If the form of the distribution of data is unknown, the Bayesian method fails in the parametric inference because there is no posterior distribution of the parameter. In this paper, a theoretical framework of Bayesian likelihood is introduced via the Hilbert space method, which is free of the distributions of data and the parameter. The posterior distribution and posterior score function based on given inner products are defined and, consequently, the quasi posterior distribution and quasi posterior score function are derived, respectively, as the projections of the posterior distribution and posterior score function onto the space spanned by given estimating functions. In the space spanned by data, particularly, an explicit representation for the quasi posterior score function is obtained, which can be derived as a projection of the true posterior core function onto this space. The methods of constructing conservative quasi posterior score and quasi posterior log-likelihood are proposed. Some examples are given to illustrate the theoretical results. As an application, the quasi posterior distribution functions are used to select variables for generalized linear models. It is proved that, for linear models, the variable selections via quasi posterior distribution functions are equivalent to the variable selections via the penalized residual sum of squares or regression sum of squares.

In order to construct estimating functions in some parametric models, this paper introduces two classes ofinformation matrices. Some necessary and suficient conditions for the information matrices achieving their upper bounds are given. For the problem of estimating the median, some optimum estimating functions based on the informa tion matrices are acquired. Under some regularity conditions, an approachtocarrying outthebestbasisfunct ionisintro duced. In nonlinear regression models, an optimum estim ating function based on the inform ation m atrices isobtained. Some exam ples aregivento illustratethe results. Finally, theco ncept of optimum estim atingfunct ionan dthem ethodsofco ns tructing optimum estimating functionar edeveloped in more general statistical models.

In the nonparametric regression models, the original regression esti-mators including kernel estimator, Fourier series estimator and wavelet estimator are always constructed by weighted sum of data and the weights depend only on the distance between the design points and estimation points. As a result these esti-mators are not robust to the perturbations in data. In order to avoid this problem, a new nonparametric regression model, called depth-weighted regression model, is introduced and then depth-weighted wavelet estimation is defined. The new estima-tion is robust to the perturbations in data, which attains very high breakdown value close to 1/2. On the other hand some asymptotic behaviours such as asymptotic normality are obtained. But the asymptotic normality indicates that, as a price to pay for robustness, the depth-weighted wavelet estimation is less efficient than the original wavelet estimation.