentially the (random) number 9T of items to be inspected. If the lot is sent out after inspecting n items, the cost of inspection incurred is nc and the expected loss due to the defectives remaining in the lot is (N - n)pk. Then the risk is (N - n)pk + nc = Npk + nk(c/k -p), and hence the risk associated with the plan At is given by

Limit distributions of the minimax of independent identically distributed random variables

Abstract: Summary. The class of limiting distributions of the normalized minimax (or maximin) of independent identically distributed random variables is obtained and the domains of attraction of the three limiting types are characterized. Asymptotic independence of the minimax and maximin is also demonstrated. 1. Introduction. Consider a sequence of independent identically distributed random variables on some probability space with P the probability measure thereupon. These will be double indexed in the fashion