TESTING THE EQUALITY OF THE MEANS OF TWO NORMAL DISTRIBUTIONS WHEN THE SAMPLES ARE TRUNCATED

Given a pair of random samples (x(,1), x(,2), . . . x(,n(,1))) and (y(,1), y(,2), . . . y(,n(,2))) from normal populations that have been truncated on the left at different points, the problem is to find a statistic for testing the null hypothesis that the underlying means of the two populations are equal, assuming the variances and truncation points are known. The likelihood ratio method was selected as providing the best test for this problem. In order to specify the statistic it is necessary to develop algorithms for computing the maximum likelihood estimators of both the individual population means and the joint mean when equality is assumed. The estimator for the mean of a single population has been given as a rational polynomial approximation to the solution of the maximum likelihood equation. The estimator for the joint mean under the null hypothesis is approximated by a linear combination of the individual estimators weighted by the ratio of their variances. Both estimators are shown to be asymptotically normal and Monte Carlo procedures are used to determine the required sample size for convergence to normality for a range of truncation points from -2 to +2. The formula for the likelihood ratio statistic, , and the function (lamda) = -2 1n are derived. Computer simulation is used to generate the empirical distribution of (lamda) for selected values of the sample size and truncation points. The empirical distributions are compared to the limiting chi-square distribution with one degree of freedom. Although the overall fit is generally good, examination of the percentage points of the empirical distribution of (lamda) show that use of the chi-square approximation would result in an increase in actual test size, particularly if the sample size is as small as 6. The asymptotic normality of the estimators led to consideration of an alternate test statistic derived from the usual normal theory, namely the difference of the sample estimators from each population. This statistic was shown by simulation to fit a standard normal distribution over the same range of values that were tested for the likelihood statistic. The percentage points for this statistic were very close to the percentage points of the standard normal. Moreover, the distribution of the square of this statistic was shown to be equivalent to the distribution of (lamda) except in the right tail of the distribution. Since the second test statistic is much easier to compute than (lamda), and shows a better fit to the limiting distribution in the tails, it may be preferred for use.