| Likelihood plays a critical role in addressing various types of questions in statistical inference, but difficulty arises when nuisance parameters are present. In this dissertation, we define any non-negative valued function of the data and the parameter of interest as a generic likelihood and attempt to identify what conditions will ensure the validity of a generic likelihood for statistical inference.; We define the concept of unbiasedness of a generic likelihood, and then extend it into asymptotic local (AL-) unbiasedness. In an axiomatic development based on an extension of the likelihood principle, we show that a generic likelihood is a valid basis for statistical inference if and only if it is unbiased. This axiomatic argument is supported by a number of desirable properties we derive for exactly or AL-unbiased generic likelihoods. Those properties provide further validation for unbiased generic likelihoods, mostly in probabilistic terms. In particular, we develop the theories of consistency and asymptotic normality for AL-unbiased generic likelihoods. These results are used to give a relatively simple proof for consistency and asymptotic normality for the partial likelihood in proportional hazards models.; It is found that many commonly used statistical approaches including those based on marginal, conditional and partial likelihoods, generalized estimating functions, and Edgeworth expansion, can be validated in terms of unbiasedness of the corresponding generic likelihoods. The concept of unbiasedness is also used to justify a latent variables approach that is used for inference from incomplete data when the missing data are not missing at random. Also, we propose a relatively general method for eliminating nuisance parameters to construct an unbiased generic likelihood. This method is illustrated with examples of generalized estimating functions and the pseudo-likelihood method in spatial statists.; Some fundamental issues in statistical inference including building and checking a statistical model, and obtaining useful, valid bases for statistical inference from a given statistical model are also discussed. To this end, we study decomposition of data generating mechanisms and establish the relationship between decomposition and unbiased generic likelihoods. Generalized residual squares are proposed for model checking.; Based on the developments in this dissertation, finally, we suggest representing the evidence of the observed data about the parameter of interest by the curve of an exactly or AL-unbiased generic likelihood. |
