For a cumulative link model in the Bayesian context, the posterior distribution cannot be obtained in closed form, and we have to resort to an approximation method. A simple data-augmentation strategy is widely used for that purpose but is known to work poorly. The marginal augmentation procedure and the parameter-expanded data-augmentation procedure are considered to be remedies, but such strategies are still not free from poor convergence. In this paper, we propose a kind of the hybrid Markov chain Monte Carlo strategy. To evaluate the efficiency, a local non-degeneracy is introduced, and we also provide a numerical simulation to show the effect.
A geographical weighted regression model can be used for visualizing or interpreting the covariate effects that vary with location. This model is usually estimated by a locally weighted regression or a kernel smoothing method, but we can regard the regression coefficients as varying linear coefficients that can be obtained from a global linear regression. There are two types of design vectors, one of which expresses linearity and the other is prepared for nonlinearity, i.e., it assumes a semiparametric surface with varying coefficients. Ridge estimators can then be used to suppress overfitting of the nonlinear part. With a mixed effects model, optimization of the ridge parameters and estimation of the regression parameters can be simultaneously executed. The linear structure of the varying coefficients then provides an asymptotic confidence interval as a function of location, but it is wider than a common pointwise confidence interval. We derive some tests for the varying coefficients and offer two examples using real data to illustrate our methodology. The results of the applied tests are summarized as the uniformity and the linearity of the varying coefficients.
We consider a linear regression model with a spatially correlated error term on a lattice. When estimating coefficients in the linear regression model, the generalized least squares estimator (GLSE) is used if the covariance structures are known. However, the GLSE for large spatial data sets is computationally expensive because of the matrix inversion. To reduce the computational complexity, we propose a pseudo best estimator (PBE) using spatial covariance structures approximated by separable covariance functions and derive its asymptotic covariance matrix. Monte Carlo simulations demonstrate that our proposed PBE performs well.
Proportions based on the binominal distribution are often compared in clinical tests. Biostatisticians often use the Fisher exact test in order to show the superiority of the binominal proportions of a test drug. Kawasaki and Miyaoka (2012) derived an accurate expression for a new index: θ = P(π1, post > π2, post | X1, X2) within a Bayesian framework. In this paper, we investigate the relationship between θ proposed by Kawasaki and Miyaoka (2012) and the p-value of Fisher's exact test (Fisher (1934)). We show that these two indicators are equivalent under certain conditions.
This study proposes a class of realized non-linear stochastic volatility models with asymmetric effects and generalized Student's t-error distributions by applying three families of power transformation—exponential, modulus, and Yeo-Johnson—to lagged log volatility. The proposed class encompasses a raw version of the realized stochastic volatility model. In the Markov chain Monte Carlo algorithm, an efficient Hamiltonian Monte Carlo (HMC) method is developed to update the latent log volatility and transformation parameter, whereas the other parameters that could not be sampled directly are updated by an efficient Riemann manifold HMC method. Empirical studies on daily returns and four realized volatility estimators of the Tokyo Stock Price Index (TOPIX) over 4-year and 8-year periods demonstrate statistical evidence supporting the incorporation of skew distribution into the error density in the returns and the use of power transformations of lagged log volatility.