On a generalized Birnbaum Saunders Distribution

09/01/2021
by   Beenu Thomas, et al.
0

In this paper, a generalization for the Birnbaum Saunders distribution, which has been applied to the modelling of fatigue failure times and reliability studies, is considered. The maximum likelihood estimators and statistical inference for the distribution parameters are presented. Corresponding bivariate and multivariate distributions are proposed. The proposed distribution is applied to model real data sets.

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1 Introduction

Normal distribution is the commonly used statistical distribution. Several distributions have been evolved by making some transformations on Normal distribution. Two-parameter Birnbaum-Saunders(BS) distribution is one such distribution introduced by Birnbaum and SaundersBS(1969a) , which have been developed by employing a monotone transformation on the standard normal distribution. Bhattacharyya and FriesBF(1982) established that a BS distribution can be obtained as an approximation of an inverse Gaussian (IG) distribution. DesmondD(1986) examined an interesting feature that a BS distribution can be viewed as an equal mixture of an IG distribution and its reciprocal. This property becomes helpful in deriving some properties of the BS distribution using properties of IG distribution.

Many different properties of BS distribution have been discussed by a number of authors. It has been observed that the probability density function of the BS distribution is unimodal. The shape of the hazard function (HF) plays an important role in lifetime data analysis. Kundu et al.

kkb(2008) and Bebbington et al.BLZ(2008) proved that HF of BS distriution is an unimodal function. The maximum likelihood estimator(MLE)s of the shape and scale parameters based on a complete sample were discussed by Birnbaum and SaundersBS(1969b) . The estimators of the unknown parameters in the case of a censored sample was first developed by RieckR(1995) . Ng et al.nkb(2003) provided modified moment estimator(MME)s of the parameters in the case of a complete sample which are in explicit form. Lifl(2006) suggested four different estimation techniques for both complete and censored samples.

Several other models associated with BS distribution and their properties have been discussed in literature. For example, Rieck and NedelmanRN(1991) considered the log-linear model for the BS distribution. Desmond et al.DRL(2008) considered the BS regression model. Lemonte and CordeiroLC(2009)

considered the BS non-linear regression model. Kundu et al.

kbj(2010) introduced the bivariate BS distribution and studied some of its properties and characteristics. The multivariate generalized BS distribution has been introduced, by replacing the normal kernel by an elliptically symmetric kernel, by Kundu et al.kbj(2010) . A generalization of BS distribution is done by Chacko et al.cmb(2015) . Further research in inference and its generalization to bivariate and multivariate framework is still not addressed.

In this paper, the generalization of Birnbaum Saunders distribution is considered. Section 2 reviewed univariate, bivariate and multivariate Birnbaum Saunders distribution. Section 3 discussed the univariate generalization of Birnbaum Saunders distribution. Section 4 discussed the bivariate generalization of Birnbaum Saunders distribution. Section 5 discussed the multivariate generalization of Birnbaum Saunders distribution. Application to real data set is given in section 6. Conclusions are given at the last section.

2 Birnbaum Saunders Distribution

2.1 Univariate Birnbaum Saunders Distribution

A random variable following the BS distribution is defined through a standard Normal random variable. Therefore the probability density function(pdf) and cumulative distribution function(cdf) of the BS model can be expressed in terms of the standard Normal pdf and cdf.


The cdf of a two parameter BS random variable T for can be written as

(1)

where is the standard Normal cdf. The pdf of BS distribution is

(2)

Here and are the shape and scale parameters respectively.

2.2 Bivariate Birnbaum Saunders Distribution

The bivariate Birnbaum-Saunders (BVBS) distribution was introduced by Kundu et.alkbj(2010)

. The bivariate random vector

is said to have a BVBS distribution with parameters if the cumulative distribution function of can be expressed as

(3)

for . Here is the cdf of standard bivariate normal vector with correlation coefficient . The corresponding pdf is

for .

2.3 Multivariate Birnbaum Saunders Distribution

Kundu et al.kbj(2013) introduced the multivariate BS distribution. Let , where and , with for . Let be a positive-definite correlation matrix. Then, the random vector is said to have a p-variate BS distribution with parameters if it has the joint CDF as

for Here, for denotes the joint cdf of a standard normal vector with correlation matrix . The joint pdf of can be obtained from the above equation as

for here, for

is the pdf of the standard Normal vector with correlation matrix .
Now consider the generalization of BS distribution.

3 Univariate -Birnbaum Saunders Distribution

Following Chacko et al. cmb(2015) , consider the univariate BS distribution.
Let instead of in the univariate BS distribution. The cdf of a univariate -BS random variable T can be written as

(4)

where is the standard normal cdf. The parameters and in 4 are the shape and scale parameters, respectively. The parameter governs both scale and shape.
If the random variable T has the BS distribution function in 4, then the corresponding pdf is

(5)

The pdf is unimodal. Now the moments can be obtained as below.

3.1 Moments

If T is a - BS distribution, denoted as , then the moments of this distribution are obtained by making transformation

Using this, we get the moments.

3.2 Property of Birnbaum Saunders Distribution

Theorem 1.

If T has a -BS distribution with parameters and then also has a -BS distribution with parameters and

Proof.

Put ,     ,

3.3 Estimation

The estimation of parameters can be done by method of maximum likelihood.

3.3.1 Maximum Likelihood Estimates

Let , ,…, be the random sample of size n. Based on a random sample, the MLEs of the unknown parameters can be obtained by maximising the log-likelihood function. The likelihood function is

The log-likelihood is

Equating the partial derivative of log-likelihood function with respect to parameters, to zero, we get,

and

The equations can be solved numerically.

4 Bivariate - Birnbaum Saunders Distribution

The bivariate random vector is said to have a bivariate BS distribution with parameters if the joint cdf of can be expressed as

Here and is cdf of standard bivariate Normal vector with correlation coefficient . The corresponding joint pdf of and is given by

where denotes the joint pdf of and given by

Now

Theorem 2.

If BV -BS then .

Proof.

Theorem 3.

If then

4.0.1 Estimation

Based on a bivariate random sample from the distribution, the MLEs of the unknown parameters can be obtained by maximizing the log likelihood function. If we denote then the likelihood function is

The log-likelihood function is

5 Multivariate - Birnbaum Saunders Distribution

Along the same lines as the univariate and bivariate BS distribution, the multivariate BS distribution can be defined.

Then the multivariate BS distribution is as follows:

Definition 1.

Let , where and , with for . Let be a positive definite correlation matrix. Then, the random vector is said to have a m-variate BS distribution with parameters if it has the joint CDF as

for and . Here, for denotes the joint cdf of a standard Normal vector with correlation matrix .

5.1 Applications

Birnbaum and Saunders (1958) obtained fatigue life real data corresponding to cycles until failure of aluminum specimens of type 6061-T6, see Table 1. These specimens were cut parallel to the direction of rolling and oscillating at 18 cycles per seconds. They were exposed to a pressure with maximum stress 31,000 pounds per square inch (psi) for specimens for each level of stress. All specimens were tested until failure.

70 90 96 97 99 100 103 104 104 105 107 108 108 108 109
109 112 112 113 114 114 114 116 119 120 120 120 121 121 123
124 124 124 124 124 128 128 129 129 130 130 130 131 131 131
131 131 132 132 132 133 134 134 134 134 134 136 136 137 138
138 138 139 139 141 141 142 142 142 142 142 142 144 144 145
146 148 148 149 151 151 152 155 156 157 157 157 157 158 159
162 163 163 164 166 166 168 170 174 196 212
Table 1: Fatigue lifetime data

For this data set the point estimates of and obtained by the method of maximum likelihood are given in the table 2.

1.509180e+00 1.179090e-06 6.198936e+00

Table 2: Point estimates of and

Here Kolmogrov-Smirnov test statistic is 0.9703 and p-value is 0.3088. Since p-value is greater than significance level we can conclude that univariate

Birnbaum-Saunders distribution is a good fit for given data.

5.2 Conclusion

This paper discussed a generalization of BS distribution.This three parameter distribution is more plausible model for the distribution of fatigue failure. There is flexibility in selection of models. Peakedness depends on the value of

. It is a good model for distributions with smaller variances. Moreover the shape of density curve with various skewness and kurtosis provide a well defined class of life distributions useful in reliability and social sciences. Snedecors F distribution has the property that reciprocal is also F, similar property holds for

Birnbaum Saunders distribution. But computation of MLE is a complicated one. But numerical procedure is applied for computation of MLE.

References

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