Abstract

A new class of bivariate distributions is deduced by specifying its conditional hazard functions (hfs) which are Kumaraswamy distribution. The interest of this model is positively, negatively, or zero correlated. Properties and local measures of dependence of the bivariate Kumaraswamy conditional hazard (BKCH) distribution are studied. The estimation of type parameters is considered by used the maximum likelihood and pseudolikelihood of the new class. A simulation study was performed to inspect the bias and mean squared error of the maximum likelihood estimators. Finally, an application is obtained to clarify our results with the maximum likelihood and pseudolikelihood. Also, the results are used to compare BKCH distribution with bivariate exponential conditionals (BEC) and bivariate Lindley conditionals hazard (BLCH) distributions.

1. Introduction

The problem of constructing new families of continuous bivariate distributions is one of the important and continue active research problem in statistics. This is due from one side to the limitations of the existing distributions and its lack in modeling some stochastic phenomena and from the other side to meet the needs of modern and developed applied field. Balakrishnon et al. [1] obtained a new technology to build the bivariate probability density functions (pdfs) with specified conditional hfs, and the development of the models determines many of the several properties, including survival function (sf), hazard bivariate function, the Clayton–Oakes measure, and the measure conditional pdfs and hfs of the marginal and conditional pdfs. Moreover, Navarro and Sarabia [2] studied the reliability properties for the series and parallel systems with component lifetimes having these dependence models.

Therefore, the authors introduce a new form of the bivariate distributions with Kumaraswamy conditional hfs, and we studied some of the characteristics and discussed the estimation procedures of the parameters of this distribution. Moreover, we estimate the parameters of this distribution by two methods, e.g., maximum likelihood estimation (MLE) and maximum pseudolikelihood estimation (MPLE) procedures. Finally, this distribution is compared with other distributions considering a real dataset collected from a secondary source (p. 374 of Johnson and Wichern [3]).

A continuous random variable is Kumaraswamy distribution if its pdf (Kumaraswamy [4] is

The corresponding sf isand the hf is

We note that

This distribution is applicable to some scientific, practical, and hydrological applications (Dey et al. [5]; Ishaq et al. [6]).

2. BKCH Model

Suppose that the conditional hfs (Kumaraswamy) are distributed aswhere and are function of and , respectively.

Then, the bivariate distribution from (6) will be called a bivariate BKCH distribution.

Theorem 1. The general bivariate distribution with conditional hfs as and iswhere is the normalizing constant such that integrates to 1.

Proof. The conditional pdfs from (6) using (5) areThen, the identity yields the following relation:where and denote the marginal pdfs.
Denoting then taking logarithms in (9) and using (10) and (11) WITH then taking logarithms in (10), using (11) and (12)which is a functional equation of the form , whose most general solution is given by Aczel [7] asFrom (11) and (12), we have the corresponding marginal pdfs given byandEquation (7) describes the complete class of BKCH distribution that has the three parameters , and .
Figure 1 shows the BKCH given by (7) for the special cases for any , , and .
The conditional hfs for the BKCH distribution areThe compatibility of (17) and (18), Balakrishnan et al. [1] secures the existence of BKCH distribution. Figure 2 shows hf of the BKCH given by (7) for the special cases for and Similarly,

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Figure 1 

The pdf of the BKCH distribution for selected values of , and . (a) . (b) . (c) . (d) .

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Figure 2 

The hf of the BKCH distribution for selected values of and . (a) . (b) . (c) . (d) .

3. BKCH Properties

3.1. The Conditional Distributions

The conditional pdfs of the BKCH distribution arei.e.,

When , the conditional pdfs are identical.

and given by (19) and (20) satisfy the compatibility conditions studied by Arnold et al. [8] and secure existence of BKCH distribution.

3.2. Regression Function

Now, using (19) and (20), the conditional moments are

For we haveand

We notice that the regression function for is nonlinear increasing for and decreasing for (Figure 3). Additionally, the regression function for is nonlinear increasing for and decreasing for (Figure 4).

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Figure 3 

of the BKCH distribution for selected values. (a) . (b) .

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Figure 4 

of the BKCH distribution for selected values of and (a) . (b) .

3.3. Stochastic Ordering

Let and be two-bivariate BKCH distribution, and such that and and increasing in x, for all Also, decreases in y, for all Based on the results obtained by Shanthikumar [9], then we obtain likelihood ratio order (, hazard rate order , and stochastic order , for all .

4. Local Measures of Dependence

Holland and Wang [10, 11] introduced a local dependence function given by the following formula:where is the joint pdf of x and y.

Theorem 2. The BKCH distribution given by (8) is TP₂ (totally positive of order 2) iff and TN₂ (totally negative of order 2) iff .

Proof. From (7), we obtainHence, . If , then BKCH is , and . If , then BKCH is .

Remark 1. (1)It follows from theorem (2) that if the BKCH distribution, given by (7), is or , then X and Y are positively or negatively, respectively, correlated.(2)The conditional pdfs in (19) and (20) have the local measures of dependence:(3) iff ; furthermore, the two random variables X and Y are independent iff Balakrishnan and Lai [12].The class has two parameters only by choosing ; the joint pdf (7) can be written asFigure 5 shows the joint pdf given by (29) for the special cases for and .
The conditional hfs for the joint pdf given by (29) areand the conditional pdfs areNow, the marginal pdfs of X and Y using (29) areAnd the moment of joint pdf given by (29) is

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Figure 5 

The pdf given by (19) for selected values of and (a) . (b) . (c) . (d) .

5. Estimation of BKCH Parameters

Suppose that are observed from BKCH distribution.

5.1. MLE of BKCH Parameters and Asymptotic Confidence Intervals

From the BKCH distribution, the log-likelihood function for the sample is