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IRJET- Testing Exponentiality Against New Better Than Used Average Renewal Failure Rate Ageing Class
1.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 08 | Aug 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1276 Testing Exponentiality Against New Better than used Average Renewal Failure Rate Ageing Class U. Rizwan1, M. Apsal2 1,2Department of Mathematics, Islamiah College, Vaniyambadi. ------------------------------------------------------------------------***----------------------------------------------------------------------- Abstract: - In this paper, a new class of life distribution namely New Better than Used Average Renewal Failure Rate ageing class (NBUARFR) is discussed. The Pitman asymptotic efficiency of the test are calculated. Power estimates of the test is also evaluated. Keywords: Classes of Life Distribution Ageing class, efficiency, estimates. 1. Introduction Testing exponentiality against various classes of Life distribution has got a deal life attention with respect to the testing exponential against NBUE, NBUFR and NBUARFR classes. In this paper, real data is given and we desire to test : data is exponential versus the alternative hypothesis : data is not exponential to choose between and or to make a decision, we need to the determine how close a specific sample result falls to one of the hypothesis being tested. These comparisons produce new NBU type classes including New Better than Renewal of Used (NBRU), Renewal New is Better than Used (RNBU) and Renewal New is Better than Renewal Used Expectation (RNBRUE) when comparing the residual equilibrium life at a certain age and its equilibrium (stationary) life in expectation. 2. Preliminaries In this section, we give some definitions that are necessary for further discussion. Definition 2.1. A life distribution F is New Better than Used in Average Renewal Failure Rate (NBUARFR), if (0) ∫ ( ) . Equivalently, (0) ̅ ( ) where ̅ ( ) ∫ ̅ ( ) . Where is the failure rate of a new system is than the average renewal failure rate system. Definition 2.2. A random variable X is said to be NBRUL, if ∫ ̅ ( ) ̅ ( ) ∫ ̅( ) . It is obvious that NBRUL contains the NBRU class, that is NBRUL NBRUL. Also, NBRU NBRUL NBRUE. 3. Main Results In this section, we gives some results related to our study. Theorem 3.1. The life distribution F or the its survival F having NBUARFR, if and only if ∫ ̅ ( ) ( ) ( ) . Proof. Let F be an distribution with failure rate r( ), F is NBUARFR means (0) ̅ ( ) Then ( ) ̅ ( ) ̅ ( ) ( ) This is equivalent to the form ∫ ̅ ( ) ( ) ( ) The above equation is satisfied, then it is easy to prove the NBUARFR property.
2.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 08 | Aug 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1277 Theorem 3.2. If X is a random variable with distribution function F and F belongs to NBRUL class, then ( ) ( ) ( ) . (1) Where ( ) ∫ ( ) And ∫ ̅( ) Proof. Since F is NBRUL then ∫ ̅ ( ) ̅ ( ) ∫ ̅( ) . Integrating both sides with respect to t over [0;1), gives ∫ ∫ ̅ ( ) ∫ ∫ ̅( ) (2) setting ∫ ∫ ̅ ( ) can be rewritten as ∫ 0 1 (3) So, ( ) ( ) Similarly, if we write ∫ ∫ ̅( ) ̅ ( ) Then ( ) ( ) ( ). (4) Substituting the equations (3) and (4) in equation (2), we get ( ) ( ) ( ). Which is completes the proof. Theorem 3.3. As , ( ) ( )- is asymptotically normal with mean 0 and variance [( ) ( ) ( ) ] U and variance to ( ) ( )( ) Proof. Using standard U-statistic theory =V* , ( ) - , ( ) -+ Then E( ( ) ( )) . / ∫ ( ) , and E( ( ) ( )) ∫ ( ) ( ) therefore [( ) ( ) ( ) ] Under ( ) ( )( ) 4 The Pitman Asymptotic Efficiency (PAE) To judge on the quality of this procedure Pitman Asymptotic Efficiencies (PAE) are computed and compared with some other tests for the following alternative distribution (i) The Weibull distribution ( ) .
3.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 08 | Aug 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1278 (ii) The Linear Failure Rate distribution ( ) . (iii) ( ) ( ) Note that if and ̅ goes to the exponential distribution then if and ̅ ̅ reduce to the exponential distribution. The PAE is defined by ( ( )) ( ) | ( )| When s =2.5, this leads to , ( ) - . , ( ) - .956183. And , ( ) -= 0.278887. Where ( ) Test Weibull LFR Makeham Ahmed [2] 0.132 0.433 0.144 Bryson and Siddique [3] 0.170 0.408 0.039 Abdul- Aziz [1] 0.223 0.535 0.184 Kango [7] 0.050 0.217 0.144 Our test ( ) 1.172 0.956 0.279 Table 1. Compare between the PAE of our test and some others tests. From table 1, it is obvious that ( ) is better than the others based on PAE. 5. Power Estimate of the Test ( ) In the paper, the power of our ( ) is estimated at ( ) confidence level, =0.05 with suitable parameters values of at n=10, 20 and 30 with respect to three alternatives Weibull, Linear Failure Rate (LFR) and Gamma distribution based on 10000 samples. N Weibull LFR Gamma 10 2 3 4 0.8043 0.9972 0.9999 0.2511 0.2850 0.2821 0.2737 0.7407 0.9149 20 2 3 4 0.9851 1.0000 1.0000 0.4897 0.5571 0.5892 0.5689 0.9502 0.9976 30 2 3 4 0.9996 1.0000 1.0000 0.6864 0.7682 0.8033 0.7393 0.9929 0.9997 Table 2. The power estimates of ( ). Table 2, when that the power estimates of test ( ) are good for all alternative and increase when the value of the parameters and the samples size increase. 6. Conclusion Testing exponentiality against NBUARFR distribution is considered. A new class of the distribution is added to the family of renewal classes of life distribution. The Pitman asymptotic efficiency of the test are calculated compared with other test PAE and power estimates of the test. Quality criteria of the test is shown by the famous criteria which is Pitman efficiency. References 1) Abdul Aziz.A.A (2007) On testing against RNBRUE alternatives, Applied Mathematics Science. 1, pp 1725-1736 2) Ahemd.I.A (1975) A nonparametric test for the monotonicity of a failure rate average function, Communication in Statistics. 4, pp 967-974. 3) Bryson.M.C and Siddique.M.M (1969) Some criteria for ageing, Journal of the American Statistics Association. 64, pp 1472-1483.
4.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 08 | Aug 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 1279 4) Deshpande.J.V and Kocher and Singh.C.H (1986) Aspects of positive ageing Journal of Applied Probability. 28, pp 773-779. 5) Elbatal.I (2009) On testing statistics of Renewal New Better than Renewal Used Class of Life Distribution Int. J. Contempt. Math Science, Vol.4, no.1, pp 17-29. 6) Hendi.M.I (1999) Testing general harmonic better than used in expectation using kernel method, Egypt. Computer. Sc.J.21, pp, 1-16. 7) Kango.A.I (1993) Testing for New is Better than Used, Communication in statistics theory. Method 12, pp 311-321.
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