Studies Of Some Statistical Issues In Loglinear Models

When the data under considered are nominal or even ordinal,categorical data analysis is an essential and efficient tool to study them.It is a strongpoint for loglinear models that it is a useful tool to identify the association between various variables in multi-dimensional contingency tables.TheΦ-divergence measure which is used to measure the difference between two distributions is introduced by Csisz(?)r(1967) and Ali et al.(1966) respectively. In recent years,Φ-divergence measure has been used to various(regression) models incluing loglinear models under multinomial sampling by many statistics who defined the minimumΦ-divergence estimator(MΦE) and studied its properties.The minimumΦ-divergence estimator is a generalization of the MLE and has some kind of robustness.For the strongpoint of loglinear models and the robustness of the minimumΦ-divergence estimator,we forecast that the study in this field would last out for a period of time.For this point,we shall apply theΦ-divergence measure to product-multinomial loglinear models and could divide our work into three main parts.Firstly,we define the minimumΦ-divergence estimator under loglinear models with product-multinomial sampling,and study its properties and several kinds of hypothesis test problems including the goodness-of-fit test,the nested hypothesis test and the contiguous hypothesis test.Under certain conditions,we present the asymptotic expansion and normality of the minimumΦ-divergence estimator;based on the MΦE andΦ-divergence measure, various statistics are constructed and used to test whether the data are sampled from loglinear models with product-multinomial sampling and to decide which hypothesis is true in the nested hypotheses.An approximation to the power function of the goodness-of-fit test is given and these tests are consistent.Under a sequence of contiguous hypotheses,the asymptotic distribution of the statistics isχ~2 with some non-centrality parameter.Secondly,we also defined the restricted minimumΦ-divergence estimator(RMΦDE) under loglinear models with product-multinomial sampling,and study its properties and some kinds of hypothesis problems along with model diagnostic study.Under some conditions, the asymptotic expansion and normality of the restricted minimumΦ-divergence estimator is presented;based on the MΦE andΦ-divergence measure,various statistics are constructed and used to test whether the data are sampled from loglinear models with constraints under product-multinomial.In order to evaluate the power of the goodness-of-fit test an approximation to the power function is given and in conclusion the goodness-of-fit test is consistent.Under a sequence of contiguous hypotheses,the asymptotic distribution of the statistics isχ~2 with some non-centrality parameter.Further more,we use restricted minimumΦ-divergence estimator to do diagnostic study.Finally,we use the minimumΦ-divergence estimator andΦ-divergence measure to study the nonadditivity and model selection of loglinear models.Considering that loglinear models with product-multinomial sampling may not be adequate for our data under considered,we construct three kinds of statistics based on theΦ-divergence measure and minimumΦ-divergence estimator and use them to test the nonadditivity of loglinear models.Based on theΦ-divergence measure and minimumΦ-divergence estimator,a model selection procedure is putted forward and proven to be strong consistent.Moreover,the missing detection probability of this model selection procedure has an upper bound in an exponential version.