1 MLE for Goodness - of Fit

“Chi-squared tests” refers to a family of statistical test whose large-sample asymptotics are related in one way or another to the χ family of distributions. These are our first examples of “non-parametric procedures.” Let me mention one particularly useful chi-squared test here before we get started properly. Consider a population whose distribution is believed to follow a known density function f—say, exponential(12). Then there is a chi-squared test that can be used to check our hypothesis. This chi-squared test is easy to implement, and works well for large-sized samples. First, we study a related parametric estimation problem. Then we find a consistent MLE. And finally, we use our estimate to devise a test for our hypothesis. This will be done in several stages. It is likely that you already know something of the resulting “χ-test” (page 7). But here you will find a rigorous proof, in line with the spirit of this course, of the fact that the said χ-test actually works. A [convincing but] non-rigorous justification of the same fact can be found in pages 314–316 of the excellent monograph of P. Bickel and K. Doksum [Mathematical Statistics, Holden Day, First edition, 1977]. There, a likelihood principle argument is devised, which “usually” works.