Testing BRNBU Ageing Class of Life-Time Distribution Based on Moment Inequality

In this paper, new moment inequality is derived for Bivariate Renewal New Better than Used (BRNBU) ageing class of life-time distribution. This inequality demonstrates that if the mean life is finite, then all higher order moments exist. Based on the Moment inequality, new testing procedures for testing bivariate exponentiality against BRNBU ageing class of life-time distribution is introduced.The asymptotic normality of the test statistic and its consistency are studied. Using Monte Carlo Method, critical values of the proposed test are calculated for n = 5(5)100 and tabulated. Finally, the theoretical results are applied to analyze real-life data sets.

ISSN: 2456-8686, 5(1), 2021:001-010 https://doi.org/10.26524/cm87 of shocks they receive. In such cases, the time to failure is often more appropriately represented by the number of times they are used before they fail, which is a discrete random variable. Bivariate distributions are observed in different practical situations, for example, in the survival of the paired organs such as persons eyes,ears, kidneys and lungs, or in the survival of two-engine airplane.
Testing bivariate exponentiality against some bivariate ageing classes of life-time distributions has seen a good deal of attention. The moment inequality for the Bivariate Renewal New Better than Used (BRNBU) Ageing Class of life-time distribution can be found in the work of [6]. Now we propose a test statistic testing Bivariate Exponentiality Against Bivariate Renewal New Better than Used (BRNBU) Ageing Class of life-time distribution, based on the moment inequality.
The rest of the paper is arranged as follows; In section 2, the preliminaries required for the further discussion is given. In section 3, moment inequality for the Bivariate Renewal New Better than Used (BRNBU) Ageing Class of life-time distribution is derived. A new test statistic for Bivariate Renewal New Better than Used (BRNBU) Ageing Class of life-time distribution based on moment inequality is proposed in section 4. Using Monte Carlo Method critical values of the proposed test statistic are calculated for n = 5(5)100 and tabulated in section 5. The application of the proposed test to real data sets is discussed in section 6. Finally, conclusion is given in section 7.

Preliminaries
Let (X, Y ) denote the survival time of a device having a joint distribution function F (x, y). The bivariate joint survival function is given by where it is assumed that F (0, 0) = 1.
The following definition of Bivariate ageing classes of life-time distributions appeared in [6].

Moment Inequality
A moment inequality is derived for the system whose life-time distribution is a Bivariate Renewal New Better than Used ageing class of life-time distribution.
Theorem 3.1 If F is a Bivariate Renewal New Better than Used ageing class of life-time distribution, then moments of all order r, for all integer r ≥ 0, exist and are finite. Here µ denotes the mean of F . .
Proof: Since F is a Bivariate Renewal New Better than Used ageing class of life-time distribution, we have for all x, y, t, s ≥ 0, where Multiplying both side by (x + t) r (y + s) r and integrating, we get where I is a indicator function. Further, Using the equation (2) and (3), the inequality (1) becomes .
This completes the proof of theorem.

Testing Bivariate Exponentiality against Bivariate Renewal New Better than Used Ageing Class
Using inequality (4), we test the null hypothesis H 0 : F is Bivariate exponential against the alternative hypothesis H 1 : F is BRNBU and not bivariate exponential. Consider the bivariate exponential distribution introduced by [5], given by F (x, y) = exp(−λ 1 x − λ 2 y − λ 12 max(x, y)), for all x, y, λ 1 , λ 2 > 0 and λ 12 ≥ 0, where Note that under H 0 , δ E = 0, while under H 1 , δ E > 0. Let (X 1 , X 2 ), (X 3 , X 4 ), . . . , (X i−1 , X i ), . . . (X n−1 , X n ) be a bivariate random sample from a distribution F. Taking r = 1 in equation (5), we get The estimate δ E of δ E can be obtained as To make the test statistic scale invariant, let Then ∆ E is equivalent to the classical U-statistic, [4] and is given by U n = 1 ( n 4 ) i<j ≤n φ(X i , X j ). The asymptotic normality of ∆ E is summarized in the following theorem.
Here µ denotes the mean of F . Under H 0 , the variance reduces to σ 2 = 56. Proof: From the standard theory of U-statistic, [4], we have Thus From the equation (6), it is clear that E[ς 0 (X i , X j )] = 0 and This completes the proof of the theorem.

Corollary 4.2
Under H 0 , the limiting distribution of U n is normal with mean ∆ E . The variance of √ n(U n ) is a function of λ 1 , λ 2 and λ 12 . Proof: Since the variances of √ n(U n ) is very complicated under H 0 and since U n is a function of U -statistic, jackknifing would not only reduce the bias, but also enable us to estimate the variance of V ( √ nU n ).
This completes the proof of the corollary.

Monte Carlo Simulations
In this section the Monte Carlo null distribution critical points of ∆ E are simulated based on 50000 generated samples of size n = 5(5)100. Table 1 gives the upper percentile points of statistic ∆ E for different confidence levels 90%, 95%, 98% and 99%.  Figure 1, that the critical values increase as the confidence level increases and almost increases as the sample size increase.

Application to Real Life Data
Here, we present a real life example to illustrate the use of our test statistics ∆ E . We consider the example given by [2] which is a list of paired first failure time (in hours) of the transmission and the transmission pump on 15 Caterpillar tractors. We use our test to detect whether these failure times follow a bivariate exponential ISSN: 2456-8686, 5(1), 2021:001-010 https://doi.org/10.26524/cm87

Conclusion
The Bivariate Renewal New Better than Used Ageing Class of life-time distribution is considered. The moment inequality is derived. A new test statistic for testing Bivariate Exponentiality against Bivariate Renewal New Better than Used Ageing class of life-time distributions based on the moment inequality is proposed. Using Monte Carlo Method, critical values of the proposed test are calculated for n = 5(5)100 and tabulated. Finally, application to real-life data is given.