Research On Some Topics Related To Maximum Likelihood Estimation

Maximum likelihood (ML) estimation is an important and classical method of parameter estimation, and is always an active research field. This dissertation investigates some topics related to ML from three aspects, which enrich the research contents of ML by using various techniques. The main research works and contributions of this dissertation are outlined as follows.Firstly, we consider the singular normal linear modelwhere σ~2Σ is nonnegatively definite and |σ~2Σ|=0 (the parameters are β and σ~2), and the singular multivariate normal distributionwhere Σ is nonnegatively definite and |Σ| =0 (the parameters are β and Σ ). Usually,we obtain the ML estimation by solving the likelihood equations. However, the probability density functions of model (1) and (2) are hard to show directly. We notethat in probability 1, y - Xβ falls in the subspace span(Σ) spanned by the columnsof Σ in model (1) and y -μ. falls in the subspace span(Σ) spanned by thecolumns of Σ in model (2). So, by this way, we give the ML estimates of model (1) and model (2) respectively. And we also demonstrate the feasibility and effectivity of the estimators by simulated experiment.Secondly, we carry out researches on large sample size property of the root of log likelihood equation for Cauchy distribution. Taking into account that the ML estimation of Cauchy distribution have no explicit expression, we, based on the likelihood equation, prove that there exists a strongly consistent and asymptoticallynormal estimator (θ|^)_n, which, in probability 1, is the root of the log-likelihood...