Hypothesis Testing in the 20th Century with a Special Reference to Testing with Misspecified Models

This century and the history of modern statistics began with Karl Pearson's, [181], (1900)goodness-of-fit test, one of the most important breakthroughs in science. The basic motivation behind this test was to see whether an assumed probability model adequately described the data at hand. Then, over the first half of this century we saw the developments of some general principles of testing, such as Jerzy Neyman and Egon Pearson's,[159], (1928) likelihood ratio test, Neyman's,[155], (1937) smooth test, Abraham Wald's test in 1943 and C.R. Rao's score test in 1948. All these tests were developed under the assumption that the underlying model is correctly specified. Trygve Haavelmo,[99], (1944) termed this underlying model as the priori admissible hypothesis. Although Ronald Fisher,[80], (1922) identified the "Problem of Specification" as one of the most fundamental problems in statistics much earlier, Haavelmo was probably the first to draw the attention to the consequences of misspecification of the priori admissible hypothesis on the standard hypothesis testing procedures. We will call this the type-III error. In this paper, we will deal with a number of ways that an assumed probability model can be misspecified, and discuss how some of the standard tests could be modified to make them valid under various misspecification