# Accession Number:

## AD0431814

# Title:

## SEQUENTIAL TESTS FOR THE MEAN OF A NORMAL DISTRIBUTION IV (DISCRETE CASE),

# Descriptive Note:

# Corporate Author:

## STANFORD UNIV CALIF

# Personal Author(s):

# Report Date:

## 1964-01-31

# Pagination or Media Count:

## 32.0

# Abstract:

The problem of sequentially testing whether the mean of a normal distribution is positive has been approximated by the continous analogue where one must decide whether the mean drift of a Wiener-Levy process is positive or negative. The asymptotic behavior of the solution of the latter problem was studied as t approaches infinity and zero. 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. The results 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. The associated problem is the following a WienerLevy process Zt starting at a point z,t, t is less than 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 the quantity squared if it is less than 0 and 0 if it is more than or equal to 0. Stopping is permitted at times t -1, -2,... . The conditional risk for the optimal procedure, given Ztz is studied. Bounds and limiting properties are derived using Spitzers results on the solution of certain Wiener-Hopf equations. Author