Revisit of relationships and models for the BirnbaumSaunders and inverseGaussian distributions
 William Jason Owen^{1} and
 Hon Keung Tony Ng^{2}Email author
https://doi.org/10.1186/s4048801500348
© Owen and Ng. 2015
Received: 29 May 2015
Accepted: 22 October 2015
Published: 3 November 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.
Keywords
Bimodality Cycles to failure Fatigue Generalized BirnbaumSaunders distribution Momentratio diagramIntroduction
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 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).
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).
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).
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}.
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
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

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.
Coefficient of variation γ _{2}, coefficient of skewness γ _{3}, and kurtosis γ _{4} for BS and IG distributions
Distribution  γ _{2}  γ _{3}  γ _{4} 

BS(α, β)  \(\frac {\alpha \sqrt {4 + 5 \alpha ^{2}}}{2 + \alpha ^{2}}\)  \(\frac {4 \alpha (6 + 11 \alpha ^{2})}{(4 + 5 \alpha ^{2})^{3/2}}\)  \(3 + \frac {6 \alpha ^{2} (40 + 93 \alpha ^{2})}{(4 + 5 \alpha ^{2})^{2}}\) 
IG(μ, λ)  \(\sqrt {\frac {\mu }{\lambda }}\)  \(3\sqrt {\frac {\mu }{\lambda }}\)  \(3 + 15 \left (\frac {\mu }{\lambda }\right)\) 
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).
A generalized BS distribution
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
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.
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).
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
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.
Note that this equation is equivalent to the estimate obtained based on the likelihood equation for α.
and ε _{ i } is the error term. Nonlinear leastsquares estimates of the parameters α and ν of the above model can be obtained.

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).
Simulated biases and MSEs for different estimation procedures for parameter setting α=1.0,ν=0.5
β  α  ν  

Bias  MSE  Bias  MSE  Bias  MSE  
n=20  β=1.0  Method 1  0.0195  0.0425  0.9742  4.0217  0.2869  0.3524 
Method 2  0.0434  0.0813  0.4214  2.2961  0.0974  0.2553  
Method 3  0.0195  0.0425  0.1487  2.1084  –0.0322  0.2548  
Method 4  0.0434  0.0813  –0.4201  1.3537  –0.2534  0.2354  
Method 5  0.0307  0.0529  1.3228  7.3012  0.3447  0.4640  
β=1.5  Method 1  0.0381  0.0932  0.9322  3.2543  0.2855  0.3089  
Method 2  0.0650  0.1731  0.3446  1.7079  0.0759  0.2060  
Method 3  0.0381  0.0932  0.0979  1.6965  –0.0443  0.2157  
Method 4  0.0650  0.1731  –0.4908  1.1100  –0.2836  0.2131  
Method 5  0.0408  0.1285  1.4301  7.0455  0.3891  0.4982  
β=0.5  Method 1  0.0108  0.0102  0.9442  4.3086  0.2758  0.3304  
Method 2  0.0217  0.0216  0.4038  2.0323  0.0936  0.2282  
Method 3  0.0108  0.0102  0.1228  2.1729  –0.0445  0.2402  
Method 4  0.0217  0.0216  –0.4360  1.2374  –0.2616  0.2193  
Method 7  0.0095  0.0125  1.4200  7.2533  0.3673  0.4795  
n=40  β=1.0  Method 1  0.0174  0.0208  0.4064  1.1396  0.1259  0.1394 
Method 2  0.0267  0.0434  0.1390  0.8548  0.0183  0.1206  
Method 3  0.0174  0.0208  –0.0899  0.9007  –0.0928  0.1441  
Method 4  0.0267  0.0434  –0.4459  0.8461  –0.2485  0.1733  
Method 5  0.0112  0.0220  0.5577  1.6258  0.1622  0.1789  
β=1.5  Method 1  0.0130  0.0433  0.3903  1.1135  0.1147  0.1417  
Method 2  0.0244  0.0859  0.1178  0.7991  0.0066  0.1238  
Method 3  0.0130  0.0433  –0.0822  0.8952  –0.0911  0.1470  
Method 4  0.0244  0.0859  –0.4382  0.7508  –0.2415  0.1621  
Method 5  0.0140  0.0495  0.5002  1.5122  0.1441  0.1717  
β=0.5  Method 1  0.0037  0.0050  0.3697  1.0955  0.1061  0.1347  
Method 2  0.0060  0.0093  0.1313  0.8476  0.0104  0.1214  
Method 3  0.0037  0.0050  –0.1212  0.8887  –0.1097  0.1481  
Method 4  0.0060  0.0093  –0.4268  0.8502  –0.2423  0.1716  
Method 5  –0.0001  0.0056  0.5676  1.5449  0.1693  0.1721  
n=60  β=1.0  Method 1  –0.0023  0.0120  0.2280  0.6084  0.0645  0.0888 
Method 2  0.0046  0.0267  0.0521  0.5265  –0.0104  0.0873  
Method 3  –0.0023  0.0120  –0.1512  0.5816  –0.1065  0.1105  
Method 4  0.0046  0.0267  –0.4006  0.6432  –0.2208  0.1402  
Method 5  0.0132  0.0138  0.3006  0.7754  0.0893  0.1049  
β=1.5  Method 1  0.0089  0.0309  0.2217  0.6390  0.0642  0.0901  
Method 2  0.0178  0.0562  0.0497  0.5494  –0.0094  0.0881  
Method 3  0.0089  0.0309  –0.1560  0.6107  –0.1074  0.1129  
Method 4  0.0178  0.0562  –0.3966  0.6474  –0.2171  0.1368  
Method 5  0.0044  0.0330  0.3469  0.8634  0.1039  0.1131  
β=0.5  Method 1  0.0018  0.0032  0.2649  0.6466  0.0822  0.0934  
Method 2  0.0026  0.1010  0.0850  0.5690  0.0053  0.0911  
Method 3  0.0018  0.0032  –0.1172  0.5847  –0.0889  0.0145  
Method 4  0.0026  0.1010  –0.3739  0.6695  –0.2078  0.1414  
Method 5  0.0009  0.0033  0.3197  0.7679  0.0986  0.1013 
Simulated biases and MSEs for different estimation procedures for parameter setting α=2.0,ν=0.5
β  α  ν  

Bias  MSE  Bias  MSE  Bias  MSE  
n=20  β=1.0  Method 1  0.0386  0.1125  1.3195  7.9982  0.1482  0.0990 
Method 2  0.1371  0.4192  0.0557  2.9805  –0.0424  0.0740  
Method 3  0.0386  0.1125  0.4002  4.6121  –0.0118  0.0910  
Method 4  0.1371  0.4192  –1.1470  3.0691  –0.3047  0.1604  
Method 5  0.0749  0.1563  1.6102  12.0105  0.1629  0.1341  
β=1.5  Method 1  0.0741  0.2669  1.2255  6.2592  0.1490  0.0933  
Method 2  0.2483  0.9636  0.0169  2.5749  –0.0439  0.0760  
Method 3  0.0741  0.2669  0.3387  3.8775  –0.0086  0.0843  
Method 4  0.2483  0.9636  –1.1137  3.0016  –0.2946  0.1566  
Method 5  0.0736  0.2879  1.8437  14.1580  0.1863  0.1367  
β=0.5  Method 1  0.0300  0.0311  1.1648  6.3143  0.1345  0.0929  
Method 2  0.0830  0.1013  0.0232  2.9223  –0.0484  0.0799  
Method 3  0.0300  0.0311  0.2871  3.8638  –0.0234  0.0886  
Method 4  0.0830  0.1013  –1.0694  3.2768  –0.2920  0.1621  
Method 5  0.0712  0.2610  1.7594  12.6750  0.1843  0.1315  
n=40  β=1.0  Method 1  0.0283  0.0510  0.4013  1.8797  0.0427  0.0391 
Method 2  0.0675  0.1607  –0.1425  1.3239  –0.0564  0.0424  
Method 3  0.0283  0.0510  –0.0800  1.4982  –0.0522  0.0469  
Method 4  0.0675  0.1607  –0.9059  2.0129  –0.2316  0.1069  
Method 5  0.0253  0.0554  0.6419  2.4061  0.0782  0.0458  
β=1.5  Method 1  0.0386  0.1025  0.5161  1.9371  0.0593  0.0384  
Method 2  0.1448  0.4378  –0.0506  1.2901  –0.0414  0.0429  
Method 3  0.0386  0.1025  0.0286  1.5041  –0.0350  0.0453  
Method 4  0.1448  0.4378  –0.8061  1.8517  –0.2089  0.0993  
Method 5  0.0369  0.1242  0.6034  2.3951  0.0697  0.0439  
β=0.5  Method 1  0.0111  0.0126  0.5099  2.0122  0.0580  0.0370  
Method 2  0.0450  0.0478  –0.0599  1.4283  –0.0459  0.0443  
Method 3  0.0111  0.0126  0.0191  1.5585  –0.0372  0.0440  
Method 4  0.0450  0.0478  –0.8308  2.0662  –0.2190  0.1056  
Method 5  0.0121  0.0142  0.7498  2.9983  0.0894  0.0511  
n=60  β=1.0  Method 1  0.0101  0.0312  0.3484  1.2761  0.0380  0.0300 
Method 2  0.0454  0.1185  –0.0580  0.9684  –0.0351  0.0315  
Method 3  0.0101  0.0312  0.0024  1.0757  –0.0282  0.0327  
Method 4  0.0454  0.1185  –0.6773  1.4934  –0.1730  0.0763  
Method 5  0.0117  0.0341  0.4055  1.3271  0.0490  0.0275  
β=1.5  Method 1  0.0288  0.0703  0.2911  0.9903  0.0348  0.0239  
Method 2  0.0813  0.2419  –0.0925  0.7792  –0.0361  0.0269  
Method 3  0.0288  0.0703  –0.0473  0.8886  –0.0320  0.0278  
Method 4  0.0813  0.2419  –0.6837  1.3505  –0.1694  0.0701  
Method 5  0.0366  0.0835  0.3219  1.1432  0.0369  0.0272  
β=0.5  Method 1  0.0120  0.0091  0.3565  1.1113  0.0423  0.0255  
Method 2  0.0365  0.0323  –0.0721  0.8640  –0.0365  0.0296  
Method 3  0.0120  0.0091  0.0150  0.9128  –0.0228  0.0269  
Method 4  0.0365  0.0323  –0.6760  1.3800  –0.1699  0.0720  
Method 5  0.0051  0.0081  0.3621  1.2234  0.0432  0.0260 
Simulated biases and MSEs for different estimation procedures for parameter setting α=1.0,ν=0.9
β  α  ν  

Bias  MSE  Bias  MSE  Bias  MSE  
n=20  β=1.0  Method 1  0.0098  0.0132  0.8859  4.1859  0.4528  1.0031 
Method 2  0.0126  0.0243  0.4174  2.6917  0.1578  0.7571  
Method 3  0.0098  0.0132  0.0205  2.1971  –0.1634  0.7715  
Method 4  0.0126  0.0243  –0.4101  1.4524  –0.4618  0.7328  
Method 5  0.0083  0.0153  1.4731  7.7400  0.6839  1.6105  
β=1.5  Method 1  0.0130  0.0309  0.8992  3.9446  0.4732  1.0613  
Method 2  0.0220  0.0545  0.4182  2.1596  0.1773  0.7605  
Method 3  0.0130  0.0309  0.0643  2.2646  –0.1191  0.7975  
Method 4  0.0220  0.0545  –0.4120  1.3491  –0.4478  0.7151  
Method 5  0.0174  0.0324  1.3639  6.1369  0.6680  1.4928  
β=0.5  Method 1  0.0016  0.0033  0.8285  3.4397  0.4223  0.9522  
Method 2  0.0022  0.0060  0.3769  1.9818  0.1481  0.6989  
Method 3  0.0016  0.0033  –0.0263  1.8751  –0.1866  0.7413  
Method 4  0.0022  0.0060  –0.4537  1.2252  –0.4826  0.6884  
Method 5  0.0041  0.0039  1.3170  6.0487  0.6453  1.4899  
n=40  β=1.0  Method 1  0.0038  0.0066  0.3101  0.9566  0.1507  0.4135 
Method 2  0.0058  0.0128  0.1039  0.7441  0.0009  0.3774  
Method 3  0.0038  0.0066  –0.1887  0.7752  –0.2426  0.4574  
Method 4  0.0058  0.0128  –0.4548  0.7507  –0.4519  0.5265  
Method 5  0.0029  0.0070  0.5476  1.5316  0.2904  0.5658  
β=1.5  Method 1  0.0002  0.0144  0.3060  1.0136  0.1518  0.4263  
Method 2  0.0002  0.0270  0.1067  0.8040  0.0045  0.3930  
Method 3  0.0002  0.0144  –0.2057  0.8406  –0.2554  0.4841  
Method 4  0.0002  0.0270  –0.4648  0.8255  –0.4624  0.5604  
Method 5  0.0102  0.0149  0.5592  1.5119  0.3069  0.5646  
β=0.5  Method 1  –0.0010  0.0015  0.3810  1.0063  0.2074  0.4302  
Method 2  –0.0004  0.0029  0.1759  0.7747  0.0596  0.3760  
Method 3  –0.0010  0.0015  –0.1378  0.7647  –0.1964  0.4470  
Method 4  –0.0004  0.0029  –0.4162  0.7652  –0.4181  0.5223  
Method 5  0.0015  0.0017  0.5205  1.5198  0.2768  0.5461  
n=60  β=1.0  Method 1  0.0012  0.0042  0.1727  0.5760  0.0825  0.2853 
Method 2  0.0017  0.0076  0.0400  0.5214  –0.0212  0.2874  
Method 3  0.0012  0.0042  –0.2268  0.5858  –0.2465  0.3796  
Method 4  0.0017  0.0076  –0.4125  0.6330  –0.4002  0.4547  
Method 5  0.0013  0.0041  0.3112  0.7285  0.1738  0.3287  
β=1.5  Method 1  –0.0019  0.0094  0.2492  0.6194  0.1329  0.2882  
Method 2  –0.1581  0.0429  0.1031  0.5236  0.0221  0.2745  
Method 3  –0.0019  0.0094  –0.0013  0.5552  –0.0017  0.3187  
Method 4  –0.1581  0.0429  –0.3658  0.5916  –0.3655  0.4219  
Method 5  0.0072  0.0104  0.3172  0.7224  0.1742  0.3264  
β=0.5  Method 1  0.0017  0.0010  0.1963  0.5976  0.0976  0.2855  
Method 2  0.0032  0.0020  0.0502  0.5026  –0.0130  0.2738  
Method 3  0.0017  0.0010  –0.2188  0.5991  –0.2424  0.3867  
Method 4  0.0032  0.0020  –0.4192  0.6177  –0.4049  0.4456  
Method 5  0.0010  0.0011  0.3104  0.8103  0.1656  0.3594 
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
Simulated dataset from a twofolded Weibull mixture model
0.5446  0.5934  0.5958  0.6106  0.6335  0.6945  0.7097  0.7431  0.7587  0.7851 
0.8162  0.8450  0.8502  0.8503  0.8549  0.8743  0.8801  0.9335  0.9833  0.9836 
1.0268  1.0394  1.0412  1.0472  1.0732  1.0849  1.0948  1.1129  1.1673  1.1824 
1.3267  1.4230  1.4706  1.4995  1.5185  1.5287  1.5353  1.5362  1.5797  1.5995 
1.7477  1.7985  1.8231  1.8341  1.8701  1.8793  1.9394  1.9522  2.0401  2.1118 
Estimates of the parameters in GSB_{2} for dataset presented in Table 5
Estimate of  

α  β  ν  
Method 1  2.9565  1.1256  2.7307 
Method 2  3.5729  1.0790  3.0254 
Method 3  3.4970  1.1256  3.0461 
Method 4  3.6660  1.0790  3.1143 
Method 5  3.6325  1.1065  3.1135 
95 % CI based on MLE (Method 5)  (1.5859, 8.3186)  (0.9193, 1.3319)  (2.0135, 4.8140) 
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
Random sample of size 100 from daily observations on the spot exchange rate of the Euro into sterling pound during the period August 29, 2000 to August 15, 2015
1.4404  1.1897  1.3923  1.6333  1.2857  1.1726  1.2251  1.4574  1.4757  1.5566 
1.1298  1.1787  1.2004  1.2074  1.4873  1.4725  1.4547  1.2513  1.1411  1.3908 
1.5754  1.5698  1.1461  1.6241  1.4463  1.4910  1.4633  1.4837  1.4326  1.2799 
1.4767  1.4536  1.2907  1.1647  1.1254  1.4639  1.1938  1.2684  1.4454  1.4352 
1.2393  1.6595  1.4127  1.1448  1.4688  1.4623  1.1615  1.2122  1.1044  1.5635 
1.2000  1.4872  1.2342  1.1908  1.4856  1.4803  1.6295  1.6159  1.6568  1.1092 
1.2565  1.2472  1.4729  1.4348  1.1293  1.3284  1.1209  1.6139  1.5138  1.4868 
1.2270  1.5058  1.3918  1.1451  1.3256  1.4337  1.2467  1.5673  1.1771  1.2635 
1.1433  1.3280  1.4605  1.4453  1.6184  1.0856  1.2671  1.1414  1.3252  1.5695 
1.1762  1.3044  1.7034  1.6238  1.5790  1.4554  1.2554  1.3363  1.2449  1.6246 
1.1963  1.2469  1.6014  1.1681  1.1846  1.3171  1.1761  1.1405  1.6248  1.4294 
Estimates of the parameters in GSB_{2} for dataset presented in Table 7
Estimate of  

α  β  ν  
Method 1  4.9277  1.3487  11.2675 
Method 2  3.5225  1.3324  9.2726 
Method 3  5.5220  1.3487  11.9603 
Method 4  4.7636  1.3324  11.0038 
Method 5  5.6042  1.3529  12.0478 
95 % CI based on MLE (Method 5)  (3.2669, 9.6136)  (1.2985, 1.4095)  (9.4995, 15.2797) 
−∞<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
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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