An Application of EM Algorithm to a Change-Point Problem

We consider the problem of change-point in a classical framework while assuming a probability distribution for the change-point. An EM algorithm is proposed to estimate the distribution of the change-point. A change-point model for multiple profiles is also proposed, and EM algorithm is presented to estimate the model. Two examples of Illinois traffic data and Dow Jones Industrial Averages are used to demonstrate the proposed methods.     

Bayesian decision theoretic approach to hypothesis problems with skewed alternatives

Many hypothesis problems in practice require the selection of the left side or the right side alternative when the null is rejected. For parametric models, this problem can be stated as _H0:θ=_θ0$H_0:\theta ={\theta }_0$vs.  _H−:θ<_θ0$H_{-}:\theta <{\theta }_0$ or _H+:θ>_θ0$H_{+}:\theta >{\theta }_0$. Frequentists use Type-III error (directional error) to develop statistical methodologies. This approach and other approaches considered in the literature do not take into account the situations where the selection of one side may be more important or when one side may be more probable than the other. This problem can be tackled by specifying a loss function and/or by specifying a hierarchical prior structure with allowing the skewness in the alternatives. Based on this, we develop a Bayesian decision theoretic methodology and show that the resulted Bayes rule perform better in the side of the alternatives which is more probable. The methodology can be also used in a frequentist's framework when it is desired to discover an alternative that is more important. We also consider the multiple hypotheses problem and develop new false discovery rates for the selection of the left and the right sides of alternatives. These discovery rates would be useful in the situations when one side of the alternatives are more important or more probable than the other.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

ROBUST M-ESTIMATION OF THE INTRACLASS CORRELATION COEFFICIENT

Robust M-estimators of intraclass correlation coefficient, location and scale parameters are defined for familial data. It is shown that these estimators are strongly consistent. Also the asymptotic distributions of these estimators are derived when the underlying distribution is elliptically and permutationally symmetric.     

Bayes Estimation of Intraclass Correlation Coefficients Under Unequal Family Sizes

Bayesian inference for the intraclass correlation ρ is considered under unequal family sizes. We obtain the posterior distribution of ρ and then compare the performance of the Bayes estimator (posterior mean of ρ) with that of Srivastava's (1984) estimator through simulation. Simulation study shows that the Bayes estimator performs better than the Srivastava's estimator in terms of lower mean square error. We also obtain large sample posteriors of ρ based on the asymptotic posterior distribution and based on the Laplace approximation.     

Bayesian analysis of hypothesis testing problems for general population: A Kullback–Leibler alternative

We consider a hypothesis problem with directional alternatives. We approach the problem from a Bayesian decision theoretic point of view and consider a situation when one side of the alternatives is more important or more probable than the other. We develop a general Bayesian framework by specifying a mixture prior structure and a loss function related to the Kullback–Leibler divergence. This Bayesian decision method is applied to Normal and Poisson populations. Simulations are performed to compare the performance of the proposed method with that of a method based on a classical z-test and a Bayesian method based on the “0–1” loss.     

Detection of outliers in familial data

Procedures for detection of outliers in familial data is given for mean-slippage and dispersion-slippage model of outliers for equal and unequal family sizes. The distributions of the test statistics are derived and Bonferroni's bounds for the values of significant probabilities are given.