A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

Nearly optimal covariate designs—Part II

This is a continuation to Part I toward our efforts for providing illustrative examples in the context of analysis of covariance (ANCOVA) models and related analyses of data. We discuss four more examples here, and these are derived from standard textbooks. We re-visit these examples with a view to suggest optimal/nearly optimal designs for estimation of the covariate parameter(s).     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.     

On a Skew Bimodal Normal–Normal distribution fitted to the Old-Faithful geyser data

The Bimodal Normal distribution introduced by Alavi (2011) is a symmetric distribution where its variance is three times the variance of the corresponding normal distribution. Azzalini (1985) introduced the univariate Skew Normal distribution to model asymmetry data. In this paper the Skew Bimodal Normal–Normal distribution is introduced as a skew-symmetric distribution generated by the cumulative function of standard normal. Some properties of the distribution and some methods for generating data from this distribution are introduced. The maximum likelihood estimation of parameters is obtained. The distribution is fitted to the Old Faithful Geyser data.     

Data‐driven choice of the smoothing parametrization for kernel density estimators

There are several levels of sophistication when specifying the bandwidth matrix H to be used in a multivariate kernel density estimator, including H to be a positive multiple of the identity matrix, a diagonal matrix with positive elements or, in its most general form, a symmetric positive‐definite matrix. In this paper, the author proposes a data‐based method for choosing the smoothing parametrization to be used in the kernel density estimator. The procedure is fully illustrated by a simulation study and some real data examples. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Automatic and asymptotically optimal data sharpening for nonparametric regression

In this article we consider data-sharpening methods for nonparametric regression. In particular modifications are made to existing methods in the following two directions. First, we introduce a new tuning parameter to control the extent to which the data are to be sharpened, so that the amount of sharpening is adaptive and can be tuned to best suit the data at hand. We call this new parameter the sharpening parameter. Second, we develop automatic methods for jointly choosing the value of this sharpening parameter as well as the values of other required smoothing parameters. These automatic parameter selection methods are shown to be asymptotically optimal in a well defined sense. Numerical experiments were also conducted to evaluate their finite-sample performances. To the best of our knowledge, there is no bandwidth selection method developed in the literature for sharpened nonparametric regression.     

Functional principal component analysis via regularized Gaussian basis expansions and its application to unbalanced data

This paper introduces regularized functional principal component analysis for multidimensional functional data sets, utilizing Gaussian basis functions. An essential point in a functional approach via basis expansions is the evaluation of the matrix for the integral of the product of any two bases (cross-product matrix). Advantages of the use of the Gaussian type of basis functions in the functional approach are that its cross-product matrix can be easily calculated, and it creates a much more flexible instrument for transforming each individual's observation into a functional form. The proposed method is applied to the analysis of three-dimensional (3D) protein structural data that can be referred to as unbalanced data. It is shown that our method extracts useful information from unbalanced data through the application. Numerical experiments are conducted to investigate the effectiveness of our method via Gaussian basis functions, compared to the method based on B-splines. On performing regularized functional principal component analysis with B-splines, we also derive the exact form of its cross-product matrix. The numerical results show that our methodology is superior to the method based on B-splines for unbalanced data.