Likelihood ratio statistics for (i) testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped (cf. Shaked, Ann. Statist., 1979), and for (ii) testing the null hypothesis that this parameter vector is starshaped, are considered. For both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the Chi-bar-square distribution. Analogous tests on a collection of Poisson means are also found to have asymptotic Chi-bar-square distributions.
"Order Restricted Statistical Tests on Multinomial and Poisson Parameters: The Starshaped Restriction." Ann. Statist. 10 (4) 1246 - 1252, December, 1982. https://doi.org/10.1214/aos/1176345989