Inference on order restricted means of inverse Gaussian populations under heteroscedasticity

Abstract

The hypothesis testing problem of homogeneity of k(≥2) inverse Gaussian means against ordered alternatives is studied when nuisance or scale-like parameters are unknown and unequal. The maximum likelihood estimators (MLEs) of means and scale-like parameters are obtained when means satisfy some simple order restrictions and scale-like parameters are unknown and unequal. An iterative algorithm is proposed for finding these estimators. It has been proved that under a specific condition, the proposed algorithm converges to the true MLEs uniquely. A likelihood ratio test and two simultaneous tests are proposed. Further, an algorithm for finding the MLEs of parameters is given when means are equal but unknown. Using the estimators, the likelihood ratio test is developed for testing against ordered alternative means. Using the asymptotic distribution, the asymptotic likelihood ratio test is proposed. However, for small samples, it does not perform well. Hence, a parametric bootstrap likelihood ratio test (PB LRT) is proposed. Therefore, the asymptotic validity of the bootstrap procedure has been shown. Using the Box-type approximation method, test statistics are developed for the two-sample problem of equality of means when scale-like parameters are heterogeneous. Using these, two PB-based heuristic tests are proposed. Asymptotic null distributions are derived and PB accuracy is also developed. Two asymptotic tests are also proposed using the asymptotic null distributions. To get the critical points and test statistics of the three PB tests and two asymptotic tests, an ‘R’ package is developed and shared on GitHub. Applications of the tests are illustrated using real data.