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Simulated percentage points for the null distribution of the likelihood ratio test for a mixture of two normals. (English) Zbl 0715.62040
Summary: We find the percentage points of the likelihood ratio test of the null hypothesis that a sample of n observations is from a normal distribution with unknown mean and variance against the alternative that the sample is from a mixture of two distinct normal distributions, each with unknown mean and unknown (but equal) variance. The mixing proportion \(\pi\) is also unknown under the alternative hypothesis. For 2,500 samples of sizes \(n=15\), 20, 25, 40, 50, 70, 75, 80, 100, 150, 250, 500, and 1000, we calculated the likelihood ratio statistic, and from these values estimated the percentage points of the null distributions. Our algorithm for the calculation of the maximum likelihood estimates of the unknown parameters included precautions against convergence of the maximization algorithm to a local rather than global maximum. Investigations for convergence to an asymptotic distribution indicated that convergence was very slow and that stability was not apparent for samples as large as 1000. Comparisons of the percentage points to the commonly assumed chi- squared distribution with 2 degrees of freedom indicated that this assumption is too liberal; i.e., one’s P-value is greater than that indicated by \(\chi^ 2_ 2\). We conclude then that one would need what is usually an unfeasible large sample size \((n>1000)\) for the use of large-sample approximations to be justified.

MSC:
62F03 Parametric hypothesis testing
62E15 Exact distribution theory in statistics
62P10 Applications of statistics to biology and medical sciences; meta analysis
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