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.