 Review
 Open Access
 Published:
Revisit of relationships and models for the BirnbaumSaunders and inverseGaussian distributions
Journal of Statistical Distributions and Applications volume 2, Article number: 11 (2015)
Abstract
The BirnbaumSaunders distribution was derived in 1969 as a lifetime model for a specimen subjected to cyclic patterns of stresses and strains, and the ultimate failure of the specimen is assumed to be due to the growth of a dominant crack in the material. The inverse Gaussian distribution is used to describe the first passage time for a particle (moving with constant velocity) that is subject to linear Brownian motion. These two models have a rich history, and they have been shown to be much related. In this article, these two models will be reviewed and comparisons will be made. Specifically, two momentratio diagrams will be presented that gives insight to the reason of both distributions often achieve similar fits to experimental data. Next, a generalized BirnbaumSaunders distribution, will be presented and several properties will be derived. In particular, it will be shown that this generalized model can be expressed as a mixture of inverse Gaussiantype random variables (similar to the twoparameter BirnbaumSaunders model). Estimation of the parameters in the generalized BirnbaumSaunders distribution will be discussed. Lastly, some conclusions from this investigation are presented.
Introduction
Birnbaum and Saunders (1969a) derived a lifetime distribution, known as the BirnbaumSaunders distribution (abbreviated as BS hereafter), which is founded on modeling the failure of a specimen that is subjected to a cyclic pattern stresses and strains. The ultimate failure is due to the growth of a dominant crack in the material. At each increment of load, this dominant crack extends by a random, nonnegative amount. Another model, the inverse Gaussian distribution (abbreviated by IG hereafter), was originally used to describe the first passage time in a Brownian motion. However, the inverse Gaussian distribution has been used quite frequently to model reliability data, and the relationship between the BS and the IG distributions was first established by Bhattacharyya and Fries (1982). Starting in Section 2, these two distributions will be presented, and the main results, extensions, and summaries from an extensive literature review will be provided. Following that section, the main purpose of this paper is twofold and is introduced in the two following sections. In Section 3, comparisons of these two models will be made by providing two momentratio diagrams for both the BS and IG distributions. The first diagram is a graph of the coefficient of variation (CV) versus skewness and the second diagram is a graph of the skewness versus the kurtosis. While functionally these standardized moments are very different, when they are graphed they prove to be very similar. This will further support the claims that the BS and IG distributions are nearly interchangeable. Following this, a threeparameter extension to the BS distribution proposed by DíazGarcía and DomínguezMolina (2006) is presented in Section 4. Some very unique properties and summary values will be presented, and it is shown that the distribution has relationship to other models found in the literature. In addition, this model has the ability to exhibit a bimodal shape for certain ranges of parameter values. We also discuss different estimation methods for the parameters in the threeparameter BS distribution and study their performances in Section 4. Two numerical examples are given in Section 5 to illustrate the usefulness of the generalized distribution and the estimation methods. Lastly, Section 6 presents some conclusions from this investigation.
The twoparameter BS and IG distributions
The precise derivation of the BS model is given in Birnbaum and Saunders (1969a) and also summarized in Owen (2006). The cumulative distribution function (CDF) for a random variable S that follows the BS distribution is given by
where α>0, β>0, and Φ represents the standard normal CDF. To use shorthand, we describe the random variable S that has distribution function (1) as S∼BS(α,β). The distribution (1) has some interesting properties given in Johnson et al. (1995b). The parameter β is a scale parameter since by definition S/β∼BS(α,1). In addition, β is the median for the distribution as from (1),
The parameter α is a shape parameter. The BS distribution exhibits the wellknown reciprocal property: the random variable S ^{−1}∼BS(α,1/β), so it is in the same family of distributions (see Saunders 1974).
If Z is a standard normal random variable, then
has the BS distribution in (1). This presentation is also given in Owen (2006) but it is slightly different from Birnbaum and Saunders (1969a). In addition, the expression in (2) is useful for random number generation and for deriving integer moments. The mean and variance for S are
respectively. Higher order moments will come into play in the subsequent sections. Since these higher order moments cannot be found in the literature, they are provided here. The third and fourth moments (both about zero) are given by
respectively. These can be expressed by using the relationship (2) above. The expected value for a standard normal variable that is raised to an integer power is given in Zacks (1992).
The probability density function (PDF) of the random variable S following the BS distribution in (1) is given by
where α>0, β>0 and it is characteristically rightskewed when graphed. As α decreases, particularly for values less than unity, the density becomes nearly symmetric as the curve spread (variance) decreases. This twoparameter family of distributions has various applications in reliability and life testing. For example, Birnbaum and Saunders (1969b) consider an example to model the strength of aluminum coupons subjected to cyclic stresses and strains. Three data sets are considered, each having different levels of maximum stress per cycle. The parameters in (1) are estimated using maximum likelihood (ML) estimation. The likelihood equations are given in Birnbaum and Saunders (1969b) and the ML estimates for α and β need to be found using numerical techniques. Dupuis and Mills (1998) consider an alternative to ML estimation for the BS distribution that is shown to be robust to the presence of contaminated data. Ng et al. (2006) considered modified moment techniques for estimation and biasreduction methods for estimation of the parameters in the BS model. Ashcar (1993) derived the approximate Fisher information matrix for the parameters α and β and considered Bayesian inference using noninformative and Jeffrey’s priors. Other important efforts include a loglinear model for the BS distribution, which was derived by Rieck and Nedleman (1991).
In the past decade, considerable research has been dedicated to the generalizations and applications of the BS distribution. Numerous authors have investigated different aspects related to the BS distribution. For instance, various generalizations and extensions of the BS distribution are proposed and discussed; see, for example, DíazGarcía and LeivaSánchez (2005), DíazGarcía and DomínguezMolina (2006), Owen (2006), Leiva et al. (2008), Sanhueza et al. (2008), Gómez et al. (2009), Guiraud et al. (2009), Leiva et al. (2009), Castillo et al. (2011), Cordeiro and Lemonte (2011), Genç (2013), Cordeiro and Lemonte (2014) and Cordeiro, et al.: A model with longterm survivors: Negative binomial BirnbaumSaunders (to appear). These generalized BS distributions have been applied to model data obtained from different disciplines including environmental sciences, reliability engineering and biomedical studies. For example, Leiva et al. (2008) used generalized BS distributions to model air pollutant concentration, Leiva et al. (2009) proposed a lengthbiased BS distribution for water quality data, Guiraud et al. (2009) used a noncentral BS distribution for reliability analysis and Cordeiro, et al.: A model with longterm survivors: Negative binomial BirnbaumSaunders (to appear) proposed the negative binomial BS distribution as a cure rate survival model. Software packages written in R (R Core Team 2015) for BS and generalized BS distributions have been developed by Leiva et al. (2006) and Barros et al. (2009).
In particular, Owen (2006) derived a threeparameter model based on the BS model in (7), and this was from a consequence of relaxing the assumption of independent crack extensions and viewing critical crack growth as a longmemory process. The threeparameter BirnbaumSaunders distribution derived in Owen (2006) has PDF
where a>0,b>0 and κ>0 are the parameters. To avoid confusion in a literature review, this model in (8) is referred to as the GBS_{1}.
The IG distribution, another important distribution in reliability analysis, has PDF
where μ>0 and λ>0 are the parameters. We denote a random variable X having distribution (9) as X∼IG(μ,λ). Both μ and λ behave as shapetype parameters. When graphed, (9) exhibits a rightskewed shape. Interested readers can refer to Johnson et al. (1995a) for a discussion on the IG distribution and for more information, see Chhikara and Folks (1989). Of interest here is the scalechange property: if X∼IG(μ,λ), then a X∼IG(a μ,a λ) for a constant a>0. In addition, the mean, variance, third and fourth moments (again, both about zero) are given by
respectively. The IG distribution is a twoparameter exponential family (see Lehmann and Casella 1998), and this fact has often made the IG distribution more attractive for the development of exact statistical procedures.
Relationships between and comparisons of the BS and IG models
Of utmost importance here are the work published in two papers that relate the BS and IG models. In this section, relationships between these two models will be established.
3.1 Literature review
In Bhattacharyya and Fries (1982), the identical approach in Birnbaum and Saunders (1969a) was viewed as a Wiener process of accumulated fatigue in time (with positive drift parameter μ and diffusion constant τ ^{2}), and this leads directly to an IG distribution. From this derivation, it is shown that a BS random variable or “event” is actually contained within the IG model since only positive increments in the growth of the dominant crack are allowed in Birnbaum and Saunders (1969a) (a “negative growth” can be viewed as a repair in the dominant crack, but this event is rare whenever μ≫τ). Therefore, the BS distribution can be considered as an approximation of the IG distribution, but Bhattacharyya and Fries (1982) argued that this approximation is not necessary since the IG distribution benefits from its exponential family structure. Still, the observation in Bhattacharyya and Fries (1982) has elucidated the observation that (7) and (9) can give similar shapes when fitting observed failure data. Following this result, Desmond (1986) demonstrated that the BS distribution can be written as an equal mixture of an IG distribution and the distribution of the reciprocal of an IG random variable. That is, for the mutually statistically independent random variables X _{1}, X _{2}, and B, where X _{1}∼IG(β,β/α ^{2}), \(X_{2}^{1} \sim \text {IG}(1/\beta, 1/(\alpha ^{2} \beta))\), and B follows Bernoulli distribution with Pr(B=0)= Pr(B=1)=0.5. Then,
Desmond (1986) argued that in a stochastic modeling point of view, the BS distribution is often more appropriate but the IG is preferred for statistical analysis (see, for example, Durham and Padgett 1997, Owen 2007). In addition, Desmond (1986) observed that the hazard functions for the BS and IG distributions are very similar, giving further evidence that the two models are nearly identical.
3.2 Alternative approach to relate the BS and IG distributions
Construction of moment ratio diagrams has provided alternative ways for comparing univariate distributions and illustrating their differences. By selecting two standardized moments for a host of distributions, these can be graphed and used to not only compare and contrast distribution qualities but also the diagrams give a means for selecting potential models based on sample data. The “standardized moments” that are being referred to are the coefficient of variation (CV) γ _{2}, coefficient of skewness (or third standardized moment) γ _{3}, and the kurtosis (or fourth standardized moment) γ _{4}. Following Cox and Oates (1984) (see also Johnson et al. 1995a; 1995b), the moment ratio diagrams are given by:

plotting γ _{2} on the horizontal axis (abscissa) and γ _{3} on the vertical axis (ordinate)

plotting γ _{3} on the horizontal axis and γ _{4} on the vertical axis (the classical presentation of this graph is given upside down)
To identify a potential distribution to consider when modeling a dataset, sample standardized moments can be calculated and plotted as points in either (I) and/or (II) – thus, probability distributions that are “close” to the point estimates can be considered as candidates for probability models. A recent article by Vargo et al. (2010) revisited the moment ratio diagrams and presented comprehensive graphs of (I) and (II) above for over 30 commonly used univariate distributions. Limiting relationships between several wellknown families (e.g., t and chisquare) were also considered. In addition, to identify potential distributions to model a dataset, the authors presented a novel application of bootstrapping. Therein, bootstrap samples are generated and the sample estimates of CV, skewness, and kurtosis (represented as \({\hat \gamma }_{2}\), \({\hat \gamma }_{3}\) and \({\hat \gamma }_{4}\), respectively) are calculated in order to generate the “concentration ellipse” to graph on the moment ratio diagrams. In this way, distributions that are close to the concentration ellipse should be considered as candidates. Since point estimates of higher moments can be highly variable, this bootstrap approach includes the sampling error.
Several distributions are presented in Vargo et al. (2010), but the BS and IG distributions are absent while the Wald distribution is included. The Wald distribution is an ambiguous model; there are some references that state that the Wald distribution is identical to the IG distribution, but other references claim that the Wald distribution is a special case of the IG distribution with μ=1 (see, page 262 of Johnson et al. 1995a). Therefore, in this section the two momentratio diagrams (I) and (II) will be developed for both the twoparameter IG and BS distributions.
For a (general) random variable Y with mean μ and standard deviation σ, the three standardized moments (and their relationship to moments taken about zero) are given by
Taking these expressions along with the moments for BS distribution presented in (3)–(6) and the moments for IG distribution presented in (10)–(13), the results are summarized in Table 1.
The results in Table 1 correct the mistakes in the coefficient of skewness and kurtosis that appeared in the literature (see, for example, Ng et al. 2006, Balakrishnan et al. 2011, Lemonte and Ferrari 2011).
There are two remarks from Vargo et al. (2010) that are applicable here: (i) since β is a scale parameter in the BS model, all of the standardized moments are free of β; and (ii) since μ and λ are both shape parameters in the IG model, they both appear in the standardized moments. However, since all of the standardized moments are common functions of the quotient μ/λ, this will simplify the display of the momentdiagrams. Note that in Vargo et al. (2010), it is most typical that the plotted points seen in momentratio diagrams lie not far from the origin. Figures 1 and 2 are the moment ratio diagrams (I) and (II), respectively, for the BS and IG distributions. The similarity of the shapes provides more evidence that the two distributions are quite similar and even in many cases comparable.
A generalized BS distribution
The BS distribution in (1) can be made more general by allowing the exponent (presently set to the value 1/2) to take on other values. In so doing, it will be shown that this extension exhibits some very interesting properties while still retaining a functional relationship to the IG. We define the generalized BirnbaumSaunders distribution by its CDF representation given by
where α>0, β>0, and ν>0 make this a valid CDF. This is refereed to as the “second” generalized BS distribution as to not confuse with another, but very different, unrelated generalized BS distribution in Owen (2006). We denote this generalized BS distribution as GBS_{2}. This generalization of BS distribution was proposed by DíazGarcía and DomínguezMolina (2006) (see also Eq. (B9.2) of Sanhueza et al. 2008). This model includes (1) as a special case when ν=0.5. Here, we describe a random variable T that has distribution function (18) as T∼GBS_{2}(α,β,ν). The PDF for the GBS_{2}(α,β,ν) distribution is given by
This generalized BirnbaumSaunders distribution is a flexible threeparameter family of distributions, and the shape of the density varies widely with different values of the parameters. Interestingly, the density is bimodal if both α>2 and ν>2, and the major mode is always less than the minor mode. In Figs. 3 and 4, we present various graphs of the PDF in (19); since β is a scale parameter, without loss of generality, it is fixed at unity for all cases. For the reader’s interest, we also provide the values of the mean and standard deviation for each distribution that were calculated using (24).
As it can be seen, the densities in Fig. 4 are interesting since that while the medians are equal the means and standard deviations are quite different. Lastly, the ability for (19) to achieve a bimodal shape truly expands the flexibility for the model; often, when dealing with a dataset with two modes a mixture model is the standard approach (see, for example, Chen et al. 2008).
4.1 Properties and related distributions
The distribution in (18) has some very interesting properties, and many similar properties of the twoparameter model (1) still hold for (18). Namely, β remains a scale parameter and is also the median for the distribution. Both α and ν are shapetype parameters. If Z is a standard normal variable, then the random variable
follows GBS_{2}(α,β,ν). Thus, random variates from this generalized distribution can easily be simulated. Clearly, the reciprocal property is also preserved since a trivial calculation shows that
In fact, (18) generally describes the family of distributions of power transformations for random variables distributed as (1); it is straightforward to show that if S∼BS(α,β), then for any nonzero realvalued constant k, S ^{k}∼GBS_{2}(α,β ^{k},0.5k^{−1}), where · represents the absolute value function. In addition, when k is a positive integer, the GBS_{2} variable S ^{k} can be expressed as a mixture of IGtype random variables. Following the work of Desmond (1986) as described in Section 3.1, the binomial theorem can be applied. Since all crossproduct terms are zero, we can obtain the following relationship:
where X _{1}, X _{2}, and B are as defined in Section 3.1. In (21), the random variables X _{1} and X _{2} are raised to the power k, which are called power IG distribution. One may refer to Hossain et al. (1997) for a description of the power IG distribution as well as properties of mixtures of IG random variables.
Closedform expressions of moments for (18) do not exist for all values of the parameters but they can be evaluated using the following relationship. If T∼GBS_{2}(α,β,ν), then it can be shown that the transformation U= lnT has the following distribution function
where δ= lnβ and η=1/ν. The distribution (22) is the threeparameter form of the sinhnormal (here, abbreviated by SN) distribution (see, for example, Johnson et al. 1995b), and we denote a random variable U following distribution (22) as U∼SN(α,δ,η). The parameter α>0 is a shape parameter, −∞<δ<∞ is a location parameter (also the mean of the distribution), and η>0 is a scale parameter. The PDF of the SN distribution is always symmetric about δ and it is mound shaped if α>2. For values of α>2, the PDF is bimodal. This distribution is also referred to as the “central” SN distribution. For a detail description of the SN distribution and a normal approximation to the SN distribution for small values of α, see Rieck (1999).
The relationship between the twoparameter BS distribution and a twoparameter SN distribution was established in Rieck and Nedleman (1991): if S∼BS(α,β), then U= lnS∼SN(α, lnβ,2). Later, Rieck (1999) obtained expressions for integer and fractional moments for distribution (22) using the moment generating function (MGF) of the twoparameter SN distribution via the relation
This method is applicable to the GBS_{2} distribution as well. Following Rieck (1999), the MGF for the threeparameter SN distribution (21) is given by
where r is any real number and K _{ ω }(z) is a modified Bessel function of the third kind of order ω (Watson 1995). This function K _{ ω }(z) can be expressed in an integral form as
Numerous software packages (e.g., R, SPLUS, EXCEL) can evaluate K _{ ω }(z) for specific parameter values of ω and z. Of interest here is the result from Rieck (1999) that the denominator in (23) reduces to \(\alpha \sqrt {2 \pi } \exp ( \alpha ^{2})\). Using this result, we obtain the expression
Thus, (24) can be used to calculate the mean and variance for the GBS_{2}(α,β,ν) distribution analogous to the formulae provided in (3) and (4) for the BS model.
4.2 Estimation of parameters
Suppose T _{1},T _{2},…,T _{ n } are independent and identically distributed random variables from the GBS_{2} distribution with CDF in (18) and PDF in (19). We denote the observed values of T _{1},T _{2},…,T _{ n } by t _{1},t _{2},…,t _{ n }. The ML estimation method can be used here. The likelihood function can be expressed as
Thus, the loglikelihood function is
Taking derivatives of the loglikelihood function with respect to the parameters and setting these derivatives to zeros, we have the likelihood equations (see Appendix):
The maximum likelihood estimators (MLEs) of β and ν, denoted by \({\hat \beta }\) and \(\hat \nu \), respectively, can be obtained by solving Eqs. (26) and (27) simultaneously. Numerical methods for twodimensional optimizations can be employed here. Then, the MLEs of α, \(\hat \alpha \), can be obtained by Eq. (25). Then, the observed Fisher information matrix is given by
where
Hence, a local estimate of the asymptotic variancecovariance matrix of the MLEs can be obtained by inverting the observed Fisher information matrix as
Following the asymptotic theory of MLEs and the fact that α, β and ν are positive parameters, use log transformations to obtain approximate confidence intervals for these parameters (see, for example, Meeker and Escobar 1998). Specifically, for parameter α, we can approximate the distribution of \(\frac {\ln ({\hat \alpha })  \ln (\alpha)}{\sqrt {Var(\ln ({\hat \alpha }))}}\) by a standard normal distribution, where the variance of the logtransformed MLE, \(Var(\ln ({\hat \alpha }))\), can be approximated by delta method as
A twosided 100(1−δ) % normalapproximation confidence interval for α obtained in this manner is then given by
where z _{ q } is the qth upper percentile of a standard normal distribution. Following the same procedure, normalapproximation confidence intervals for β and ν can be constructed.
Beside the use of ML estimation, moment and quantile based estimators can also be considered. Based on the first moments of the random variable T and 1/T, we have
Therefore, the parameter β can be estimated by using the first sample moments of T and 1/T, i.e.,
Consider the monotone transformation
which follows a normal distribution with mean 0 and variance α ^{2}/4. For known values of β and ν, by equating the sample second moment of W to α ^{2}/4, an estimator of the parameter α can be expressed as
Note that this equation is equivalent to the estimate obtained based on the likelihood equation for α.
Since the parameter β is the median of the distribution, therefore, the median of T _{1},T _{2},…,T _{ n } can be used as an estimate of β. Let us denote this estimator as
After obtaining the estimate of β, we can consider the transformed data \(x_{i} = t_{i}/{\tilde \beta }\), i=1,2,…,n and denote the corresponding order statistics as x _{1:n }<x _{2:n }<…<x _{ n:n }. Consider the following nonlinear regression model,
where
and ε _{ i } is the error term. Nonlinear leastsquares estimates of the parameters α and ν of the above model can be obtained.
In order to evaluate the performance of different estimation procedures for the parameters of the GBS_{2}(α,β,ν) distribution, a Monte Carlo simulation study is conducted. We consider the following estimation procedures:

Method 1. Estimate β by \({\tilde \beta }_{1}\), then obtain an estimate of ν, say \({\tilde \nu }_{1}\) by Eq. (27) by substituting \(\beta = {\tilde \beta }_{1}\). Use Eq. (25) to obtain an estimate of α, say \({\tilde \alpha }_{1}\) with \(\beta = {\tilde \beta }_{1}\) and \(\nu = {\tilde \nu }_{1}\).

Method 2. Estimate β by \({\tilde \beta }_{2}\), then obtain an estimate of ν, say \({\tilde \nu }_{2}\) by Eq. (27) by substituting \(\beta = {\tilde \beta }_{2}\). Use Eq. (25) to obtain an estimate of α, say \({\tilde \alpha }_{2}\) with \(\beta = {\tilde \beta }_{2}\) and \(\nu = {\tilde \nu }_{2}\).

Method 3. Estimate β by \({\tilde \beta }_{1}\), then obtain estimates of ν and α, say \({\tilde \nu }^{*}_{1}\) and \({\tilde \alpha }^{*}_{1}\), by using the nonlinear leastsquares method.

Method 4. Estimate β by \({\tilde \beta }_{2}\), then obtain estimates of ν and α, say \({\tilde \nu }^{*}_{2}\) and \({\tilde \alpha }^{*}_{2}\), by using the nonlinear leastsquares method.

Method 5. Maximum likelihood estimation based on solving Eqs. (25) – (27).
In the simulation study, 1,000 simulations are used to estimate the biases and mean squared errors (MSEs) of these estimators. We consider the parameter settings α=1,ν=0.5,β=0.5,1.0,1.5; α=1,ν=0.9,β=0.5,1.0,1.5; α=2,ν=0.5,β=0.5,1.0,1.5. The simulation results are presented in Table 2, Table 3 and Table 4 for sample sizes n=20, 40 and 60.
From Table 2, Table 3 and Table 4, we observed that the ML estimators (Method 5) do not perform well in terms of MSE for small to moderate sample sizes (say, n=20 and n=40), especially for the estimation of the parameter α. Even for large sample size (n=60), the MSEs of the ML estimators are larger than those obtained by other methods in most cases. Therefore, we would not consider the ML estimators in the subsequence comparisons.
Based on the simulation results, for parameter β, the estimator based on the first sample moments of T and 1/T, i.e., \({\tilde \beta }_{1}\) (Method 1 and Method 3) gives smallest MSEs in most situations. However, the estimators for α and ν based on the value of \({\tilde \beta }_{1}\) are not the best among all the methods considered here. For estimation of parameters α and ν with small sample sizes (n=20), we observed that Method 4 performs better when the true value of α=1.0 and Method 2 performs better when the true value of α=2.0. It is interesting to point out that even Method 4 does not perform as well as Method 2 when the true value of α=2.0, the variances of the estimators from Method 4 are much smaller than that of Method 2. For moderate and large sample sizes (n=40 and 60), Method 2 gives the smaller MSEs in most cases.
Overall speaking, we would recommend the use of momentbased estimator \({\tilde \beta }_{1}\) for the estimation of the parameter β for any sample sizes. For small sample sizes, we would suggest the use of nonlinear leastsquared method by setting \(\beta = {\tilde \beta }_{2}\) (i.e., Method 4) to estimate the parameters α and ν. For moderate to large sample sizes, the use of likelihood equations by setting \(\beta = {\tilde \beta }_{2}\) (i.e., Method 2) is recommended.
Illustrative examples
5.1 Example 1: simulated data from twofold Weibull mixture
In this subsection, we use a simulated dataset to illustrate the usefulness of the generalized BirnbaunSaunders distribution in modeling bimodal data and the estimation procedures studied in Section 4. We consider a twofold Weibull mixture model which is commonly used to model twofold competing risk failure mechanism involving two failure modes in reliability engineering. The PDF of the twofold Weibull mixture is given by (see, for example, Murthy et al. 2004, Razal and Salih 2009):
where
is the density of the Weibull distribution with shape parameter a>0 and scale parameter b>0, and 0<w<1 is the mixing parameter. A sample of size 50 is generated from the twofold Weibull mixture in (29) with w=0.6, a _{1}=6, b _{1}=1, a _{2}=6, b _{2}=1. The dataset is presented in Table 5 and the GSB_{2} distribution is used to model this dataset. The estimates of the model parameters based on different methods studied in Section 4 are presented in Table 6 and the histogram of the dataset with the fitted probability density functions are plotted in Fig. 5. We also presented the 95 % confidence intervals for the model parameters based on normalapproximation of the logtransformed MLEs that are discussed in Section 4.
Here, we also compare the model studied in Section 4 to the threeparameter BirnbaumSaunders distribution studied in Owen (2006) with PDF in (8). For the dataset presented in Table 5, the maximum likelihood estimates of the parameters a, b and κ are 0.3856, 1.1336 and 0.4536, respectively, which give the maximum loglikelihood to be −63.80. Comparing to the maximum loglikelihood based on the proposed threeparameter GSB_{2} distribution which is −23.69, the proposed model clearly provides a better fit for this dataset.
Although the fiveparameter model in (29) is more flexible than the threeparameter GSB_{2} distribution, from this example, we can see that the GSB_{2} distribution can be a simpler and effective alternative to model the bimodal behavior in the density function.
5.2 Example 2: spot exchange rate of euro into sterling pound
To further illustrate the usefulness of the GSB_{2} distribution in modeling bimodal data, we consider a real data example of spot exchange rate of the Euro into sterling pound. The data for this example were downloaded from the Bank of England Statistical Interactive Database (http://www.bankofengland.co.uk/boeapps/iadb) which consist of 3,786 daily observations on the spot exchange rate of the Euro into sterling pound during the period August 29, 2000 to August 15, 2015. A random sample of size 100 of these 3,786 daily spot exchange rate observations are presented in Table 7 and the histogram of the data is presented in Fig. 6, where it is readily seen that the distribution is bimodal. The GSB_{2} distribution is used to model this dataset. The estimates of the model parameters based on different methods studied in Section 4 are presented in Table 8 and the fitted probability density functions are plotted in Fig. 6. We also presented the 95 % confidence intervals for the model parameters based on normalapproximation of the logtransformed MLEs that are discussed in Section 4.
The study of Boothe and Glassman (1987) showed that the mixture of two normal distributions is one of the best models to describe exchange rate data, hence we fit the exchange rate data in Table 7 with a mixture of two normal distributions with PDF
where 0≤λ≤1 and
−∞<μ _{ j }<∞ and σ _{ j }>0 for j=1,2. The above normal mixture model is fitted by using the R package normalmixEM (Benaglia et al. 2009). The MLEs of the model parameters based on the exchange rate data in Table 7 are \({\hat \lambda } = 0.4681\), \({\hat \mu }_{1} = 1.2002\), \({\hat \mu }_{2} = 1.4992\), \({\hat \sigma }_{1} = 0.0623\) and \({\hat \sigma }_{2} = 0.0951\) with the maximum loglikelihood value to be −52.2517. Comparing to the maximum loglikelihood based on the proposed threeparameter GSB_{2} distribution which is −52.5557, the GSB_{2} model provides similar maximum likelihood in fitting this dataset using three parameters instead of five parameters. Note that the Akaike information criterion (AIC) of the GSB_{2} distribution is 94.5034 for fitting the data set in Table 7, while the AIC of the mixture of two normal distributions is 99.1114. Therefore, the generalized BirnbaumSaunders GSB_{2} distribution could be chosen as a better model based on AIC.
Conclusions
The BirnbaumSaunders and inverseGaussian distributions have a long, rich history in statistical literature. They have often deemed as interchangeable, and this article brought to light more comparisons of their utility and similarities. The momentratio diagrams showed another way that the densities and higher moments are quite similar. A generalized BirnbaunSaunders distribution, the GBS_{2} model, is a threeparameter distribution that not only includes the usual twoparameter BS distribution as a special case but also shares some unique relationships with the IG model. Lastly, the fact that the GBS_{2} model can exhibit bimodality with certain parameter values makes it a very flexible distribution. It is hoped that this paper generates increased interests in the BirnbaumSaunders, inverseGaussian and the threeparameter generalized BirnbaumSaunders GBS_{2} models.
Appendix: likelihood equations for GBS_{2} distribution
Suppose t _{1},t _{2},…,t _{ n } is a random sample of size n from the GBS_{2} distribution with CDF in (18) and PDF in (19). The loglikelihood function is
Taking derivatives of the loglikelihood function with respect to the parameters, we have
By setting these derivatives to zeros, we have the following likelihood equations
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Acknowledgements
The authors gratefully thank two associate editors and two anonymous reviewers for their suggestions that substantially improved the article. H.K.T. Ng’s work was supported by a grant from the Simons Foundation (#280601). H.K.T. Ng would like to pay his respect to his great coauthor, Professor William Jason Owen, who passed away during the preparation of this manuscript.
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HKTN and WJO contributed equally to the manuscript. HKTN is the only author who read and approved the final manuscript since WJO passed away before the manuscript was submitted.
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Keywords
 Bimodality
 Cycles to failure
 Fatigue
 Generalized BirnbaumSaunders distribution
 Momentratio diagram