Estimation of the Inverse Weibull Distribution Parameters under Type-I Hybrid Censoring

The hybrid censoring is a mixture of type-I and type-II censoring schemes. This paper presents the statistical inferences of the inverse Weibull distribution parameters when the data are type-I hybrid censored. First, we consider the maximum likelihood estimates of the unknown parameters. It is observed that the maximum likelihood estimates can not be obtained in closed form. We further obtain the Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters under the assumption of independent gamma priors using the importance sampling procedure. We also compute the approximate Bayes estimates using Lindley's approximation technique. The performance of the Bayes estimates have been compared with maximum likelihood estimates through the Monte Carlo Markov chain techniques. Finally, a real data set have been analysed for illustration purpose.

The Kumaraswamy Pareto IV Distribution

We introduce a new model named the Kumaraswamy Pareto IV distribution which extends the Pareto and Pareto IV distributions. The density function is very flexible and can be left-skewed, right-skewed and symmetrical shapes. It hasincreasing, decreasing, upside-down bathtub, bathtub, J and reversed-J shaped hazard rate shapes. Various structural properties are derived including explicit expressions for the quantile function, ordinary and incomplete moments,Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, probability weighted moments and generating function. We provide the density function of the order statistics and their moments. The Renyi and q entropies are also obtained. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of three real-life data sets. In fact, our proposed model provides a better fit to these data than the gamma-Pareto IV, gamma-Pareto, beta-Pareto,exponentiated Pareto and Pareto IV models.

Different Classical Methods of Estimation and Chi-squared Goodness-of-fit Test for Unit Generalized Inverse Weibull Distribution

In this paper, we try to contribute to the distribution theory literature by incorporating a new bounded distribution, called the unit generalized inverse Weibull distribution (UGIWD) in the (0, 1) intervals by transformation method. The proposed distribution exhibits increasing and bathtub shaped hazard rate function. We derive some basic statistical properties of the new distribution. Based on complete sample, the model parameters are obtained by the methods of maximum likelihood, least square, weighted least square, percentile, maximum product of spacing and Cram`er-von-Mises and compared them using Monte Carlo simulation study. In addition, bootstrap confidence intervals of the parameters of the model based on aforementioned methods of estimation are also obtained. We illustrate the performance of the proposed distribution by means of one real data set and the data set shows that the new distribution is more appropriate as compared to unit Birnbaum-Saunders, unit gamma, unit Weibull, Kumaraswamy and unit Burr III distributions. Further, we construct chi-squared goodness-of-fit tests for the UGIWD using right censored data based on Nikulin-Rao-Robson (NRR) statistic and its modification. The criterion test used is the modified chi-squared statistic Y^2, developedby Bagdonavi?ius and Nikulin, 2011 for some parametric models when data are censored. The performances of the proposed test are shown by an intensive simulation study and an application to real data set

A Bivariate Index Vector to Measure Departure from Quasi-symmetry for Ordinal Square Contingency Tables

This study proposes a bivariate index vector to concurrently analyze both the degree and direction of departure from the quasi-symmetry (QS) model for ordinal square contingency tables. The QS model and extended QS (EQS) models identify the symmetry and asymmetry between the probabilities of normal circulation and reverse circulation when the order exists for arbitrary three categories. The asymmetry parameter of the EQS model implies the degree of departure from the QS model; the EQS model is equivalent to the QS model when the asymmetry parameter equals to one. The structure of the EQS model differs depending on whether the asymmetry parameter approaches zero or infinity. Thus, the asymmetry parameter of the EQS model also implies the direction of departure from the QS model. The proposed bivariate index vector is constructed by combining existing and original sub-indexes that represent the degree of departure from the QS model and its direction. These sub-indexes are expressed as functions of the asymmetry parameter under the EQS model. We construct an estimator of the proposed bivariate index vector and an approximate confidence region for the proposed bivariate index vector. Using real data, we show that the proposed bivariate index vector is important to compare degrees of departure from the QS model for plural data sets.

Biclustering Analysis Using Plaid Model on Gene Expression Data of Colon Cancer

Unlike other typical clustering analysis, which considers column only, biclustering analysis processes a matrix into sub-matrices based on rows and columns simultaneously. One method of bicluster analysis uses the probabilistic model, like the plaid model, that provides overlapping bicluster. The plaid model calculates the value of an element given from a particular sub-matrix for each cell; thus, the value can be seen as the number of contributions of a particular bicluster. The algorithm begins with preparing the input data as a matrix, then an initial model is assessed and makes a residual matrix from the model. After that, we determine bicluster candidates, which are evaluated for its effect parameters and bicluster membership parameters. Finally, the bicluster candidate is pruned to give the optimal bicluster. We implemented the algorithm on gene expression dataset of colon cancer, where the rows and columns contain observations and types of genes, respectively. We carried out in six distinct scenarios in which each scenario uses different model parameters and threshold values. We measured the results using Jaccard index and coherence variance. Our experiments show that biclustering analysis on a model with mean, row, and column effects of colon cancer data output low coherence variance.

Generalized Topp-Leone-Weibull AFT Modelling: A Bayesian Analysis with MCMC Tools using R and Stan

The generalized Topp-Leone-Weibull (GTL-W) distribution is a generalization of Weibull distribution which is obtained by using generalized Topp-Leone (GTL) distribution as a generator and considering Weibull distribution as a baseline distribution. Weibull distribution is a widely used survival model that has monotone- increasing or decreasing hazard. But it cannot accommodate bathtub shaped and unimodal shaped hazards. As a survival model, GTL-W distribution is more flexible than the Weibull distribution to accommodate different types of hazards. The present study aims at fitting GTL-W model as an accelerated failure time (AFT) model to censored survival data under Bayesian setting using R and Stan languages. The GTL-W AFT model is compared with its sub-model and the baseline model. The Bayesian model selection criteria LOOIC and WAIC are applied to select the best model.

Construction of Windows for Pharmacokinetic Sampling

This paper describes a method for the construction of pharmacokinetic sampling windows so that they are around the $D$-optimum time points. Here we consider the situation where a pharmacokinetic (PK) study is accompanied by a dose-finding study in phase I clinical trial. The D-optimal criterion is often used to determine the optimal time for collecting blood samples so that they provide maximum information regarding the population PK parameters. However, collecting blood samples at the D-optimal time points is often difficult. Instead, the sampling time point chosen from a suitable time interval or window can ease the process. The proposed method is conceptually simple and considers the average value and standard deviation of D-optimal time points up to create sampling windows. Random time points can be chosen from these windows then to collect blood samples from the next cohort. The nonlinear random-effects model has been used to model the PK data. Also, we employ the continual reassessment method for dose allocation to the patients. Comparisons of the accuracy and precision for the PK parameter estimates obtained at the D-optimal and random time points are also presented. The results are convincing enough to suggest the proposed method as a useful tool for blood sample collection.

On the Mean Residual Life of Cantor-Type Distributions: Properties and Economic Applications

In this paper, we consider the mean residual life (MRL) function of the Cantor distribution and study its properties. We show that the MRL function is continuous at all points, locally decreasing at all points outside the Cantor set and has a unique fixed point which we explicitly determine. These properties readily extend to the parametric family of p-singular, Cantor type distributions introduced by Mandelbrot (1983). The findings offer evidence that, contrary to common perceptions, Cantor-type distributions are tractable enough to be considered for practical applications. We provide such an example from the field of economics in which Cantor-type distributions can be used to model markets with recurrent bandwagon effects and show that earlier anticipated bandwagon effects lead to higher monopolistic prices. We conclude with a simple implementation of the algorithm by Chalice (1991) to plot Cantor-type distributions.

A Generalized Estimating Equations Approach for Modeling Spatially Clustered Data

Clustering in spatial data is very common phenomena in various fields such as disease mapping, ecology, environmental science and so on. Analysis of spatially clustered data should be different from conventional analysis of spatial data because of the nature of clusters in the data. Because it is expected that the observations of same cluster are more similar than the observations from different clusters. In this study, a method has been proposed for the analysis of spatially clustered areal data based on generalized estimating equations which were originally developed for analyzing longitudinal data. The performance of the model for known clusters is tested in terms of how well it estimates the regression parameters and how well it captures the true spatial process. These results are presented and compared with the conditional auto-regressive model which is the most frequently used spatial model. In the simulation study, the proposed generalized estimating equations approach yields better results than the popular conditional auto-regressive model from the both perspectives of parameter estimation and spatial process capturing. A real life data on the vitamin A supplement coverage among postpartum women in Bangladesh is then analyzed for demonstration of the method. The existing divisional clustering behavior of vitamin A supplement coverage in Bangladesh is identified more accurately by the proposed approach than that by the conditional auto-regressive model.

New Distribution for Fitting Discrete Data: The Poisson-Gold Distribution and Its Statistical Properties

Motivated mainly by lifetime issues, a new lifetime distribution coined ``Discrete Poisson-Gold distribution'' is introduced in this paper. Different structural properties of the new distribution are derived including moment generating function and the $r^{th}$ moment and others are presented. In addition, we discussed various important mathematical properties of the new distribution including estimation procedures for estimating the distribution parameters using the maximum likelihood and method of moments. The usefulness and credibility of the distribution are illustrated by means of two real-data applications to show its superior performance over some other well-known lifetime distributions and to prove its versatility in practical applications.