Mixture Of Inverse Gaussian Distributions Research Articles

On Extended Normal Inverse Gaussian Distribution: Theory, Methodology, Properties and Applications

In this article, the Normal Inverse Gaussian Distribution model (NIGDM) is extended to a new Extended Normal Inverse Gaussian Distribution (ENIGDM) and its derivate models find many applications. The author proposes a new model ENIGDM, which generalizes the models of normal inverse Gaussian distribution. This class of ENIGDM is to approximate an unknown risk-neutral density. The paper discusses different properties of the ENIGDM. In particular, the applicability of this new general model with five parameters is well justified by more results which represent mixtures of inverse Gaussian distributions. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Levy’s process for modeling and analyzing statistical data, with a particular reference to extensive sets of observations and applications in wide varieties.

Constructing normalcy and discrepancy indexes for birth weight and gestational age using a threshold regression mixture model.

Birth weight and gestational age are important measures of a newborn's intrinsic health, serving both as outcome measures and explanatory variables in health studies. The measures are highly correlated but occasionally inconsistent. We anticipate that health researchers and other scientists would be helped by summary indexes of birth weight and gestational age that give more precise indications of whether the birth outcome is healthy or not. We propose a pair of indexes that we refer to as the birth normalcy index or BNI and birth discrepancy index or BDI. Both indexes are simple functions of birth weight and gestational age and in logarithmic form are orthogonal by construction. The BNI gauges whether the birth weight and gestational age combination are in a normal range. The BDI gauges whether birth weight and gestational age are consistent. We present a three-component mixture model for BNI, with the components representing premature, at-risk, and healthy births. The BNI distribution is derived from a stochastic model of fetal development proposed by Whitmore and Su (2007, Lifetime Data Analysis 13, 161-190) and takes the form of a mixture of inverse Gaussian distributions. We present a noncentral t-distribution as a model for BDI. BNI and BDI are also well suited for making comparisons of birth outcomes in different reference populations. A simple z-score and t-score are proposed for such comparisons. The BNI and BDI distributions can be estimated for births in any reference population of interest using threshold regression.

On Banerjee and Bhattacharyya (1976), “A Purchase Incidence Model With Inverse Gaussian Interpurchase Times,” Journal of the American Statistical Association, 71, 823–829

This article corrects two errors that appear in the study by Banerjee and Bhattacharyya (1976). It shows that the mixture of inverse Gaussian distributions can look like a better model of purchase frequencies than implied by Banerjee and Bhattacharyya's work.

Predicting the Time to Quasi-extinction for Populations far below their Carrying Capacity

Populations threatened by extinction are often far below their carrying capacity. A population collapse or quasi-extinction is defined to occur when the population size reaches some given lower density. If this density is chosen to be large enough for the demographic stochasticity to be ignored compared to environmental stochasticity, then the logarithm of the population size may be modelled by a Brownian motion until quasi-extinction occurs. The normal-gamma mixture of inverse Gaussian distributions can then be applied to define prediction intervals for the time to quasi-extinction in such processes. A similar mixture is used to predict the population size at a finite time for the same process provided that quasi-extinction has not occurred before that time. Stochastic simulations indicate that the coverage of the prediction interval is very close to the probability calculated theoretically. As an illustration, the method is applied to predict the time to extinction of a declining population of white stork in southwestern Germany.

Explicit M/G/1 waiting-time distributions for a class of long-tail service-time distributions

O.J. Boxma and J.W. Cohen recently obtained an explicit expression for the M/G/1 steady-state waiting-time distribution for a class of service-time distributions with power tails. We extend their explicit representation from a one-parameter family of service-time distributions to a two-parameter family. The complementary cumulative distribution function (ccdf's) of the service times all have the asymptotic form F c( t)∼ αt −3/2 as t→∞, so that the associated waiting-time ccdf's have asymptotic form W c( t)∼ βt −1/2 as t→∞. Thus the second moment of the service time and the mean of the waiting time are infinite. Our result here also extends our own earlier explicit expression for the M/G/1 steady-state waiting-time distribution when the service-time distribution is an exponential mixture of inverse Gaussian distributions (EMIG). The EMIG distributions form a two-parameter family with ccdf having the asymptotic form F c( t)∼ αt −3/2e − ηt as t→∞. We now show that a variant of our previous argument applies when the service-time ccdf is an undamped EMIG, i.e., with ccdf G c( t)=e ηt F c( t) for F c( t) above, which has the power tail G c( t)∼ αt −3/2 as t→∞. The Boxma–Cohen long-tail service-time distribution is a special case of an undamped EMIG.

An operational calculus for probability distributions via Laplace transforms

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.

An operational calculus for probability distributions via Laplace transforms

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.

A reliability application of a mixture of inverse gaussian distributions

AbstractA mixture of inverse Gaussian distributions (IGDs) is examined as a model for the lifetime of components. The components differ in one of three ways: in their initial quality, rate of wear, or variability of wear. These three cases are well represented by the parameters of the IGD model. The mechanistic interpretation of the IGD as the first passage time of Brownian motion with positive drift is adopted. The parameters considered are either dichotomous or continuous random variables. Parameter estimation is also examined for these two cases. The model seems to be most appropriate when the single IGD model fails due to heterogeneity of the initial component quality.

Modelling Task Completion Data with Inverse Gaussian Mixtures

Whitmore introduced mixtures of inverse Gaussian distributions and fitted them to several data sets involving duration phenomena. In this paper we model task completion data from a large automobile plant in southern Ontario, with a mixed inverse Gaussian distribution. The mixing is over a parameter that can be naturally identified with the intensity with which the work crew approaches its task. It is found that, depending on the nature of the task, the mixed distribution may provide an improvement over the familiar pure inverse Gaussian distribution. A secondary consideration is the study of asymptotic likelihood inference for such mixtures

Lagrange Distributions and their Limit Theorems

Suppose that $( {f_n } )$ and $( {g_n } )$ are probability distributions on the nonnegative integers with means $\alpha \leqq 1$ and $\beta < \infty $ respectively. We show that \[l_n = \sum\limits_{k = 0}^n {\frac{k}{n}f_{n - k}^{( k )} g_k ,\qquad n = 0,1,2, \cdots ,} \] defines a probability distribution which is one of the so-called Lagrangian distributions. Using techniques and results from the theory of limit distributions including those of random sums of random variables, we prove local and global limit theorems for random variables with distribution $( {l_n } )$ in the following cases: \[\alpha < 1,\beta \to \infty ;\quad \alpha = 1,\quad \beta \to \infty ;\quad {\text{and}}\quad \alpha \to 1,\quad \beta \to \infty ,\quad \beta ( {1 - \alpha } ) \to c \in ( {0,\infty } ).\] In the first two cases the limit laws are mixtures of Gaussian, respectively stable distributions, whilst in the last case a mixture of inverse Gaussian distributions appears. Examples illustrate the fact that further limit la...