Model discrimination is one of the most extensively discussed problems in statistics. In this thesis, we have defined and calculated the Chernoff index for two-sample problems, in order to tackle the non-parametric terms in retrospective sampling. We develop the profile likelihood l*(θ;x₁,⋅⋅⋅ ,x_n) and use it as test statistic for semi-parametric model discrimination tests. As for such tests, it is proved that in non-directional cases, the two types of errors would decay exponentially at the same rate. Moreover the asymptotic distribution for the test statistic is found, which enables one to construct directional tests for large samples according to a given significance level α. Then the type I and type II error probabilities for the directional tests are also studied. And it is fo...[ Read more ]

Model discrimination is one of the most extensively discussed problems in statistics. In this thesis, we have defined and calculated the Chernoff index for two-sample problems, in order to tackle the non-parametric terms in retrospective sampling. We develop the profile likelihood l*(θ;x₁,⋅⋅⋅ ,x_n) and use it as test statistic for semi-parametric model discrimination tests. As for such tests, it is proved that in non-directional cases, the two types of errors would decay exponentially at the same rate. Moreover the asymptotic distribution for the test statistic is found, which enables one to construct directional tests for large samples according to a given significance level α. Then the type I and type II error probabilities for the directional tests are also studied. And it is found that when symmetry is broken, not only the Chernoff efficiency could no longer be achieved, but also the decay rates of two types of errors become unequal. An example of logistic model versus probit model is investigated and some simulations are done to illustrate the theoretical results. Lastly two potential topics are proposed for further research.