An algorithm for the Buckley–James estimator with interval-censored data

The Buckley–James estimator (BJE) [J. Buckley and I. James, Linear regression with censored data, Biometrika 66 (1979), pp. 429–436] has been extended from right-censored (RC) data to interval-censored (IC) data by Rabinowitz et al. [D. Rabinowitz, A. Tsiatis, and J. Aragon, Regression with interval-censored data, Biometrika 82 (1995), pp. 501–513]. The BJE is defined to be a zero-crossing of a modified score function H(b), a point at which H(·) changes its sign. We discuss several approaches (for finding a BJE with IC data) which are extensions of the existing algorithms for RC data. However, these extensions may not be appropriate for some data, in particular, they are not appropriate for a cancer data set that we are analysing. In this note, we present a feasible iterative algorithm for obtaining a BJE. We apply the method to our data.     

Kolmogorov–Smirnov test for life test data with hybrid censoring

This work considers goodness-of-fit for the life test data with hybrid censoring. An alternative representation of the Kolmogorov–Smirnov (KS) statistics is provided under Type-I censoring. The alternative representation leads us to approximate the limiting distributions of the KS statistic as a functional of the Brownian bridge for Type-II, Type-I hybrid, and Type-II hybrid censored data. The approximated distributions are used to obtain the critical values of the tests in this context. We found that the proposed KS test procedure for Type-II censoring has more power than the available one(s) in literature.     

Expectation–maximization algorithm for system-based lifetime data with unknown system structure

AStA Advances in Statistical Analysis - In science and engineering, we are often interested in learning about the lifetime characteristics of the system as well as those of the components that made...     

Time-varying coefficient models with ARMA–GARCH structures for longitudinal data analysis

Time-varying coefficient models with autoregressive and moving-average–generalized autoregressive conditional heteroscedasticity structure are proposed for examining the time-varying effects of risk factors in longitudinal studies. Compared with existing models in the literature, the proposed models give explicit patterns for the time-varying coefficients. Maximum likelihood and marginal likelihood (based on a Laplace approximation) are used to estimate the parameters in the proposed models. Simulation studies are conducted to evaluate the performance of these two estimation methods, which is measured in terms of the Kullback–Leibler divergence and the root mean square error. The marginal likelihood approach leads to the more accurate parameter estimates, although it is more computationally intensive. The proposed models are applied to the Framingham Heart Study to investigate the time-varying effects of covariates on coronary heart disease incidence. The Bayesian information criterion is used for specifying the time series structures of the coefficients of the risk factors.     

Estimation of parameters for a Birnbaum–Saunders regression model with censored data

Little work has been published on the analysis of censored data for the Birnbaum–Saunders distribution (BISA). In this article, we implement the EM algorithm to fit a regression model with censored data when the failure times follow the BISA. Three approaches to implement the E-Step of the EM algorithm are considered. In two of these implementations, the M-Step is attained by an iterative least-squares procedure. The algorithm is exemplified with a single explanatory variable in the model.     

A semi-parametric method for transforming data to normality

A non-parametric transformation function is introduced to transform data to any continuous distribution. When transformation of data to normality is desired, the use of a suitable parametric pre-transformation function improves the performance of the proposed non-parametric transformation function. The resulting semi-parametric transformation function is shown empirically, via a Monte Carlo study, to perform at least as well as any parametric transformation currently available in the literature.     

A new local estimation method for single index models for longitudinal data

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They are very useful for longitudinal data analysis. In this paper, we develop a new efficient estimation procedure for single index models with longitudinal data, based on Cholesky decomposition and local linear smoothing method. Asymptotic normality for the proposed estimators of both the parametric and nonparametric parts will be established. Monte Carlo simulation studies show excellent finite sample performance. Furthermore, we illustrate our methods with a real data example.     

Multiple comparisons with ‘best’ for multivariate normal populations

Comparisons of multivariate normal populations are made using a mul-tivariate approach (instead of reducing the problem to a univariate one). A rather negative finding is that, for comparisons with the ‘best’ of each variate, repeated univariate comparisons appear to be almost as efficient as multivariate comparisons, at least for the bivariate case and, under certain circumstances, for higher dimensional cases. Investigations are done on comparisons with the ‘MAX-best’ population (that one having the largest maximum of the marginal means), the ‘MIN-best’ (having the largest minimum) and the ‘O-best’ (being closest to largest in all marginal means). Detailed results are given for the bivariate normal with extensions indicated for the multivariate.     

Constant-partially accelerated life tests for Marshall–Olkin exponential series system with dependent masked data

This paper considers the constant-partially accelerated life tests for series system products, where dependent M-O bivariate exponential distribution is assumed for the components.

Based on progressive type-II censored and masked data, the maximum likelihood estimates for the parameters and acceleration factors are obtained by using the decomposition approach. In addition, this method can also be applied to the Bayes estimates, which are too complex to obtain as usual way. Finally, a Monte Carlo simulation study is carried out to verify the accuracy of the methods under different masking probabilities and censoring schemes.     

Heterogeneous data modeling with two-component Weibull–Poisson distribution

The mixture distribution models are more useful than pure distributions in modeling of heterogeneous data sets. The aim of this paper is to propose mixture of Weibull–Poisson (WP) distributions to model heterogeneous data sets for the first time. So, a powerful alternative mixture distribution is created for modeling of the heterogeneous data sets. In the study, many features of the proposed mixture of WP distributions are examined. Also, the expectation maximization (EM) algorithm is used to determine the maximum-likelihood estimates of the parameters, and the simulation study is conducted for evaluating the performance of the proposed EM scheme. Applications for two real heterogeneous data sets are given to show the flexibility and potentiality of the new mixture distribution.     

A generalized Q–Q plot for longitudinal data

Most biomedical research is carried out using longitudinal studies. The method of generalized estimating equations (GEEs) introduced by Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] and Zeger and Liang [Longitudinal data analysis for discrete and continuous outcomes, Biometrics 42 (1986), pp. 121–130] has become a standard method for analyzing non-normal longitudinal data. Since then, a large variety of GEEs have been proposed. However, the model diagnostic problem has not been explored intensively. Oh et al. [Modeldiagnostic plots for repeated measures data using the generalized estimating equations approach, Comput. Statist. Data Anal. 53 (2008), pp. 222–232] proposed residual plots based on the quantile–quantile (Q–Q) plots of the χ²-distribution for repeated-measures data using the GEE methodology. They considered the Pearson, Anscombe and deviance residuals. In this work, we propose to extend this graphical diagnostic using a generalized residual. A simulation study is presented as well as two examples illustrating the proposed generalized Q–Q plots.     

A Mann–Whitney scan statistic for continuous data

ABSTRACT

A new method is proposed for identifying clusters in continuous data indexed by time or by space. The scan statistic we introduce is derived from the well-known Mann–Whitney statistic. It is completely non parametric as it relies only on the ranks of the marks. This scan test seems to be very powerful against any clustering alternative. These results have applications in various fields, such as the study of climate data or socioeconomic data.     

Generalized Poisson–Lindley linear model for count data

The purpose of this paper is to develop a new linear regression model for count data, namely generalized-Poisson Lindley (GPL) linear model. The GPL linear model is performed by applying generalized linear model to GPL distribution. The model parameters are estimated by the maximum likelihood estimation. We utilize the GPL linear model to fit two real data sets and compare it with the Poisson, negative binomial (NB) and Poisson-weighted exponential (P-WE) models for count data. It is found that the GPL linear model can fit over-dispersed count data, and it shows the highest log-likelihood, the smallest AIC and BIC values. As a consequence, the linear regression model from the GPL distribution is a valuable alternative model to the Poisson, NB, and P-WE models.     

Inference for the common mean of several Birnbaum–Saunders populations

The Birnbaum–Saunders distribution is a widely used distribution in reliability applications to model failure times. For several samples from possible different Birnbaum–Saunders distributions, if their means can be considered as the same, it is of importance to make inference for the common mean. This paper presents procedures for interval estimation and hypothesis testing for the common mean of several Birnbaum–Saunders populations. The proposed approaches are hybrids between the generalized inference method and the large sample theory. Some simulation results are conducted to present the performance of the proposed approaches. The simulation results indicate that our proposed approaches perform well. Finally, the proposed approaches are applied to analyze a real example on the fatigue life of 6061-T6 aluminum coupons for illustration.     

The Cox–Aalen model for left-truncated and right-censored data

We analyze left-truncated and right-censored (LTRC) data using an additive-multiplicative Cox–Aalen model proposed by Scheike and Zhang (2002), which extends the Cox regression model as well as the additive Aalen model. Based on the conditional likelihood function, we derive the weighted least-squared (WLS) estimators for the regression parameters and cumulative intensity functions of the model. The estimators are shown to be consistent and asymptotically normal. A simulation study is conducted to investigate the performance of the proposed estimators.     

A modified Hosmer–Lemeshow test for large data sets

The Hosmer–Lemeshow test is a widely used method for evaluating the goodness of fit of logistic regression models. But its power is much influenced by the sample size, like other chi-square tests. Paul, Pennell, and Lemeshow (2013 Paul, P., M. L. Pennell, and S. Lemeshow. 2013. Standardizing the power of the Hosmer–Lemeshow goodness of fit test in large data sets. Statistics in Medicine 32:67–80.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) considered using a large number of groups for large data sets to standardize the power. But simulations show that their method performs poorly for some models. In addition, it does not work when the sample size is larger than 25,000. In the present paper, we propose a modified Hosmer–Lemeshow test that is based on estimation and standardization of the distribution parameter of the Hosmer–Lemeshow statistic. We provide a mathematical derivation for obtaining the critical value and power of our test. Through simulations, we can see that our method satisfactorily standardizes the power of the Hosmer–Lemeshow test. It is especially recommendable for enough large data sets, as the power is rather stable. A bank marketing data set is also analyzed for comparison with existing methods.     

The homotopy perturbation method for the Black–Scholes equation

The homotopy perturbation method is designed to obtain a quick and accurate solution to the Black–Scholes equation and boundary conditions for a European option pricing problem. The problem of pricing a European option can be cast a partial differential equation. The analytical solution of the equation is calculated in the form of a convergent power series with easily computable components.