Analysis of marginal and conditional density functions for separate inference

Summary This paper discusses, with measure-theoretical rigor, some basic aspects of the theory of separate inference. To analyze densities of marginal and conditional submodels, certain operators are introduced. First a general concept of decomposition of a model is proposed, and the corresponding factorization of densities of the model is established. Next it is shown that the property of smoothness of a family of densities is retained in the operation of conditioning, and therefore it yields the differentiability of the conditional expectation of a real-valued statistic in a certain sense. On the basis of this result, two measures of the effectiveness of a submodel in separate inference are investigated. The Institute of Statistical Mathematics     

Estimation for partially linear models with missing responses: the fixed design case

Suppose that we have a partially linear model Yi=x′iβ+g(ti)+εi with independent zero mean errorsεi,where{xi,ti,i=1,···,n}are non-random and observed completely and{Yi,i=1,···,n}are missing at random(MAR).Two types of estimators ofβand g(t)for fixed t are investigated:estimators based on semiparametric regression and inverse probability weighted imputations.Asymptotic normality of the estimators is established,which is used to construct normal approximation based confidence intervals onβand g(t).Results are reported of a simulation study on the finite sample performance of the estimators and confidence intervals proposed in this paper.     

Inference for earthquake models: A self-correcting model

Questions of asymptotic inference are discussed for a point process model in which the conditional intensity function increases monotonically between events and drops by determined (nonrandom) amounts after each event. Parameter estimates are shown to be consistent and, except under the null hypothesis of a Poisson process, normally distributed. Under the null hypothesis, however, the Hessian matrix is not asymptotically constant, and the limiting distribution of the likelihood ratio statistics is not χ², but has a form related to that of the Cramer-von Mises ω² statistic for the test of goodness of fit.