Estimation for the three-parameter inverse Gaussian distribution under progressive Type-II censoring

Inverse Gaussian distribution has been used widely as a model in analysing lifetime data. In this regard, estimation of parameters of two-parameter (IG2) and three-parameter inverse Gaussian (IG3) distributions based on complete and censored samples has been discussed in the literature. In this paper, we develop estimation methods based on progressively Type-II censored samples from IG3 distribution. In particular, we use the EM-algorithm, as well as some other numerical methods for determining the maximum-likelihood estimates (MLEs) of the parameters. The asymptotic variances and covariances of the MLEs from the EM-algorithm are derived by using the missing information principle. We also consider some simplified alternative estimators. The inferential methods developed are then illustrated with some numerical examples. We also discuss the interval estimation of the parameters based on the large-sample theory and examine the true coverage probabilities of these confidence intervals in case of small samples by means of Monte Carlo simulations.     

On the bias and mean-square error of order-restricted maximum likelihood estimators

The inverse Gaussian family (IG) (μ,λ) is a versatile family for modelling nonnegative right-skewed data. In this paper, we propose robust methods for testing homogeneity of the scale-like parameters λ_i from k independent IG populations subject to order restrictions. Robustness of the procedures is examined for a variety of IG-symmetric alternatives including lognormal and the recently introduced contaminated inverse Gaussian populations. Our study shows that these inference procedures for the inverse Gaussian scale-like parameters and their properties exhibit striking similarities to those of the scale parameters of the normal distribution.     

Effect of a new selection procedure on adaptive estimation

Adaptive estimation of parameters of some failure time distributionsis considered. A new procedure named the F-procedure has beendeveloped for selecting an appropriate model out of two possible models by Pandey et.al. (1991). Applying this F-procedure adaptive estimatorsof parameters of exponential, Wei bull, inverse Gaussian (IG) and Wald failure time distributions have been proposed in this paper. Comparison of these estimators has been undertaken with MLE's of the respective parameters and with some previous adaptiveestimators by simulation of samples using the Monte Carlo method.Adaptive estimation of parameters of some failure time distributions is considered. A new procedure named the F-procedure has been developedfor selecting an appropriate model out of two possible models by Pandey et.al. (1991). Applying this F-procedure adaptive estimators of parameters of exponential, Wei bull, inverse Gaussian (IG) and Wald failure time distributions have been proposed in this paper. Comparison of these estimators has been undertaken with MLE's of the respective parameters and with some previous adaptive estimators by simulation of samples using the Monte Carlo method.     

Inferences on the difference and ratio of the means of two inverse Gaussian distributions

Methods for interval estimation and hypothesis testing about the ratio of two independent inverse Gaussian (IG) means based on the concept of generalized variable approach are proposed. As assessed by simulation, the coverage probabilities of the proposed approach are found to be very close to the nominal level even for small samples. The proposed new approaches are conceptually simple and are easy to use. Similar procedures are developed for constructing confidence intervals and hypothesis testing about the difference between two independent IG means. Monte Carlo comparison studies show that the results based on the generalized variable approach are as good as those based on the modified likelihood ratio test. The methods are illustrated using two examples.     

Multivariate distributions with generalized inverse gaussian marginals,and associated poisson mixtures

Several types of multivariate extensions of the inverse Gaussian (IG) distribution and the reciprocal inverse Gaussian (RIG) distribution are proposed. Some of these types are obtained as random-additive-effect models by means of well-known convolution properties of the IG and RIG distributions, and they have one-dimensional IG or RIG marginals. They are used to define a flexible class of multivariate Poisson mixtures.     

A method for estimating parameters and quantiles of the three-parameter inverse Gaussian distribution based on statistics invariant to unknown location

The inverse Gaussian (IG) distribution, also known as the Wald distribution, is a long-tailed positively skewed distribution and a well-known lifetime distribution. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter IG distribution, which is based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared with other prominent methods in terms of bias and variance. Finally, we present two illustrative examples.     

The Likelihood Ratio Tests for Homogeneity of Inverse Gaussian Means Under Simple Order and Simple Tree Order

Abstract

The inverse Gaussian (IG) family is now widely used for modeling non negative skewed measurements. In this article, we construct the likelihood-ratio tests (LRTs) for homogeneity of the order constrained IG means and study the null distributions for simple order and simple tree order cases. Interestingly, it is seen that the null distribution results for the normal case are applicable without modification to the IG case. This supplements the numerous well known and striking analogies between Gaussian and inverse Gaussian families     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.     

A new look at the inverse Gaussian distribution with applications to insurance and economic data

Insurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models.     

Survival analysis for the inverse Gaussian distribution with the Gibbs sampler

This paper describes a comprehensive survival analysis for the inverse Gaussian distribution employing Bayesian and Fiducial approaches. It focuses on making inferences on the inverse Gaussian (IG) parameters μ and λ and the average remaining time of censored units. A flexible Gibbs sampling approach applicable in the presence of censoring is discussed and illustrations with Type II, progressive Type II, and random rightly censored observations are included. The analyses are performed using both simulated IG data and empirical data examples. Further, the bootstrap comparisons are made between the Bayesian and Fiducial estimates. It is concluded that the shape parameter ( ϕ=λ/μ) of the inverse Gaussian distribution has the most impact on the two analyses, Bayesian vs. Fiducial, and so does the size of censoring in data to a lesser extent. Overall, both these approaches are effective in estimating IG parameters and the average remaining lifetime. The suggested Gibbs sampler allowed a great deal of flexibility in implementation for all types of censoring considered.     

IG-symmetry and R-symmetry: Interrelations and applications to the inverse Gaussian theory

The two parameter inverse Gaussian (IG) distribution is often more appropriate and convenient for modelling and analysis of nonnegative right skewed data than the better known and now ubiquitous Gaussian distribution. Its convenience stems from its analytic simplicity and the striking similarities of its methodologies with those employed with the normal theory models. These, known as the G–IG analogies, include the concepts and measures of IG-symmetry, IG-skewness and IG-kurtosis, the IG-analogues of the corresponding classical notions and measures. The new IG-associated entities, although well defined and mathematically transparent, are intuitively and conceptually opaque. In this paper, we first elaborate the importance of the IG distribution and of the G–IG analogies. Then we consider the IG-related root-reciprocal IG (RRIG) distribution and introduce a physically transparent, conceptually clear notion of reciprocal symmetry (R-symmetry) and use it to explain the IG-symmetry. We study the moments and mixture properties of the R-symmetric distributions and the relationship of R-symmetry with IG-symmetry and note that RRIG distribution provides a link, in addition to Tweedie's Laplace transform link, between the Gaussian and inverse Gaussian distributions. We also give a structural characterization of the unimodal R-symmetric distributions. This work further expands the long list of G–IG analogies. Several applications including product convolution, monotonicity of power functions, peakedness and monotone limit theorems of R-symmetry are outlined.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.     

Comparing first hitting time and proportional hazards regression models

Cox's widely used semi-parametric proportional hazards (PH) regression model places restrictions on the possible shapes of the hazard function. Models based on the first hitting time (FHT) of a stochastic process are among the alternatives and have the attractive feature of being based on a model of the underlying process. We review and compare the PH model and an FHT model based on a Wiener process which leads to an inverse Gaussian (IG) regression model. This particular model can also represent a “cured fraction” or long-term survivors. A case study of survival after coronary artery bypass grafting is used to examine the interpretation of the IG model, especially in relation to covariates that affect both of its parameters.     

Statistical inference for α-series process with the inverse Gaussian distribution

Statistical inferences for the geometric process (GP) are derived when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). An α-series process, as a possible alternative to the GP, is introduced since the GP is sometimes inappropriate to apply some reliability and scheduling problems. In this study, statistical inference problem for the α-series process is considered where the distribution of first occurrence time is IG. The estimators of the parameters α, μ, and σ² are obtained by using the maximum likelihood (ML) method. Asymptotic distributions and consistency properties of the ML estimators are derived. In order to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators, Monte Carlo simulations are performed. The results showed that the ML estimators are more efficient than the MM estimators. Moreover, two real life datasets are given for application purposes.     

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.     

Moments of nonparametric probability density functions of entropy estimators applied to testing the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model data and then it is important to develop efficient goodness of fit tests for this distribution. In this article, we introduce some new test statistics for examining the IG goodness of fit based on correcting moments of nonparametric probability density functions of entropy estimators. These tests are consistent against all alternatives. Critical points and power of the tests are explored by simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by real data examples.     

An efficient correction to the density-based empirical likelihood ratio goodness-of-fit test for the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model positively skewed data. An important issue is to develop a powerful goodness-of-fit test for the IG distribution. We propose and examine novel test statistics for testing the IG goodness of fit based on the density-based empirical likelihood (EL) ratio concept. To construct the test statistics, we use a new approach that employs a method of the minimization of the discrimination information loss estimator to minimize Kullback–Leibler type information. The proposed tests are shown to be consistent against wide classes of alternatives. We show that the density-based EL ratio tests are more powerful than the corresponding classical goodness-of-fit tests. The practical efficiency of the tests is illustrated by using real data examples.     

Poisson-mixed Inverse Gaussian Regression Model and Its Application

In this article, we have developed a Poisson-mixed inverse Gaussian (PMIG) distribution. The mixed inverse Gaussian distribution is a mixture of the inverse Gaussian distribution and its length-biased counterpart. A PMIG regression model is developed and the maximum likelihood estimation of the parameters is studied. A dataset dealing with the number of hospital stays among the elderly population is analyzed by using the PMIG and the PIG (Poisson-inverse Gaussian) regression models and it has been shown that the PMIG model fits the data better than the PIG model.     

Bayesian analysis of the scatterometer wind retrieval inverse problem: some new approaches

Summary.  The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.     

Attribute control chart for some popular distributions

The binomial distribution is often used to display attribute control data. In this paper, a statistical model is settled for attribute control chart under truncated life test. By Burr X & XII, inverse Gaussian (IG), and exponential lifetime-truncated distributions, a Shewhart-type attribute control chart is built to display the data. The performance of attributed control chart constructed on truncated life test is evaluated by average run length, which compares the performance of all distributions. Our study arranges that IG is better distribution among all.