Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

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.     

New Simple Tests for Panel Cointegration

ABSTRACT

In this paper, two new simple residual-based panel data tests are proposed for the null of no cointegration. The tests are simple because they do not require any correction for the temporal dependencies of the data. Yet they are able to accommodate individual specific short-run dynamics, individual specific intercept and trend terms, and individual specific slope parameters. The limiting distributions of the tests are derived and are shown to be free of nuisance parameters. The Monte Carlo results in this paper suggest that the asymptotic results are borne out well even in very small samples.     

A Simple Test Procedure When Data are Given by n Indepenent Blocks

In this article we consider a test procedure which is useful in the situations where data are given by n independent blocks and the experimental conditions differ between blocks. The basic idea is very simple. The significance of the sample for each block is calculated and then standardized by its null mean and variance. The sum of standardized significances is proposed as a test statistic. The normal approximation for large n and the exact method for small n are applied in the continuous case. For the discrete case, some devices are also proposed. Several examples are given in order to explain how to apply the procedure.     

Order identification for gaussian moving averages using the covariation

In the traditional Box-Jenkins procedure for fitting ARMA time series models to data, the first step is order identification. The sample autocorrelation function can be used to identify pure moving average behavior. In this paper we consider using the autocovariation function identify the order of a univariate Gaussian time series. Simulation evidence indicates the suggested method may be a superior order identification tool when at least 100 observations are taken.     

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.     

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.     

The Estimation and Testing of the Cointegration Order Based on the Frequency Domain

ABSTRACT

This article proposes a method to estimate the degree of cointegration in bivariate series and suggests a test statistic for testing noncointegration based on the determinant of the spectral density matrix for the frequencies close to zero. In the study, series are assumed to be I(d), 0 < d ? 1, with parameter d supposed to be known. In this context, the order of integration of the error series is I(d ? b), b ∈ [0, d]. Besides, the determinant of the spectral density matrix for the dth difference series is a power function of b. The proposed estimator for b is obtained here performing a regression of logged determinant on a set of logged Fourier frequencies. Under the null hypothesis of noncointegration, the expressions for the bias and variance of the estimator were derived and its consistency property was also obtained. The asymptotic normality of the estimator, under Gaussian and non-Gaussian innovations, was also established. A Monte Carlo study was performed and showed that the suggested test possesses correct size and good power for moderate sample sizes, when compared with other proposals in the literature. An advantage of the method proposed here, over the standard methods, is that it allows to know the order of integration of the error series without estimating a regression equation. An application was conducted to exemplify the method in a real context.     

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.     

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.     

More powerful sign test using median ranked set sample: Finite sample power comparison

Sign test using median ranked set samples (MRSS) is introduced and investigated. We show that, this test is more powerful than the sign tests based on simple random sample (SRS) and ranked set sample (RSS) for finite sample size. It is found that, when the set size of MRSS is odd, the null distribution of the MRSS sign test is the same as the sign test obtained by using SRS. The exact null distributions and the power functions, in case of finite sample sizes, of these tests are derived. Also, the asymptotic distribution of the MRSS sign tests are derived. Numerical comparison of the MRSS sign test power with the power of the SRS sign test and the RSS sign test is given. Illustration of the procedure, using real data set of bilirubin level in Jaundice babies who stay in neonatal intensive care is introduced.     

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.     

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.     

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.     

Multiplicative Algorithms for Maximum Penalized Likelihood Inversion with Non Negative Constraints and Generalized Error Distributions

In many linear inverse problems the unknown function f (or its discrete approximation Θ _p×1), which needs to be reconstructed, is subject to the non negative constraint(s); we call these problems the non negative linear inverse problems (NNLIPs). This article considers NNLIPs. However, the error distribution is not confined to the traditional Gaussian or Poisson distributions. We adopt the exponential family of distributions where Gaussian and Poisson are special cases. We search for the non negative maximum penalized likelihood (NNMPL) estimate of Θ. The size of Θ often prohibits direct implementation of the traditional methods for constrained optimization. Given that the measurements and point-spread-function (PSF) values are all non negative, we propose a simple multiplicative iterative algorithm. We show that if there is no penalty, then this algorithm is almost sure to converge; otherwise a relaxation or line search is necessitated to assure its convergence.     

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.     

Testing change in the variance with epidemic alternatives

ABSTRACT

In this paper we consider the dyadic increments statistics (of type DI) based on independent not identically distributed or α-mixing random variables. We obtain their limit distributions under the null hypothesis and we present application for testing epidemic change in the variance in each case. Finally, numerical simulations are done to illustrate these results.     

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.     

OPTIMAL DESIGN OF PLANS FOR ACCEPTANCE SAMPLING BY VARIABLES WITH INVERSE GAUSSIAN DISTRIBUTION

ABSTRACT

A vast majority of the literature on the design of sampling plans by variables assumes that the distribution of the quality characteristic variable is normal, and that only its mean varies while its variance is known and remains constant. But, for many processes, the quality variable is nonnormal, and also either one or both of the mean and the variance of the variable can vary randomly. In this paper, an optimal economic approach is developed for design of plans for acceptance sampling by variables having Inverse Gaussian (IG) distributions. The advantage of developing an IG distribution based model is that it can be used for diverse quality variables ranging from highly skewed to almost symmetrical. We assume that the process has two independent assignable causes, one of which shifts the mean of the quality characteristic variable of a product and the other shifts the variance. Since a product quality variable may be affected by any one or both of the assignable causes, three different likely cases of shift (mean shift only, variance shift only, and both mean and variance shift) have been considered in the modeling process. For all of these likely scenarios, mathematical models giving the cost of using a variable acceptance sampling plan are developed. The cost models are optimized in selecting the optimal sampling plan parameters, such as the sample size, and the upper and lower acceptance limits. A large set of numerical example problems is solved for all the cases. Some of these numerical examples are also used in depicting the consequences of: 1) using the assumption that the quality variable is normally distributed when the true distribution is IG, and 2) using sampling plans from the existing standards instead of the optimal plans derived by the methodology developed in this paper. Sensitivities of some of the model input parameters are also studied using the analysis of variance technique. The information obtained on the parameter sensitivities can be used by the model users on prudently allocating resources for estimation of input parameters.     

A Statistical Method for the Determination of the Appropriate Order in a General Class of Time Series Models

Abstract

We propose a method to determine the order q of a model in a general class of time series models. For the subset of linear moving average models (MA(q)), our method is compared with that of the sample autocorrelations. Since the sample autocorrelation is meant to detect a linear structure of dependence between random variables, it turns out to be more suitable for the linear case. However, our method presents a competitive option in that case, and for nonlinear models (NLMA(q)) it is shown to work better. The main advantages of our approach are that it does not make assumptions on the existence of moments and on the distribution of the noise involved in the moving average models. We also include an example with real data corresponding to the daily returns of the exchange rate process of mexican pesos and american dollars.