Use of the Likelihood Ratio Test on the Inverse Gaussian Distribution

In the following article, the likelihood ratio test is determined for four tests of hypotheses involving the inverse Gaussian distribution. For three of the hypotheses, the test produces the same statistic as the uniformly most powerful unbiased test.     

Testing Homogeneity of Inverse Gaussian Scale Parameters Based on Generalized Likelihood Ratio

This article presents a new procedure for testing homogeneity of scale parameters from k independent inverse Gaussian populations. Based on the idea of generalized likelihood ratio method, a new generalized p-value is derived. Some simulation results are presented to compare the performance of the proposed method and existing methods. Numerical results show that the proposed test has good size and power performance.     

An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

Inverse Gaussian Distribution for Modeling Conditional Durations in Finance

The durations between market activities such as trades and quotes provide useful information on the underlying assets while analyzing financial time series. In this article, we propose a stochastic conditional duration model based on the inverse Gaussian distribution. The non-monotonic nature of the failure rate of the inverse Gaussian distribution makes it suitable for modeling the durations in financial time series. The parameters of the proposed model are estimated by an efficient importance sampling method. A simulation experiment is conducted to check the performance of the estimators. These estimates are used to compute estimated hazard functions and to compare with the empirical hazard functions. Finally, a real data analysis is provided to illustrate the practical utility of the models.     

Inverse Gaussian process models for bivariate degradation analysis: A Bayesian perspective

This article conducts a Bayesian analysis for bivariate degradation models based on the inverse Gaussian (IG) process. Assume that a product has two quality characteristics (QCs) and each of the QCs is governed by an IG process. The dependence of the QCs is described by a copula function. A bivariate simple IG process model and three bivariate IG process models with random effects are investigated by using Bayesian method. In addition, a simulation example is given to illustrate the effectiveness of the proposed methods. Finally, an example about heavy machine tools is presented to validate the proposed models.     

Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non-decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY-conjugate prior ( Diaconis & Ylvisaker, 1979 ) for non-decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution ( Dawid & Lauritzen, 1993 ). In particular, if G is an incomplete prime graph it constitutes a non-trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non-decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non-decomposable zero-pattern we introduce the operation of triangular completion of an incomplete triangular matrix. Such a device turns out to be extremely useful both in the proof of theoretical results and in the implementation of the Monte Carlo procedure.     

Failure Inference and Optimization for Step Stress Model Based on Bivariate Wiener Model

In this article, we consider the situation under a life test, in which the failure time of the test units are not related deterministically to an observable stochastic time varying covariate. In such a case, the joint distribution of failure time and a marker value would be useful for modeling the step stress life test. The problem of accelerating such an experiment is considered as the main aim of this article. We present a step stress accelerated model based on a bivariate Wiener process with one component as the latent (unobservable) degradation process, which determines the failure times and the other as a marker process, the degradation values of which are recorded at times of failure. Parametric inference based on the proposed model is discussed and the optimization procedure for obtaining the optimal time for changing the stress level is presented. The optimization criterion is to minimize the approximate variance of the maximum likelihood estimator of a percentile of the products’ lifetime distribution.     

Analysis of Bivariate Survival Data using Shared Inverse Gaussian Frailty Model

In this article, we consider shared frailty model with inverse Gaussian distribution as frailty distribution and log-logistic distribution (LLD) as baseline distribution for bivariate survival times. We fit this model to three real-life bivariate survival data sets. The problem of analyzing and estimating parameters of shared inverse Gaussian frailty is the interest of this article and then compare the results with shared gamma frailty model under the same baseline for considered three data sets. Data are analyzed using Bayesian approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same group. The variance component estimation provides the estimated dispersion of the random effects. We carried out a test for frailty (or heterogeneity) using Bayes factor. Model comparison is made using information criteria and Bayes factor. We observed that the shared inverse Gaussian frailty model with LLD as baseline is the better fit for all three bivariate data sets.     

Graphical Tests for the Assumption of Gamma and Inverse Gaussian Frailty Distributions

The common choices of frailty distribution in lifetime data models include the Gamma and Inverse Gaussian distributions. We present diagnostic plots for these distributions when frailty operates in a proportional hazards framework. Firstly, we present plots based on the form of the unconditional survival function when the baseline hazard is assumed to be Weibull. Secondly, we base a plot on a closure property that applies for any baseline hazard, namely, that the frailty distribution among survivors at time t has the same form as the original distribution, with the same shape parameter but different scale parameter. We estimate the shape parameter at different values of t and examine whether it is constant, that is, whether plotted values form a straight line parallel to the time axis. We provide simulation results assuming Weibull baseline hazard and an example to illustrate the methods.     

A group sequential test for the inverse Gaussian mean

The present paper deals with the development of a group sequential test when response variable has an inverse Gaussian distribution with known scale parameter.     

Modeling of Inverse Gaussian Frailty Model for Bivariate Survival Data

Shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor (frailty) and baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and distribution of frailty. In this article, we consider inverse Gaussian distribution as frailty distribution and three different baseline distributions namely, Weibull, generalized exponential, and exponential power distribution. With these three baseline distributions, we propose three different inverse Gaussian shared frailty models. To estimate the parameters involved in these models we adopt Markov Chain Monte Carlo (MCMC) approach. We present a simulation study to compare the true values of the parameters with the estimated values. Also, we apply these three models to a real life bivariate survival data set of McGilchrist and Aisbett (1991 McGilchrist , C. A. , Aisbett , C. W. ( 1991 ). Regression with frailty in survival analysis . Biometrics 47 : 461 – 466 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to kidney infection and a better model is suggested for the data.     

Issues in Fitting Inverse Gaussian First Hitting Time Regression Models for Lifetime Data

Inverse Gaussian first hitting time regression models sometimes provide an attractive representation of lifetime data. Various authors comment that dependence of both parameters on the same covariate may imply multicollinearity. The frequent appearance of conflicting signs for the two coefficients of the same covariate may be related to this. We carry out simulation studies to examine the reality of this possible multicollinearity. Although there is some dependence between estimates, multicollinearity does not seem to be a major problem. Fitting this model to data generated by a Weibull regression suggests that conflicting signs of estimates may be due to model misspecification.     

R-Symmetry and the Power Functions of the Tests for Inverse Gaussian Means

The two-parameter Inverse Gaussian (IG) distribution is often appropriate for modeling non negative right-skewed data due to the striking similarities with the Gaussian distribution in its basic properties and inference methods. There are about 40 such G-IG analogies developed in literature and were most recently tabulated by Mudholkar and Wang. Of these, the earliest and most commonly noted similarities are the significance tests based on student's t and F distribution for the homogeneity of one, two or several means of the IG populations. However, unlike the corresponding tests in Gaussian theory, little is known about the power function of the basic tests. In this article, we employ the IG-related root-reciprocal IG distribution and a notion of Reciprocal Symmetry to establish the monotonicity of the power function of the test of significance for the IG mean.     

Estimation in inverse Gaussian regression: comparison of asymptotic and bootstrap distributions

This paper presents a brief review of the asymptotic properties of the pseudo-maximum likelihood estimator in the regression model where the reciprocal of the mean of the dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo-maximum likelihood estimator presented in Babu and Chaubey (1996) is highlighted and a simulation study is carried out to compare the approximation yielded by the bootstrap distribution to that of the asymptotic distribution.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

Inference from Accelerated Degradation and Failure Data Based on Gaussian Process Models

An important problem in reliability and survival analysis is that of modeling degradation together with any observed failures in a life test. Here, based on a continuous cumulative damage approach with a Gaussian process describing degradation, a general accelerated test model is presented in which failure times and degradation measures can be combined for inference about system lifetime. Some specific models when the drift of the Gaussian process depends on the acceleration variable are discussed in detail. Illustrative examples using simulated data as well as degradation data observed in carbon-film resistors are presented.     

Exact Likelihood Inference for Two Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of two competing products with regard to their reliability. In this article, we consider two exponential populations and when joint progressive Type-II censoring is implemented on the two samples. We then derive the moment generating functions and the exact distributions of the maximum likelihood estimators (MLEs) of the mean lifetimes of the two exponential populations under such a joint progressive Type-II censoring. We then discuss the exact lower confidence bounds, exact confidence intervals, and simultaneous confidence regions. Next, we discuss the corresponding approximate results based on the asymptotic normality of the MLEs as well as those based on the Bayesian method. All these confidence intervals and regions are then compared by means of Monte Carlo simulations with those obtained from bootstrap methods. Finally, an illustrative example is presented in order to illustrate all the methods of inference discussed here.     

Shared inverse Gaussian frailty models based on additive hazards

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), the shared frailty models were suggested. These models are based on the assumption that frailty acts multiplicatively to hazard rate. In this article, we assume that frailty acts additively to hazard rate. We introduce the shared inverse Gaussian frailty models with three different baseline distributions, namely the generalized log-logistic, the generalized Weibull, and exponential power distribution. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo technique to estimate the parameters involved in these models. We apply these models to a real-life bivariate survival dataset of McGilchrist and Aisbett (1991 McGilchrist, C.A., Aisbett, C.W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to the kidney infection data, and a better model is suggested for the data.     

An extensive power evaluation of some tests for the inverse Gaussian distribution

Many applications of the Inverse Gaussian distribution, including numerous reliability and life testing results are presented in statistical literature. The paper studies the problem of using entropy tests to examine the goodness of fit of an Inverse Gaussian distribution with unknown parameters. Some entropy tests based on different entropy estimates are proposed. Critical values of the test statistics for various sample sizes are obtained by Monte Carlo simulations. Type I error of the tests is investigated and then power values of the tests are compared with the competing tests against various alternatives. Finally, recommendations for the application of the tests in practice are presented.