Moments Inequalities for NBRUrp Distributions with Hypotheses Testing Applications

In this paper, we presented a new test statistic for testing exponentiality against new better than renewal used in the RP order NBRUrp based on moment inequality. Pitman's asymptotic efficiency, The Pitman asymptotic relative efficiency (PARE) are studied for other testes. Critical values are tabulated for sample size n=5(1)30(5)50 , the power of the test are calculate. Also we proposed a test for testing exponentiality versus NBR U rp for right censored data and the power estimates of this test are also simulated for some commonly used distributions in reliability. Finally, real data are given to elucidate the use of the proposed test statistic in the reliability analysis.


Introduction
Classes of life distributions are defined to classify the life distributions according to their aging properties. The definitions of these classes helped statisticians to define the test statistics. The test statistics are defined based on definition of the classes. The main aim of constructing new tests is to gain higher efficiencies. Many authors proposed tests for exponentiality versus some classes of life distributions based on the moment inequalities. Testing exponentiality against IFR, NBU and NBUE based on moment inequalities have been studied by (Ahmad 2001); (Ahmad and Mugdadi 2004) constructed the tests of NBUC, IFRA and DMRL depends on the moment inequalities, while testing NRBU and RNBU based on moment inequalities have been studied by (Mahmoud, EL-arishy, and Diab 2003). Using the moment inequalities of the class NBUL, (Mahmoud, Diab, and Kayid 2009) constructed a test statistic for testing exponentiality versus this class.
In this paper our theme formulates a new test statistic for testing exponentiality against NBRUL class based on the moment inequalities and discuss this test. The main classes of life distributions which have been introduced in the literature are based on new better than used (NBU), new better than used failure rate (NBUFR), new better than average failure rate (NBAFR), new better than used renewal failure rate (NBURFR), new better than used average renewal failure rate (NBARFR), new better than renewal used (NBRU) and exponential better than used in Laplace transform order (EBUL). Testing exponentiality against some classes of life distributions has been introduced by many researchers from many points of views. For more details one can refer to (Bryson and Siddiqui 1969), (Deshpande, Kochar, and Singh 1986), Ahmed 1988, 1992), (Abouammoh, Abdulghani, and Qamber 1994), (Mahmoud, Moshref, and Mansour 2015), (Kumazawa 1983), (Ahmad 1994(Ahmad , 2001. (Abouammoh and Newby 1989), (Mahmoud and Abdul Alim 2002, 2008, (Ahmad, Alwasel, and Mugdadi 2001), (Abu-Youssef 2009), (Ismail and Abu-Youssef 2012) , (Mahmoud and Rady 2013). Recently (Atallah, Mahmoud, and Al-Zahrani 2014) developed a new method for testing exponentiality which is more general and flexible than goodness approach.
The rest of this paper can be organized as follows, Section 2 gives a brief knowledge about renewal classes. In Section 3 moment inequalities for the NBRUL class are developed. In Section 4, Testing exponentiality against NBRUL is proposed based on moment inequalities. In Section 5, Pitman's asymptotic efficiency (PAE) of the test for several common alternatives will be considered. In Section 6, Monte Carlo null distribution critical points from the null distribution for sample size n = 5(5)35; 39; 40(5)50. In section 7, The power estimate for the test are calculated. Finally, the application of the proposed test for real data sets are discussed in Section 8.

Renewal classes
Let T be a random variable represents life time of a device (system or component) with a continuous life distribution F (t) . Upon arising the failure of the device, it can be substituted by a sequence of mutually independent devices which are identically distributed with the same life distribution F (t) .
The following stationary renewal distribution constitutes the remaining life distribution of the device under operation at time t.
For extra details, see (Barlow and Proschan 1981), Ahmed 1988, 1992). Now we need to present the definitions of the NBRU (NWRU) and NBRUL (NWRUL) classes of life distributions.
Definition 2.1. (Abouammoh et al. 1994) If X is a random variable with survival functionF (x) , then X is said to have new better (worse) than renewal used property, denoted by NBRU (NWRU), if Depending on the definition (2.1), (Mahmoud, EL-Sagheer, and Etman 2016) defined a new class which is called new better (worse) than renewal used in Laplace transform order NBRUL (NWRUL) as follows It is obvious that NBRU⇒NBRUL⇒NBRUE.

Moments inequalities
In this section, the moment inequalities for NBRUL class are established.
Theorem 3.1. Let F be NBRUL life distribution such that all moments exist and finite then for integers r ≥ 0 and s ≥ 0. Then where Making use of (2), yields The left hand side of (3) can be written as After some calculations, the left hand side of (3) is given by Also, the right hand side of (3) can be put in the following form After some calculations (5) can be rewritten as From (4) and (6), Eq. (1) can be proved.

Testing against NBRUL alternatives
Using Inequality (7) we can test the null hypothesis H 0 : F is exponential aganist H 1 : F is NBRUL and not exponential. δ 1 (s) has been used as follows Note that under H 0 , δ (1) (s) = 0, while under H 1 , δ (1) (s) > 0. Let X 1 , X 2 , X 3 , ........., X n be a random sample from a distribution F. The empirical estimate δ (1) n (s) of δ (1) (s) can be obtained as To make the test invarient, let ∆ (1) One can note that δ (1) (s) is an unbiased estimator of δ (1) and define the symmetric Kernel where the summation is over all arrangements of X i , X j . Then ∆ (1) n (s) in (9) is equivalent to the U n -statistic given by The asymptotic normality of ∆ (1) n (s) can be summarized in the following theorem.

The Pitman asymptotic efficiency
To judge on the quality of this procedure, Pitman asymptotic efficiencies (PAEs) are computed and compared with some other tests for the following alternative distributions: Note that For θ = 1,F 1 (x) reduces to exponential distribution while for θ = 0,F 2 (x)andF 3 (x) reduces to exponential distribution. The PAE is defined by: At s = 5, Hence, d dθ δ (1) Upon using the definition of the PAE in (14), we obtain PAE(δ (1) ) = 1  (Mahmoud and Abdul Alim 2002, 2008 0.050 0.217 0.144 Our test ∆   From Table 1, it is obvious that ∆ (1) n (5) is better than the other tests based on the PAEs.

Monte Carlo null distribution critical points
In this section the Monte Carlo null distribution critical points of ∆ (1) n (5) are simulated based on 10000 generated samples of size n = 5(5)35, 39, 40(5)50. From the standard exponential distribution by using Mathematica 8 program. Table 2 gives the upper percentile points of statistic ∆ (1) n (5) for different confidence levels 90%, 95% and 99%.
From Table 2, it is obvious that the critical values are decreasing as the samples size increasing and they are increasing as the confidence levels increasing.

Power estimates of the test ∆
(1) n (5) In this section the power of our test ∆ (1) n (5) will be estimated at (1 − α)% confidence level, α = 0.05 with suitable parameters values of θ at n = 10, 20 and 30 with respect to three alternatives Linear failure rate (LFR) , Weibull and Gamma distributions based on 10000 samples. Table 3 shows that the power estimates of our test ∆ (1) n (5) are good power for all alternatives and increases when the value of the parameter θ and the sample sizes increasing.

Applications to real data
In this section, we apply our test to some real data-sets at 95% confidence level.  96 105 107 107 107 116 150 In this case, ∆ (1) n (5) = 0.0000132958 which is less than the corresponding critical value in Table 2, then we reject H 1 which states that the data set have NBRUL property.
2-Consider the real data-set given in (Grubbs 1971) and have been used in (Shapiro 1995). This data set gives the times between arrivals of 25 customers at a facility. Since ∆ (1) n (5) = 0.00119843 and this value less than the corresponding critical value in Table 2. Then we conclude that this data set have the exponential property.

Conclusion
The NBRUL class of life distributions is considered. The moments inequalities are derived. A new test statistics for exponentiality versus NBRUL class is proposed based on the moment inequalities. Quality criteria of the test is shown by the famous criterion which is Pitman asymptotic efficiency. The upper percentiles and the power of the proposed test are calculated and tabulated. Our test is applied to some real data to show the usefulness of the test.