On a generalized Birnbaum Saunders Distribution

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

α > 0, β > 0

can be written as

F_{T} (t; α, β) = Φ [\frac{1}{α} (\frac{t}{β})^{\frac{1}{2}} - (\frac{β}{t})^{\frac{1}{2}}], t > 0

(1)

where $Φ (.)$ is the standard Normal cdf. The pdf of BS distribution is

(2)

Here $α > 0$ and $β > 0$ 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

(T_{1}, T_{2})

is said to have a BVBS distribution with parameters

α_{1}, β_{1}, α_{2}, β_{2}, ρ

if the cumulative distribution function of

(T_{1}, T_{2})

can be expressed as

F (t_{1}, t_{2})

=

(3)

for $t_{1} > 0, t_{2} > 0, α_{1} > 0, β_{1} > 0, α_{2}, β_{2} > 0 a n d - 1 < ρ < 1$ . Here $Φ_{2} (u, v; ρ)$ is the cdf of standard bivariate normal vector $(Z_{1}, Z_{2})$ with correlation coefficient $ρ$ . The corresponding pdf is

for $t_{1} > 0, t_{2} > 0, α_{1} > 0, β_{1} > 0, α_{2}, β_{2} > 0 a n d - 1 < ρ < 1$ .

2.3 Multivariate Birnbaum Saunders Distribution

Kundu et al.kbj(2013) introduced the multivariate BS distribution. Let $α - -, β - - \in R^{p}$ , where $α - - = (α_{1}, \dots, α_{p})^{T}$ and $β - - = (β_{1}, \dots, β_{p})^{T}$ , with $α_{i} > 0, β_{i} > 0$ for $i = 1, \dots, p$ . Let $Γ$ be a $p \times p$ positive-definite correlation matrix. Then, the random vector $T - - = (T_{1}, \dots, T_{p})^{T}$ is said to have a p-variate BS distribution with parameters $(α - -, β - -, Γ)$ if it has the joint CDF as

	$P (T - - \leq t -)$	$= P (T_{1} \leq t_{1}, \dots, T_{p} \leq t_{p})$
		$= Φ_{p} [\frac{1}{α_{1}} (\sqrt{(} \frac{t_{1}}{} β_{1}) - \sqrt{(} \frac{β_{1}}{t_{1}})), \dots, \frac{1}{α_{p}} (\sqrt{(} \frac{t_{p}}{β_{p}}) - \sqrt{(} \frac{β_{p}}{t_{p}})); Γ]$

for $t_{1} > 0, \dots, t_{p} > 0.$ Here, for $u - - = (u_{1}, \dots, u_{p})^{T}, Φ_{p} (u - -; Γ)$ denotes the joint cdf of a standard normal vector $Z - - = (Z_{1}, \dots, Z_{p})^{T}$ with correlation matrix $Γ$ . The joint pdf of $T - - = (T_{1}, \dots, T_{p})^{T}$ can be obtained from the above equation as

	$f_{T - -} (t -; α - -, β - -, Γ)$	$= ϕ_{p} (\frac{1}{α_{1}} (\sqrt{(} \frac{t_{1}}{} β_{1}) - \sqrt{(} \frac{β_{1}}{t_{1}})), \dots, \frac{1}{α_{p}} (\sqrt{(} \frac{t_{p}}{β_{p}}) - \sqrt{(} \frac{β_{p}}{t_{p}})); Γ)$
		$\times p \prod i = 1 \frac{1}{2 α_{i} β_{i}} {(\frac{β_{i}}{t_{i}})^{\frac{1}{2}} + (\frac{β_{i}}{t_{i}})^{\frac{3}{2}}},$

for $t_{1} > 0, \dots, t_{p} > 0;$ here, for $u - - = (u_{1}, \dots, u_{p})^{T},$

ϕ_{p} (u_{1}, \dots, u_{p}; Γ) = \frac{1}{(2 π)^{\frac{p}{2}} | Γ |^{\frac{1}{2}}} e x p {- \frac{1}{2} {u - -}^{T} Γ^{- 1} u - -}

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

3 Univariate $ν$ -Birnbaum Saunders Distribution

Following Chacko et al. cmb(2015) , consider the univariate $ν -$ BS distribution.
Let $ξ_{ν} (t) = [(\frac{t}{β})^{ν} - (\frac{t}{β})^{- ν}]$ instead of $ξ (t) = [(\frac{t}{β})^{\frac{1}{2}} - (\frac{t}{β})^{- \frac{1}{2}}]$ in the univariate BS distribution. The cdf of a univariate $ν$ -BS random variable T can be written as

F (t; α, β, ν) = Φ (\frac{1}{α} ξ_{ν} (\frac{t}{β})); t > 0, α, β > 0, 0 < ν < 1

(4)

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

f (t) = \frac{ν}{α β \sqrt{(} 2 π)} e^{- \frac{1}{2 α^{2}} [(\frac{t}{β})^{2 ν} + (\frac{t}{β})^{- 2 ν} - 2] [(\frac{t}{β})^{ν - 1} + (\frac{t}{β})^{- (ν + 1)}]} .

(5)

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

3.1 Moments

If T is a $ν$ - BS distribution, denoted as $ν - B S (α, β)$ , then the moments of this distribution are obtained by making transformation

X = \frac{1}{2} [(\frac{T}{β})^{ν} - (\frac{T}{β})^{- ν}] .

X^{2} = \frac{1}{4} [(\frac{T}{β})^{2 ν} - (\frac{T}{β})^{- 2 ν} - 2] \Rightarrow

4 X^{2} + 2 = (\frac{T}{β})^{2 ν} - (\frac{T}{β})^{- 2 ν} = (\frac{β}{T})^{2 ν} [1 + (\frac{T}{β})^{4 ν}] \Rightarrow

(\frac{T}{β})^{4 ν} - (4 X^{2} + 2) (\frac{T}{β})^{2 ν} + 1 = 0.

Using this, we get the moments.

3.2 Property of $ν -$ Birnbaum Saunders Distribution

Theorem 1.

If T has a $ν$ -BS distribution with parameters $α, β$ and $ν$ then $T^{- 1}$ also has a $ν$ -BS distribution with parameters $α, β^{- 1}$ and $ν^{- 1}$

Proof.

Put $y = \frac{1}{T} \Rightarrow T = \frac{1}{y}; \frac{d T}{d y} = - \frac{1}{y^{2}}$ , $| \frac{d T}{d y} | = \frac{1}{y^{2}}$ ,

f_{y} = f_{T} | \frac{d T}{d y} | = \frac{ν}{α β \sqrt{(} 2 π)} [((\frac{1}{β y})^{(} ν - 1)) + ((\frac{1}{β y})^{- (ν + 1)})] e^{- \frac{1}{2 α^{2}} [(\frac{1}{β y})^{(2 ν)} + (β y)^{(2 ν)} - 2]} \frac{1}{y^{2}} = \frac{ν}{α β \sqrt{(} 2 π)} [(\frac{1}{β y})^{(ν - 1)} + (β y)^{(ν + 1)} e^{- \frac{1}{2 α^{2}} [(\frac{1}{β y})^{(2 ν)} + (β y)^{(2 ν)} - 2]} \frac{1}{y^{2}}] = \frac{ν}{α β \sqrt{(} 2 π)} [\frac{1}{β^{ν - 1}} y^{ν + 1} + β^{ν + 1} y^{ν - 1}] e^{- \frac{1}{2 α^{2}} [(\frac{1}{β y})^{(2 ν)} + (β y)^{(2 ν)} - 2]}

∎

3.3 Estimation

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

3.3.1 Maximum Likelihood Estimates

Let $T_{1}$ , $T_{2}$ ,…, $T_{n}$ 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

log L = n log ν - n log α - n log β - \frac{n}{2} log (2 π) - \frac{1}{2 α^{2}} n \sum i = 1 [(\frac{t_{i}}{β})^{2 ν} + (\frac{t_{i}}{β})^{- 2 ν} - 2] + n \sum i = 1 log [(\frac{t_{i}}{β})^{ν - 1} + (\frac{t_{i}}{β})^{- (ν + 1)}] .

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

	$\frac{\partial log L}{\partial α} = 0$	$\Rightarrow \frac{- n}{α} + \frac{1}{α^{3}} n \sum i = 1 [(\frac{t_{i}}{β})^{2 ν} + (\frac{β}{t_{i}})^{2 ν} - 2] = 0$
		$\Rightarrow \frac{1}{α^{2}} n \sum i = 1 [(\frac{t_{i}}{β})^{2 ν} + (\frac{β}{t_{i}})^{2 ν} - 2] = n$
		$\Rightarrow \frac{1}{n} n \sum i = 1 [(\frac{t_{i}}{β})^{2 ν} + (\frac{β}{t_{i}})^{2 ν} - 2] = α^{2}$
		$\Rightarrow α = {\frac{1}{n} n \sum i = 1 [(\frac{t_{i}}{β})^{2 ν} + (\frac{β}{t_{i}})^{2 ν} - 2]}^{\frac{1}{2}} .$

\frac{\partial log L}{\partial β} = 0 \Rightarrow \frac{n}{β} + \frac{ν}{α^{2}} n \sum i = 1 [\frac{β^{2 ν - 1}}{t_{i}^{2 ν}} - \frac{t_{i}^{2 ν}}{β^{2 ν + 1}}] + (ν + 1) n \sum i = 1 \frac{\frac{β^{ν}}{t_{i}^{ν + 1}} - \frac{t_{i}^{ν - 1}}{β^{ν}}}{(\frac{t_{i}}{β})^{ν - 1} + (\frac{β}{t_{i}})^{ν + 1}} = 0

and

\frac{\partial log L}{\partial ν} = 0 \Rightarrow \frac{n}{ν} - \frac{1}{α^{2}} n \sum i = 1 [(\frac{t_{i}}{β})^{2 ν} log (\frac{t_{i}}{β}) + (\frac{β}{t_{i}})^{2 ν} log (\frac{β}{t_{i}})] + n \sum i = 1 \frac{(\frac{t_{i}}{β})^{ν - 1} log (\frac{t_{i}}{β}) + (\frac{β}{t_{i}})^{ν + 1} log (\frac{β}{t_{i}})}{(\frac{t_{i}}{β})^{ν - 1} + (\frac{β}{t_{i}})^{ν + 1}} = 0.

The equations can be solved numerically.

4 Bivariate $ν$ - Birnbaum Saunders Distribution

The bivariate random vector $(T_{1}, T_{2})$ is said to have a bivariate BS distribution with parameters $α_{1}, β_{1}, ν_{1}, α_{2}, β_{2}, ν_{2}, ρ$ if the joint cdf of $(T_{1}, T_{2})$ can be expressed as

	$F (t_{1}, t_{2})$	$= P (T_{1} \leq t_{1}, T_{2} \leq t_{2})$
		$= Φ_{2} [\frac{1}{α_{1}} ((\frac{t_{1}}{β_{1}})^{ν_{1}} - (\frac{β_{1}}{t_{1}})^{ν_{1}}), \frac{1}{α_{2}} ((\frac{t_{2}}{β_{2}})^{ν_{2}} - (\frac{β_{2}}{t_{2}})^{ν_{2}}); ρ]; t_{1} > 0, t_{2} > 0.$

Here $α_{1} > 0, β_{1} > 0, α_{2} > 0, β_{2} > 0, - 1 < ρ < 1$ and $Φ_{2} (u, v; ρ)$ is cdf of standard bivariate Normal vector $(z_{1}, z_{2})$ with correlation coefficient $ρ$ . The corresponding joint pdf of $T_{1}$ and $T_{2}$ is given by

f_{T_{1}, T_{2}} (t_{1}, t_{2}) = ϕ_{2} [\frac{1}{α_{1}} ((\frac{t_{1}}{β_{1}})^{ν_{1}} - (\frac{β_{1}}{t_{1}})^{ν_{1}}), \frac{1}{α_{2}} ((\frac{t_{2}}{β_{2}})^{ν_{2}} - (\frac{β_{2}}{t_{2}})^{ν_{2}}); ρ] \frac{d T_{1}}{d t} \frac{d T_{2}}{d t}

where $ϕ_{2} (u, v; ρ)$ denotes the joint pdf of $z_{1}$ and $z_{2}$ given by

ϕ_{2} (u, v; ρ) = \frac{1}{2 π \sqrt{(} 1 - ρ^{2})} e^{- \frac{1}{2 (1 - ρ^{2})} (u^{2} + v^{2} - 2 ρ u v)} .

	$\frac{d T_{1}}{d t_{1}}$	$= \frac{d}{d t_{1}} [\frac{1}{α_{1}} ((\frac{t_{1}}{β_{1}})^{ν_{1}} - (\frac{β_{1}}{t_{1}})^{ν_{1}})]$
		$= \frac{1}{α_{1}} \frac{d}{d t_{1}} [(\frac{t_{1}}{β_{1}})^{ν_{1}} - (\frac{β_{1}}{t_{1}})^{ν_{1}}]$
		$= \frac{ν_{1}}{α_{1}} [\frac{t_{1}^{ν_{1} - 1}}{β_{1}^{ν_{1}}} + \frac{β^{ν}}{t_{1}^{ν_{1} + 1}}]$
		$= \frac{ν_{1}}{α_{1} β_{1}} [(\frac{t_{1}}{β_{1}})^{ν_{1} - 1} + (\frac{β_{1}}{t_{1}})^{ν_{1} + 1}] .$

Now

	$\frac{d T_{2}}{d t}$	$= \frac{d}{d t_{2}} [\frac{1}{α_{2}} ((\frac{t_{2}}{β_{2}})^{ν_{2}} - (\frac{β_{2}}{t_{2}})^{ν_{2}})]$
		$= \frac{1}{α_{2}} \frac{d}{d t_{2}} [(\frac{t_{2}}{β_{2}})^{ν_{2}} - (\frac{β_{2}}{t_{2}})^{ν_{2}}]$
		$= \frac{ν_{2}}{α_{2} β_{2}} [(\frac{t_{2}}{β_{2}})^{ν_{2} - 1} - (\frac{β_{2}}{t_{2}})^{ν_{2} + 1}] .$

f_{T_{1}, T_{2}} (t_{1}, t_{2}) = \frac{ν_{1} ν_{2}}{2 π α_{1} α_{2} β_{1} β_{2} \sqrt{(} 1 - ρ^{2})} [(\frac{t_{1}}{β_{1}})^{ν_{1} - 1} + (\frac{β_{1}}{t_{1}})^{ν_{1} + 1}] [(\frac{t_{2}}{β_{2}})^{ν_{2} - 1} + (\frac{β_{2}}{t_{2}})^{ν_{2} + 1}] e x p {\frac{- 1}{2 (1 - ρ^{2})} [\frac{1}{α_{1}^{2}} ((\frac{t_{1}}{β_{1}})_{1}^{ν} - (\frac{β_{1}}{t_{1}})_{1}^{ν})^{2} + \frac{1}{α_{2}^{2}} ((\frac{t_{2}}{β_{2}})_{2}^{ν} - (\frac{β_{2}}{} t_{2})_{2}^{ν})^{2} - \frac{2 ρ}{α_{1} α_{2}} [(\frac{t_{1}}{β_{1}})_{1}^{ν} - (\frac{β_{1}}{t_{1}})_{1}^{ν}] [(\frac{t_{2}}{β_{2}})_{2}^{ν} - (\frac{β_{2}}{t_{2}})_{2}^{ν}]]}

Theorem 2.

If $(T_{1}, T_{2}) \sim$ BV $ν$ -BS $(α_{1}, β_{1}, ν_{1}, α_{2}, β_{2}, ν_{2}, ρ)$ then $T_{i} \sim ν - B S (α_{i}, β_{i}, ν_{i})$ .

Proof.

	$f_{T_{1}} (t_{1}, t_{2})$	$= \int_{t_{2}} f_{T_{1}, T_{2}} (t_{1}, t_{2}) d t_{2}$
		$= \frac{ν_{1} ν_{2}}{α_{1} α_{2} β_{1} β_{2} 2 π \sqrt{(} 1 - ρ^{2})} [(\frac{t_{1}}{β_{1}})^{ν_{1} - 1} + (\frac{γ_{1}}{t_{1}})^{ν_{1} + 1}] \int_{0}^{\infty} [(\frac{t_{2}}{β_{2}})^{ν_{2} - 1} + (\frac{β_{2}}{t_{2}})^{ν_{2} + 1}]$

	$P u t$	$u = (\frac{t_{2}}{β_{2}})^{ν_{2}} - (\frac{β_{2}}{t_{2}})^{ν_{2}}$
		$d u = \frac{ν_{2}}{β_{2}} [(\frac{t_{2}}{β_{2}})^{ν_{2} - 1} + (\frac{β_{2}}{t_{2}})^{ν_{2} + 1}] d t_{2} .$

∎

Theorem 3.

If $(T_{1}, T_{2}) \sim ν - B S (α_{1}, β_{1}, ν_{1}, α_{2}, β_{2}, ν_{2}, ρ)$ then

$(T_{1}^{- 1}, T_{2}^{- 1}) \sim ν - B S (α_{1}, \frac{1}{β_{1}}, ν_{1}, α_{2}, \frac{1}{β_{2}}, ν_{2}, ρ)$
$(T_{1}^{- 1}, T_{2}) \sim ν - B S (α_{1}, \frac{1}{β_{1}}, ν_{1}, α_{2}, β_{2}, ν_{2}, ρ)$
$(T_{1}, T_{2}^{- 1}) \sim ν - B S (α_{1}, β_{1}, ν_{1}, α_{2}, \frac{1}{β_{2}}, ν_{2}, ρ)$

4.0.1 Estimation

Based on a bivariate random sample ${(t_{1 i}, t_{2 i}), i = 1, 2, . . ., n}$ from the $ν - B V B S (α_{1}, β_{1}, ν_{1}, α_{2}, β_{2}, ν_{2}, ρ)$ distribution, the MLEs of the unknown parameters can be obtained by maximizing the log likelihood function. If we denote $θ = (α_{1}, β_{1}, ν_{1}, α_{2}, β_{2}, ν_{2}, ρ)$ then the likelihood function is

The log-likelihood function is

5 Multivariate $ν$ - Birnbaum Saunders Distribution

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

Then the multivariate $ν -$ BS distribution is as follows:

Definition 1.

Let $α - -, β - - \in R^{m}$ , where $α - - = (α_{1}, \dots, α_{m})^{T}$ and $β - - = (β_{1}, \dots, β_{m})^{T}$ , with $α_{1} > 0, β_{i} > 0$ for $i = 1, 2, \dots, m$ . Let $Γ$ be a $m \times m$ positive definite correlation matrix. Then, the random vector $T - - = (T_{1}, \dots, T_{m})^{T}$ is said to have a m-variate BS distribution with parameters $(α - -, β - -, Γ, ν)$ if it has the joint CDF as

	$P (T - - \leq t -)$	$= P (T_{1} \leq t_{1}, \dots, T_{m} \leq t_{m})$

for $t_{1} > 0, \dots, t_{m} > 0$ and $0 < ν < 1$ . Here, for $u - - = (u_{1}, \dots, u_{m})^{T}, Φ_{m} (u - -; Γ)$ denotes the joint cdf of a standard Normal vector $Z - - = (Z_{1}, \dots, Z_{m})^{T}$ with correlation matrix $Γ$ .

5.1 Applications

Birnbaum and Saunders (1958) obtained fatigue life real data corresponding to cycles $(\times 10^{- 3})$ 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 $n = 101$ 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 $α, β$ and $ν$ obtained by the method of maximum likelihood are given in the table 2.

$α$	$β$	$ν$
1.509180e+00	1.179090e-06	6.198936e+00

Table 2: Point estimates of

α, β

and

ν

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

ν -

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 $ν$

. 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

ν -

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

References

(1) Bebbington, M., Lai, C.D., Zitikis, R.(2008), A proof of the shape of the Birnbaum-Saunders hazard rate function, The mathematical Scientist, Vol. 33, pp. 49-56.
(2) Bhattacharyya, G.K., Fries, A.(1982), Fatigue failure models- Birnbaum Saunders versus inverse Gaussian, IEEE Transactions on Reliability, Vol. 31, pp. 439-440.
(3) Birnbaum, Z.W., Saunders, S.C.(1969a), A new family of life distribution, Journal of Applied Probability, Vol. 6, pp. 319-327.
(4) Birnbaum, Z.W., Saunders, S.C.(1969b), Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability, Vol. 6, pp. 328-347.
(5) Chacko,V.M., Mariya Jeeja, P.V., Deepa Paul(2015), p- Birnbaum-Saunders distribution: Applications to reliability and electronic banking habits, Reliability Theory and Applications, Vol. 10, pp. 70-77.
(6) Desmond, A.F.(1986), On the relationship between two fatigue-life models, IEEE Transactions on Reliability, Vol. 35, pp. 167-169.
(7) Desmond, A.F., Rodriguea-Yam, G.A., Lu,X.(2008), Estimation of parameters for a Birnbaum-Saunders regression model with censored data, Journal of Statistical Computation and Simulation, Vol. 78, pp. 983-997.
(8) From, S.G., Li, L.(2006), Estimation of the parameters of the Birnbaum-Saunders distribution, Communications in Statistics- Theory and Methods, Vol. 35, pp. 2157-2169.
(9) Kundu, D., Balakrishnan, N., Jamalizadeh, A.(2010), Bivariate Birnbaum-Saunders distribution and associated inference, Journal of Multivariate Analysis, Vol. 101, pp. 113-125
.
(10) Kundu, D., Balakrishnan, N., Jamalizadeh, A.(2013), Generalized multivariate Birnbaum-Saunders distributions and related inferential issues, Journal of Multivariate Analysis, Vol. 116, pp. 230-244.
(11) Kundu, D., Gupta, R.C.(2017), On bivariate Birnbaum-Saunders distribution, American Journal of Mathematical and Management Sciences, Vol. 36, pp. 21-33.
(12) Kundu, D., Kannan, N., Balakrishnan, N.(2008), On the hazard function of Birnbaum-Saunders distribution and associated inference, Computational Statistics and Data Analysis, Vol. 52, pp. 2692-2702.
(13) Lemonte, A. J., Cordeiro, G.M.(2009), Birnbaum-Saunders nonlinear regression models, Computational Statistics and Data Analysis, Vol. 53, pp. 4441-4452.
(14) Ng, H.K.T., Kundu, D., Balakrishnan, N.(2003), Modified moment estimation for the two-parameter Birnbaum-Saunders distribution, Computational Statistics and Data Analysis, Vol. 43, pp. 283-298.
(15) Rieck, J.R.(1995), Parametric estimation for the Birnbaum-Saunders distribution based on symmetrically censored samples, Communications in Statistics- Theory and Methods, Vol. 24, pp. 1721-1736.
(16) Rieck, J.R., Nedelman, J.R.(1991), A log-linear model for the Birnbaum-Saunders distribution, Technometrics, Vol. 33, pp. 51-60.

On a generalized Birnbaum Saunders Distribution

Authors

Bivariate Discrete Inverse Weibull Distribution

A Generalization of the Exponential-Logarithmic Distribution for Reliability and Life Data Analysis

A Class of Skewed Distributions with Applications in Environmental Data

Estimation of component reliability in repairable series systems with masked cause of failure by means of latent variables

Some Computational Aspects to Find Accurate Estimates for the Parameters of the Generalized Gamma distribution

A General Class of Trimodal Distributions: Properties and Inference

An efficient estimator of the parameters of the Generalized Lambda Distribution

1 Introduction

2 Birnbaum Saunders Distribution

2.1 Univariate Birnbaum Saunders Distribution

2.2 Bivariate Birnbaum Saunders Distribution

2.3 Multivariate Birnbaum Saunders Distribution

3 Univariate $ν$ -Birnbaum Saunders Distribution

3.1 Moments

3.2 Property of $ν -$ Birnbaum Saunders Distribution

Theorem 1.

Proof.

3.3 Estimation

3.3.1 Maximum Likelihood Estimates

4 Bivariate $ν$ - Birnbaum Saunders Distribution

Theorem 2.

Proof.

Theorem 3.

4.0.1 Estimation

5 Multivariate $ν$ - Birnbaum Saunders Distribution

Definition 1.

5.1 Applications

5.2 Conclusion

References

70	90	96	97	99	100	103	104	104	105	107	108	108	108	109
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124	124	124	124	124	128	128	129	129	130	130	130	131	131	131
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On a generalized Birnbaum Saunders Distribution

Authors

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This week in AI

1 Introduction

2 Birnbaum Saunders Distribution

2.1 Univariate Birnbaum Saunders Distribution

2.2 Bivariate Birnbaum Saunders Distribution

2.3 Multivariate Birnbaum Saunders Distribution

3 Univariate ν-Birnbaum Saunders Distribution

3.1 Moments

3.2 Property of ν−Birnbaum Saunders Distribution

Theorem 1.

Proof.

3.3 Estimation

3.3.1 Maximum Likelihood Estimates

4 Bivariate ν- Birnbaum Saunders Distribution

Theorem 2.

Proof.

Theorem 3.

4.0.1 Estimation

5 Multivariate ν- Birnbaum Saunders Distribution

Definition 1.

5.1 Applications

5.2 Conclusion

References

3 Univariate $ν$ -Birnbaum Saunders Distribution

3.2 Property of $ν -$ Birnbaum Saunders Distribution

4 Bivariate $ν$ - Birnbaum Saunders Distribution

5 Multivariate $ν$ - Birnbaum Saunders Distribution

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