Semi-parametric specification tests for mixing distributions

We present a semi-parametric method for testing mixing distributions in the mixed Poisson model. The proposed method, which is based on the generalized method of moments, does not demand the complete specification of the probability function but only requires a specification of a set of moment conditions which the model should satisfy. We demonstrate that an explicit expression for moment relations between the mixing and the mixed distributions provides a natural way in selecting moment restrictions and model parameterization. The Monte Carlo evidence suggests that the test has satisfactory performance for moderate size samples.     

Exponentiated beta distributions

Beta distributions are usually defined on the unit interval [0,1]. For wider applicability, we introduce six exponentiated beta distributions and derive several of their properties, including the moment generating function, expectation, variance, skewness, kurtosis, Shannon entropy, and the Rényi entropy.     

On the performance of the flexible maximum entropy distributions within partially adaptive estimation

The partially adaptive estimation based on the assumed error distribution has emerged as a popular approach for estimating a regression model with non-normal errors. In this approach, if the assumed distribution is flexible enough to accommodate the shape of the true underlying error distribution, the efficiency of the partially adaptive estimator is expected to be close to the efficiency of the maximum likelihood estimator based on knowledge of the true error distribution. In this context, the maximum entropy distributions have attracted interest since such distributions have a very flexible functional form and nest most of the statistical distributions. Therefore, several flexible MaxEnt distributions under certain moment constraints are determined to use within the partially adaptive estimation procedure and their performances are evaluated relative to well-known estimators. The simulation results indicate that the determined partially adaptive estimators perform well for non-normal error distributions. In particular, some can be useful in dealing with small sample sizes. In addition, various linear regression applications with non-normal errors are provided.     

The beta Burr XII distribution with application to lifetime data

For the first time, a five-parameter distribution, the so-called beta Burr XII distribution, is defined and investigated. The new distribution contains as special sub-models some well-known distributions discussed in the literature, such as the logistic, Weibull and Burr XII distributions, among several others. We derive its moment generating function. We obtain, as a special case, the moment generating function of the Burr XII distribution, which seems to be a new result. Moments, mean deviations, Bonferroni and Lorenz curves and reliability are provided. We derive two representations for the moments of the order statistics. The method of maximum likelihood and a Bayesian analysis are proposed for estimating the model parameters. The observed information matrix is obtained. For different parameter settings and sample sizes, various simulation studies are performed and compared in order to study the performance of the new distribution. An application to real data demonstrates that the new distribution can provide a better fit than other classical models. We hope that this generalization may attract wider applications in reliability, biology and lifetime data analysis.     

Inference for a simple step-stress model with competing risks for failure from the exponential distribution under time constraint

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under time constraint when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. Since it is found that the MLEs do not exist when there is no failure by any particular risk factor within the specified time frame, the exact sampling distributions of the MLEs are derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions, the parametric bootstrap method, and the Bayesian posterior distribution, we discuss the construction of confidence intervals and credible intervals for the parameters. Their performance is assessed through Monte Carlo simulations and finally, we illustrate the methods of inference discussed here with an example.     

Bivariate Beta–Inverse Weibull Distribution: Theory and Applications

Probability distributions have been in use for modeling of random phenomenon in various areas of life. Generalization of probability distributions has been the area of interest of several authors in the recent years. Several situations arise where joint modeling of two random phenomenon is required. In such cases the bivariate distributions are needed. Development of the bivariate distributions necessitates certain conditions, in a field where few work has been performed. This paper deals with a bivariate beta-inverse Weibull distribution. The marginal and conditional distributions from the proposed distribution have been obtained. Expansions for the joint and conditional density functions for the proposed distribution have been obtained. The properties, including product, marginal and conditional moments, joint moment generating function and joint hazard rate function of the proposed bivariate distribution have been studied. Numerical study for the dependence function has been implemented to see the effect of various parameters on the dependence of variables. Estimation of the parameters of the proposed bivariate distribution has been done by using the maximum likelihood method of estimation. Simulation and real data application of the distribution are presented.     

Bayesian computer-aided experimental design of heterogeneous scaffolds for tissue engineering

This paper presents a Bayesian methodology for computer-aided experimental design of heterogeneous scaffolds for tissue engineering applications. These heterogeneous scaffolds have spatial distributions of growth factors designed to induce and direct the growth of new tissue as the scaffolds degrade. While early scaffold designs have been essentially homogenous, new solid freeform fabrication (SFF) processes enable the fabrication of more complex, biologically inspired heterogeneous designs with controlled spatial distributions of growth factors and scaffold microstructures. SFF processes dramatically expand the number of design possibilities and significantly increase the experimental burden placed on tissue engineers in terms of time and cost. Therefore, we use a multi-stage Bayesian surrogate modeling methodology (MBSM) to build surrogate models that describe the relationship between the design parameters and the therapeutic response. This methodology is well suited for the early stages of the design process because we do not have accurate models of tissue growth, yet the success of our design depends on understanding the effect of the spatial distribution of growth factors on tissue growth. The MBSM process can guide experimental design more efficiently than traditional factorial methods. Using a simulated computer model of bone tissue regeneration, we demonstrate the advantages of Bayesian versus factorial methods for designing heterogeneous fibrin scaffolds with spatial distributions of growth factors enabled by a new SFF process.     

Cell population balance modeling and control in continuous bioreactors

In this work we address the problem of controlling different moments of the cell mass distribution using cell population balance modeling for the derivation of appropriate nonlinear feedback control laws. The mathematical model used for dynamic simulation and controller synthesis consists of a partial integro-differential equation describing the dynamics of the cell mass distribution and an ordinary integro-differential equation accounting for substrate consumption. Nonlinear feedback laws that induce a desired closed-loop response for the biomass concentration (first moment), the cell density (zeroth moment) and the second moment of the cell mass distribution are developed and their performance and robustness to parameter uncertainties are tested via numerical simulations.     

Exact likelihood inference for two exponential populations under joint Type-II censoring

Comparative lifetime experiments are of paramount importance when the object of a study is to ascertain the relative merits of two competing products in regard to the duration of their service life. In this paper, we discuss exact inference for two exponential populations when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the two exponential mean parameters. We then derive the moment generating functions and the exact distributions of these MLEs along with exact confidence intervals and simultaneous confidence regions. Moreover, simultaneous approximate confidence regions based on the asymptotic normality of the MLEs and simultaneous credible confidence regions from a Bayesian viewpoint are also discussed. A comparison of the exact, approximate, bootstrap and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference discussed here.     

A generalized gamma distribution with application to drought data

Gamma distributions are some of the most popular models for hydrological processes. In this paper, a very flexible family which contains the gamma distribution as a particular case is introduced. Evidence of flexibility is shown by examining the shape of its probability density function (pdf). A treatment of the mathematical properties is provided by deriving expressions for the n th moment. Estimation and simulation issues are also considered. Finally, a detailed application to drought data from the State of Nebraska is illustrated.