Large Sample Test Procedures

In Chaps. 2, 3, and 4, we discussed point estimation of a parameter and studied the large sample optimality properties of the estimators. We also discussed interval estimation for large n. The present and the next chapters are devoted to the large sample test procedures. All the results about the estimators established in Chaps. 2, 3, and 4 are heavily used in both the chapters. Most of the theory of testing of hypotheses has revolved around the Neyman-Pearson lemma, which leads to the most powerful test for simple null against simple alternative hypothesis. It also leads to the uniformly most powerful tests in certain models, in particular for exponential families. A likelihood ratio test procedure, which we discuss in the second section, is also an extension of Neyman-Pearson lemma in some sense. Likelihood ratio test procedure is the most general test procedure when the parameter space is either a subset of $$\mathbb {R}$$ R or $$\mathbb {R}^k$$ R k . Whenever an optimal test exists, such as the most powerful test, uniformly most powerful test, uniformly most powerful unbiased test, the likelihood ratio test procedure leads to the optimal test procedure. In Chap. 5, we discuss likelihood ratio test procedure when the null hypothesis is simple or composite. We derive the asymptotic null distribution of the test statistic in all such cases under the assumption that the probability law $$f(x, \theta )$$ f ( x , θ ) belongs to Cramér family. In the last section we illustrate use of R software in large sample test procedures and in likelihood ratio test procedures.