Non-Regular Statistical Estimation

1. General discussions on unbiased estimation.- 1.1. Formulation.- 1.2. Undominated case.- 1.3. The support depending on the parameter.- 1.4. Discrete parameter set.- 1.5. Discontinuous and non-differentiable density.- 1.6. Non square-integrable likelihood ratio.- 1.7. Asymptotic theory for non-regular cases.- 1.8. Asymptotic Bayes posterior distribution of the parameter in non-regular cases.- 1.9. Overview.- 2. Lower bound for the variance of unbiased estimators.- 2.1. Minimum variance.- 2.2. Bhattacharyya type bound for the variance of unbiased estimators in non-regular cases.- 2.3. Lower bound for the variance of unbiased estimators for one-directional distributions.- 2.4. A second type of one-directional distribution and the lower bound for the variance of unbiased estimators.- 2.5. Locally minimum variance unbiased estimation.- 3. Amounts of information and the minimum variance unbiased estimation.- 3.1. Fisher information and the minimum variance unbiased estimation.- 3.2. Examples on unbiased estimators with zero variance.- 3.3. A definition of the generalized amount of information.- 3.4. Examples on the generalized amount of information.- 3.5. Order of consistency.- 4. Loss of information associated with the order statistics and related estimators in the case of double exponential distributions.- 4.1. Loss of information of the order statistics.- 4.2. The asymptotic loss of information.- 4.3. Proofs of Theorems in Section 4.2.- 4.4. Discretized likelihood estimation.- 4.5. Second order asymptotic comparison of the discretized likelihood estimator with others.- 5. Estimation of a common parameter for pooled samples from the uniform distributions and the double exponential distributions.- 5.1. Estimators of a common parameter for the uniform distributions.- 5.2. Comparison of the quasi-MLE, the weighted estimator and others for the uniform distributions.- 5.3. Estimators of a common parameter for the double exponential distributions.- 5.4. Second order asymptotic comparison of the estimators for the double exponential distributions.- 6. Higher order asymptotics in estimation for two-sided Weibull type distributions.- 6.1. The 2?-th order asymptotic bound for the distribution of 2?-th order AMU estimators.- 6.2. Proofs of Lemmas and Theorem in Section 6.1.- 6.3. The 2?-th order asymptotic distribution of the maximum likelihood estimator.- 6.4. The amount of the loss of asymptotic information of the maximum likelihood estimator.- 7. “3/2-th” and second order asymptotics of the generalized Bayes estimators for a family of truncated distributions.- 7.1. Definitions and assumptions.- 7.2. Generalized Bayes estimators for a family of truncated distributions.- 7.3. Second order asymptotic bound in symmetrically truncated densities.- 7.4. Maximum probability estimation.- 7.5. Examples.- 7.6. Some remarks.- Supplement. The bound for the asymptotic distribution of estimators when the maximum order of consistency depends on the parameter.- 5.1. Order of consistency depending on the parameter.- 5.2. The bound for the asymptotic distribution of AMU estimators in the autoregressive process.- References.