To test a null hypothesis of homogeneity of parameters versus an order restricted alternative a popular method is to use a restricted likelihood ratio test. An alternative approach which oftentimes has desirable properties is a directed likelihood ratio test. In this paper, we give sufficient conditions under which these approaches coincide. We indicate cases where they do not coincide. One such case is where the parameters satisfy the lower star-shaped restriction and the underlying random variables are independent with Poisson distributions. For this model, we study and prefer the directed likelihood ratio test. We also propose estimators of the parameters which we feel are more desirable than the maximum likelihood estimators. © 2002 Elsevier Science B.V. All rights reserved.
Cohen, A., & Sackrowitz, H. B. (2002). Directed likelihood ratio tests for ordered alternatives and inference results for the star-shaped restriction. Journal of Statistical Planning and Inference, 103(1–2), 181–189. https://doi.org/10.1016/S0378-3758(01)00220-8