In this paper, we deal with modeling crisp input-fuzzy output data by constructing a MARS-fuzzy regression model with crisp parameters estimation and fuzzy error terms for the fuzzy data set. The proposed method is a two-phase procedure which applies the MARS technique at phase one and an optimization problem at phase two to estimate the center and fuzziness of the response variable. A realistic application of the proposed method is also presented in a hydrology engineering problem. Empirical results demonstrate that the proposed approach is more efficient and more realistic than some traditional least-squares fuzzy regression models.

Jaynes' principle of maximum entropy states that among all the probability distributions satisfying some constraints, one should be selected which has maximum uncertainty. In this paper, we consider the methods of obtaining maximum entropy bivariate density functions via Taneja and Burg's measure of entropy under the constraints that the marginal distributions and correlation coefficient are prescribed. Next, a numerical method is considered. Finally, each method is illustrated via a numerical example.

This paper considers the problem of simultaneous variable selection and estimation in a semiparametric mixed-effects model for longitudinal data with normal errors. We approximate the nonparametric function by regression spline and simultaneously estimate and select the variables under the optimization of the penalized objective function. Under some regularity conditions, the asymptotic behaviour of the resulting estimators is established in a high-dimensional framework where the number of parametric covariates increases as the sample size increases. For practical implementation, we use an EM algorithm to selects the significant variables and estimates the nonzero coefficient functions. Simulation studies are carried out to assess the performance of our proposed method, and a real data set is analyzed to illustrate the proposed procedure.