Abstract

In this paper, a bivariate distribution with a two-parameter exponential conditional is obtained. A multivariate form of the result is also attained under the joint independence of components assumption. A maximum Likelihood method of estimation is provided as well as the intervals of confidence for the parameters of this bivariate distribution. The pdf of the order statistics and concommitants are also derived.

1. Introduction

The univariate exponential distribution which is analytically very simple plays an important role in describing the life time of a single component [see, e.g., Balakrishnan and Basu (1995)] ¹. The reliability is the domain in which most of the bivariate distributions with exponential marginals arise. Several versions of this bivariate exponential distribution are encountered in the literature and have been used for modeling the two components systems. Indeed, a complete class of bivariate distribution respectively with normal and exponential conditional were identified, Castillo and Galombos (1987a) ⁴, Barry C. Arnold and David Strauss (1988) ³.

The marginal densities of the bivariate exponential may not be exponential. It can be a mixture of exponential. In such case the bivariate distribution is often called a bivariate exponential mixture distribution (see, Kotz et al. ⁸). Many authors proposed the multivariate form of the exponential distribution (see, Johnson et al. ⁷).

Recently Filus and Filus ⁶ have proposed for modeling lifetimes of multi-component system, a new class of probability distributions based upon a linear combination of independent random variables.

In this paper, we define a bivariate distribution with a two-parameters (a, b) exponential conditional which can be used for modeling lifetime of two component system.

The bivariate distribution with conditional a two-parameters exponential distribution is introduced in section 2 below with some characteristics such as the marginal densities, the moments, the product moments, the conditional moments, the moment generating function, the survivor distribution and the entropies. In section 3, we infer about the parameters of our bivariate distribution by giving their maximum likelihood likelihood estimators (MLEs) and intervals of confidence.

In section 4, we introduce the distribution of the concomitants of the order statistic. Finally in section 5 the multivariate case is studied with its related properties.

2. The Bivariate Distribution with Conditional a Two-parameters Exponential Distribution

Let X be a two-parameter exponential distribution random variable. The probability density function (p.d.f ) of X is given by

(2.1)

The cumulative distribution function of is given by

(2.2)

Now, let be a random variable such that the distribution of given is a two-parameters exponential distribution. The p.d.f of is given by

(2.3)

Thus the joint density of the random variables and defined above is given by

(2.4)

It can be easily verified that equation (2.4) integrates to 1, so it is a joint probability distribution.

The plot of this joint distribution for different values of a, b, and c is given in Figure 1.

Thus the cumulative distribution of the random variables X and Y is

As the marginal of X is given by (2.1), the marginal of Y is derived as follows

Theorem 2.1.

Proof.

(2.5)

Consequently the cumulative distribution of Y is

Remark 2.2. The marginal of is not an but a mixture of exponential, so is a bivariate exponential mixture distribution.

The moments of are given by

Theorem 2.3. The moments of are:

Proof.

By analogy

(2.6)

Remark 2.4. From (2.6) we deduce that:

1.

2.

The (p, q)th joint moment of (X, Y) can also be obtained as follows

Theorem 2.5.

(2.7)

Proof.

Expanding in power series and putting and we get

Let and

(2.8)

By the same way we prove that then

(2.9)

The moment generating function of (X, Y) is given as

(2.10)

The product moment exists if with

From (2.10) we can deduce:

1.

2.

3. as , X and Y are positively correlated.

4. The matrix of Variance-Covariance of X and Y is

The conditional distribution of ant that of are

(2.11)

and

(2.12)

Using (2.11) we get the pth conditional moments of X as

Theorem 2.6.

Proof.

(2.13)

Similarly, using (2.12) we get the qth conditional moments of Y as

Theorem 2.7.

Proof.

(2.14)

Remark 2.8. From (2.13) and (2.14) we can easily obtain the conditional means and variances of and

For the mixture distribution (2.4) the joint survivor function which can be used in the reliability study of systems, is given by

(2.15)

The failure rates of the random variables X and Y having p.d.f f_X(x) and f_Y(y) given by (2.1) and (2.5), respectively, are

and

The plot of the failure rate of Y for different values of a, b, and c is given in Figure 2.

In this section we introduce the entropy between X and Y which is defined as and interpreted as the quantity of information on X we gain by learning Y. So, for the bivariate mixture distribution the entropy is

(2.16)

3. Inference

We introduce here, the maximum Likelihood estimation for the bivariate model.

Let for be a sample of size n from the bivariate distribution defined in (2.4). Then the log likelihood function is

(3.17)

We have to maximize this function under the constraints for (5.14), b > 0, and c > 0.

Theorem 3.1. The maximum likelihood estimators of a, b, and c are given by

Proof. From (2:4) we deduce that

More, it will be assumed that

1. such that not all equal

2. such that (which means ).

So , and the unique constraint on a is \,for all , which can be written as .

The function is increasing linear with respect to the variable a when we fixe b>0 and c>0. Therefore its maximum is attained for So we have just to maximize the following function with respect to the variables b and c

(3.18)

This function g can be written as with

Maximize g with respect to (b, c) is equivalent to minimize g₁ with respect to b and minimize g₂ with respect to c.

Those two functions g₁ and g₂ are of the form

( for and for).

We can easily prove that h has a unique global minimum on attained at such that

So and are the global minimum for and respectively.

Therefore the function l has a global maximum (under the constraints) attained at

So and are the MLEs of , , and respectively.

Lawless (1982) ⁶ proved that and are independent with

(3.19)

Using (3.19) and we get the following results:

1. ( is a positively biased estimator of with bias equal )

2. and

3. ( is a negatively biased estimator of with bias equal

4. and

5. ( is an unbiased estimator of c)

6.

Remark 3.2.

1. , then , and are consistent estimators of a, b and c respectively.

2. and are asymptotically unbiased estimators of a and b respectively.

We introduce here, the intervals of confidence for the three parameters a, b, and c.

We can use the pivotal quantity in (3.19) to make inference on b, and a confidence interval for b is given by

It follows also from (3.19) that

By the same way, using the pivotal quantity , a confidence interval for a can be derived as

Also for n enough large , follows and then

So for as an estimator of , a confidence interval for c can be derived and it is given by

4. Concomitants of Order Statistics

In this section we introduce the distribution of the concomitants of the order statistic for the bivariate exponential mixture distribution. The density of probability of the rth concomitant is given by ⁵ as

where is the density function of the rth order statistic for the variable X given by

Given (2.1), (2.2), and (2.3), the distribution of the rth order statistic for X is

(4.20)

Theorem 4.1. The density of the rth concomitant is given by

Proof.

The pth moment of the concomitant of the order statistic is given by

Theorem 4.2.

Proof. Using the same techniques of integrations as in theorem 4.1 above we get our result.

Remark 4.3. From theorem 4.2 we can deduce the expected value and variance of

The expression of the survivor function

of is.

Theorem 4.4.

Proof. Obvious.

5. Multivariate Case

Let be random variables, the multivariate case is built as

where and are independent random variables for and and . HhUsing the same arguments as in the univariate case above, the joint component model is built and the marginal density function for each random variable is derived. In general, has the following density

(5.21)

Based on the independence assumption of the above model, the joint density of has the following form

The joint density of is obtained by integrating the joint density with respect to the variable .

Remark 5.1. For example, substituting and , into the above formula, we get:

(5.22)

that can be rewritten as

integrates to 1 so it's a legitimate distribution.

Using the density of defined by (3.9) and by analogy with theorem 2.3, the expression of the moments of is

(5.23)

Remark 5.2. From (3.11) we deduce that for all

1.

2.

The covariance between and for is derived as:

(5.24)

Bivariate case will reduce to equation (2.8).

6. Conclusion

Unlike the bivariate exponential with conditional exponential ³, and the bivariate distribution with normal conditional ⁴, the bivariate exponential distribution with conditional has the great advantage of giving us explicit, consistent, unbiased and asymptotically unbiased estimators of our parameters a, b and c with reliable confidence intervals for them.

References