bivariate birnbaum-saunders distributionhome.iitk.ac.in/~kundu/iisa-jan-2013.pdf · 2013-01-08 ·...

Bivariate Birnbaum-Saunders Distribution

Debasis Kundu

Department of Mathematics & StatisticsIndian Institute of Technology Kanpur

January 2nd. 2013

Debasis Kundu Bivariate Birnbaum-Saunders Distribution

CollaboratorsDistributions Obtained from Normal Distribution

Birnbaum-Saunders Distribution: Introduction & PropertiesBivariate Birnbaum-Saunders Distribution

Copula Structure

Outline

1 Collaborators

2 Distributions Obtained from Normal Distribution

3 Birnbaum-Saunders Distribution: Introduction &Properties

4 Bivariate Birnbaum-Saunders Distribution

5 Copula Structure




Copula Structure

Collaborators

1 N. Balakrishnan

2 R.C. Gupta

3 Ahad Jamalizadeh

4 N. Kannan

5 V. Leiva

6 H.K. T. Ng,

7 A. Sanhueza.




Copula Structure

Outline

1 Collaborators




5 Copula Structure




Copula Structure

Transformation from Normal Random Variable

Different random variables have been derived from normalrandom variables.

Log-normal random variable: If X is a normal random variablethen Y = eX has log-normal random variable.

FY (y) = Φ(ln y)

The PDF of Log-normal random distribution is;

f (x ;µ, σ) =1√

2πσxe−

(ln x−µ)2

2σ2




Copula Structure

Log-normal Random Variable

The shape of the log-normal PDF for different σ.




Copula Structure

Log-normal distribution: Closeness

The shape of the PDF of log-normal distribution is alwaysunimodal.

The PDF of the log-normal distribution is right skewed.

The PDF of log-normal distribution has been used verysuccessfully to model right skewed data.

It is observed that the PDF of log-normal distribution is verysimilar to the shape of the PDF of well known Weibulldistribution.

Quite a bit of work has been done in discriminating betweenthese two distribution functions.




Copula Structure

Inverse Gaussian random variable

Inverse Gaussian random variable has the following PDF:

f (x ;µ, λ) =

(λ

2πx3

)1/2

exp

(−λ(x − µ)2

2µ2x

)

Inverse Gaussian has the CDF;

F (x ;µ, σ) = Φ

{λ

x

(x

µ− 1

)}+ e2λ/µΦ

{−λ

x

(x

µ+ 1

)}




Copula Structure

Inverse Gaussian Distribution: PDF

The shape of the Inverse Gaussian PDF for different λ.




Copula Structure

Skew normal distribution: Introduction

So far skewed distribution has been obtained for non-negativerandom variable. Now we provide a method to introduce skewnessto a random variable which may have a support on the entire realline also.

Consider two independent standard normal independentrandom variables, say X and Y .

Then because of symmetry

P(Y < X ) =1

2




Copula Structure

Skew normal distribution: Introduction

Now suppose α > 0 and consider P(Y < αX ).

In this case also since Y − αX is a normal random variablewith mean 0, then clearly P(Y < αX ) = 1/2

On the other hand

P(Y < αX ) =

∫ ∞−∞

φ(x)Φ(αx)dx =1

2.

Therefore, ∫ ∞−∞

2φ(x)Φ(αx) = 1.




Copula Structure

Skew normal distribution

Skew normal distribution has the following PDF:

f (x ;α) = 2φ(x)Φ(αx); −∞ < x <∞

It is a skewed distribution on the whole real line




Copula Structure

Skew Normal Random Variable

The shape of the skew normal PDF for different α.




Copula Structure

Outline

1 Collaborators




5 Copula Structure




Copula Structure

Birnbaum-Saunders distribution: PDF and CDF

Birnbaum-Saunders Distribution has the following CDF:

F (x ;α, β) = Φ

[1

α

{(x

β

)1/2

−(β

x

)1/2}]

It is a skewed distribution on the positive real line.

The PDF of Birnbaum-Saunders distribution is.

f (x) =1

2√

2παβ

[(β

x

)1/2

+

(β

x

)3/2]

exp

[− 1

2α2

(x

β+β

x− 2

)]




Copula Structure

Birnbaum-Saunders Random Variable

The shape of the Birnbaum-Saunders PDF for different α.




Copula Structure

Birnbaum-Saunders distribution

Birnbaum-Saunders distribution has been developed to modelfailures due to crack.

It is a assumed that the j-th cycle leads to an increase incrack Xj amount.

It is further assumed that∑n

j=1 Xj is approximately normally

distributed with mean nµ and variance nσ2.

Then the probability that the crack does not exceed a criticallength ω is

Φ

(ω − nµ

σ√

n

)= Φ

(ω

σ√

n− µ√

n

σ

)




Copula Structure


It is further assumed that the failure occurs when the cracklength exceeds ω.

If T denotes the lifetime (in number of cycles) until failure,then the CDF of T is approximately

P(T ≤ t) ≈ 1− Φ

(ω

σ√

t− µ√

t

σ

)= Φ

(µ√

t

σ− ω

σ√

t

).

It is exactly the same form as defined before.




Copula Structure


Fatigue failure is due to repeated applications of a commoncyclic stress pattern.

Under the influence of this cyclic stress a dominant crack inthe material grows until it reaches a critical size w is reached,at that point fatigue failure occurs.

The crack extension in each cycle are random variables andthey are statistically independent.

The total extension of the crack is approximately normallydistributed.




Copula Structure


The following observations are useful:

If T a Birnbaum-Saunders distribution, say, BS(α, β) thenconsider the following transformation

X =1

2

[(T

β

)1/2

−(

T

β

)−1/2]

Equivalently

T = β(

1 + 2X 2 + 2X(1 + X 2

)1/2)

Then X is normally distributed with mean zero and varianceα2/4.




Copula Structure


The above transformation becomes very helpful:

It can be used very easily to generate samples fromBirnbaum-Saunders distribution.

It helps to derive different moments of theBirnbaum-Saunders distribution.

It helps to derive some other properties of theBirnbaum-Saunders distribution.




Copula Structure

Birnbaum-Saunders distribution: Basic Properties

Here α is the shape parameter and β is the scale parameter.

α governs the shape of PDF and hazard function.

For all values of α the PDF is unimodal.

Mode cannot be obtained in explicit form, it has to beobtained by solving a non-linear equation in α.

Clearly, the median is at β, for all α.




Copula Structure

Birnbaum-Saunders distribution: Inverse

Another interesting observation which is useful for estimationpurposes:

If T is a Birnbaum-Saunders distribution, i .e. BS(α, β), thenT−1 is also a Birnbaum-Saunders with parameters α and β−1

The above observation is very useful. Immediately we obtain

E (T−1) = β−1(1 +1

2α2), Var(T ) = α2β−2(1 +

5

4α2).




Copula Structure

Birnbaum-Saunders distribution: Hazard Function

The shape of the hazard function of Birnbaum-Saundersdistribution is unimodal.

The turning point of the hazard function can be obtained bysolving a non-linear equation involving α.

The turning point of the hazard function of theBirnbaum-Saunders distribution can be approximated verywell for α > 0.25, by

c(α) =1

(−0.4604 + 1.8417α)2.

The approximation works very well for α > 0.6.




Copula Structure

Nice representation

Now we will present one nice representation of theBirnbaum-Saunders distribution. We have already defined theInverse Gaussian random variable which has the following PDF:

f (x ;µ, σ) =

(1

2σ2πx3

)1/2

exp

(−(x − µ)2

2σ2µ2x

)We will denote this as IG(µ, σ2).

Suppose X1 ∼ IG(µ, σ2), and X−12 ∼ IG(µ−1, σ2mu2), then

consider the new random variable for 0 ≤ p ≤ 1:

X =

X1 w.p 1− p

X2 w.p. p




Copula Structure

Nice representation

Then the PDF of X can be expressed as follows:

fX (x) = pfX1(x) + (1− p)fX2(x)

Note that the PDF of X1 is a Birnbaum-Saunders PDF, and thePDF of X2 can be easily obtained.

Interestingly, when p = 1/2, the PDF of X becomes the PDF of aBirnbaum-Saunders PDF.




Copula Structure

Outline

1 Collaborators




5 Copula Structure




Copula Structure

Bivariate Birnbaum-Saunders distribution

Remember the univariate Birnbaum-Saunders distribution has beendefined as follows:

P(T1 ≤ t1) = Φ

[1

α

(√t

β−√β

t

)]

The bivariate bivariate Birnbaum-Saunders distribution can bedefined analogously as follows:




Copula Structure

Bivariate Birnbaum-Saunders distribution

Let the joint distribution function of (T1,T2) be defined as followsF (t1, t2) = P(T1 ≤ t1,T2 ≤ t2), then (T1,T2) is said to havebivariate Birnbaum-Saunders distribution with parametersα1, α2, β2, β2, ρ, if

F (t1, t2) = Φ2

[1

α1

(√t1

β1−√β1

t1

),

1

α2

(√t2

β2−√β2

t2

); ρ

]

Here Φ2(u, v) is the CDF of a standard bivariate normal vector(Z1,Z2) with the correlation coefficient ρ. Let’s denote this byBVBS(α1, β1, α2, β2, ρ).




Copula Structure


Bivariate Birnbaum-Saunders distribution has several interestingproperties.

The joint PDF of (T1,T2) can be easily obtained in terms of thePDF of bivariate normal distribution.

The joint PDF can take different shapes, but it is unimodal,skewed. Need not be symmetric.

The correlation coefficient between T1 and T2 can be both positiveand negative.




Copula Structure

Birnbaum-Saunders Random Variable

Contour plot of the Bivariate Birnbaum-Saunders PDF.




Copula Structure

BVBS: Properties

If (T1,T2) ∼ BVBS(α1, β1, α2, β2), then we have the followingresults:

T1 ∼ BS(α1, β1) and T2 ∼ BS(α2, β2).

(T−11 ,T−1

2 ) ∼ BVBS(α1, β−11 , α2, β

−12 , ρ).

(T−11 ,T2) ∼ BVBS(α1, β

−11 , α2, β2,−ρ).

(T1,T−12 ) ∼ BVBS(α1, β1, α2, β

−12 ,−ρ).




Copula Structure

Bivariate Birnbaum-Saunders Distribution: Generation

It is very easy to generate Bivariate Birnbaum-Saunders randomvariables using normal random numbers generator:Generate first U1 and U2 from N(0,1)Generate

Z1 =

√1 + ρ+

√1− ρ

2U1 +

√1 + ρ−

√1− ρ

2U2

Z2 =

√1 + ρ−

√1− ρ

2U1 +

√1 + ρ+

√1− ρ

2U2

Make the transformation:

Ti = βi

1

2αiZi +

√(1

2αiZi

)2

+ 1

2

, i = 1, 2.




Copula Structure

Bivariate Birnbaum-Saunders Distribution: Inference

The MLEs of the unknown parameters can be obtained to solve afive dimensional optimization problem.The following observation is useful:[(√

T1

β1−√β1

T1

),

(√T2

β2−√β2

T2

)]∼ N2 {(0, 0),Σ}

where

Σ =

(α2

1 α1α2ρα1α2ρ α2

2

).




Copula Structure

Bivariate Birnbaum-Saunders Distribution: Inference

1 For given β1 and β2, the MLEs of α1, α2 and ρ can beobtained in explicit forms.

2 The MLEs of β1 and β2 can be obtained by maximizing theprofile log-likelihood function.

3 Five dimensional optimization problem can be reduced to atwo dimensional optimization problem.




Copula Structure

Outline

1 Collaborators




5 Copula Structure




Copula Structure

Copula

To every bivariate distribution function FY1,Y2(·, ·) with continuousmarginals FY1(·) and FY2(·), corresponds a unique functionC : [0, 1]× [0, 1]→ [0, 1], called a copula such that for(y1, y2) ∈ (−∞,∞)× (−∞,∞)

FY1,Y2(y1, y2) = C (FY1(y1),FY2(y2))

andC (u, v) = FY1,Y2(F−1

Y1(u),F−1

Y2(v))




Copula Structure

Bivariate Gaussian Copula

CG (u, v) =

∫ Φ−1(u)

−∞

∫ Φ−1(v)

−∞φ2(x , y ; ρ)dxdy = Φ2(Φ−1(u),Φ−1(v); ρ),

where

φ2(u, v ; ρ) =1

2π√

1− ρ2exp

{− 1

2(1− ρ2)(u2 + v2 − 2ρuv)

}




Copula Structure


The bivariate Birnbaum-Saunders distribution can be written asfollows;

FT1,T2(t1, t2) = CG (FT1(t1;α1, β1),FT2(t2;α2, β2); ρ).

If (T1,T2) ∼ BVBS(α1, β1, α2, β2, ρ), then different properties canbe established using copula structure.




Copula Structure

Different Properties

1 It is TP2 for ρ > 0 and RR2 for ρ < 0.

2 The conditional failure rate of T1 given T2 = t2 is adecreasing (increasing) function of t2 for ρ > 0(ρ < 0).

3 The conditional failure rate of T1 given T2 > t2 is adecreasing (increasing) function of t2 for ρ > 0(ρ < 0).

4 T1 is stochastically increasing in T2 and T2 is stochasticallyincreasing in T1, if ρ > 0




Copula Structure

Thank You


bivariate birnbaum-saunders distributionhome.iitk.ac.in/~kundu/iisa-jan-2013.pdf · 2013-01-08 ·...

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