Small sample likelihood based inference for the normal variance ratio

Abstract

Abstract This study deals with the small sample likelihood based inference for the ratio of twonormal variances. The small sample likelihood inference is an approximation method.The signed log-likelihood ratio statistic and the modi ed signed log-likelihood ratiostatistic, which converge to standard normal distribution, are proposed for the normalvariance ratio. Through the simulation study, the coverage probabilities of con denceinterval and power of the exact, the signed log-likelihood and the modi ed signed log-likelihood ratio statistic will be compared. A real data example will be provided.Keywords: Likelihood based inference, modi ed signed log-likelihood ratio statistic,normal variance ratio, signed log-likelihood ratio statistic. 1. Introduction The normal distribution plays an important role in statistical inference, and there are somany studies about this distribution. The statistical inference for the ratio of two normalvariances arises in the areas for comparing the precision of two independent normal popula-tions. About the ratio of two normal variances, there also exists an exact statistical inferencewith Fstatistic. This problem is simple, obvious and important.When comparing the dispersion of two normal population, one can use the distributiontable of F statistic. Since F distribution depends on degrees of freedom, the distributiontable of Fdistribution is more than several pages. For the degrees of freedom exceed 30, thetable does not have the value. In this case, one must use an interpolation to obtaining thequantile.An approximation of a statistic to standard normal distribution has been developed inmany statistical models. Since the percentile of standard normal distribution is well known,a statistic, which distributes as standard normal distribution asymptotically, may be veryuseful. But these statistics have the error rate depending on sample size. When the samplesize is small, these approximation to standard normal distribution is quite inaccurate. Forexample, a signed log-likelihood ratio statistic converges to standard normal distributionwith an error of O(n