Estimation of Stress-Strength Reliability for Kumaraswamy Exponential Distribution Based on Upper Record Values

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1 International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 2, HIKARI Ltd, Estimation of Stress-Strength Reliability for Kumaraswamy Exponential Distribution Based on Upper Record Values Essam A. Amin Department of Mathematical Statistics Institute of Statistical Studies and Research (ISSR Cairo University, Egypt Copyright c 2017 Essam A. Amin. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we consider the maximum likelihood estimator for the reliability function for the kumaraswamy exponential distribution based on upper record values. Exact interval estimation for reliability function is conducted. Simulations study are conducted to investigate the theoretical results. Keywords: Kumaraswamy exponential,upper record values, Reliability function, Maximum Likelihood estimation, Confidence interval 1 Introduction In the reliability literature, the stress-strength term refers to a component which has a random strength X and is subject to a random stress Y. The component fails if the stress applied to it exceeds the strength, while the component works whenever Y < X. Thus, R = P (Y < X is a measure of component reliability. Stress-strength models have attracted many statisticians for many years due to their applicability in different and diverse areas such as engineering, economics, quality control and, in the last thirty years, there have been numerous applications to medical problems and clinical trials

2 0 Essam A. Amin (e.g. Gupta and Gupta, 1990, Adimari and Chiogna, 200 and to behavioral and educational studies (Klein and Moeschberger, 2003 where left-truncated models are applied. The reliability and its estimation have been well studied under different distributional assumptions on X and Y. Kotz et al. (2003 provide an excellent review of theory and applications in this area. Record values and associated statistics are of great interest in several reallife applications involving weather, economic, sports data Olympic records or world records in sport and life test. Also in industry, it might be interesting to determine the minimum failure stress of the products sequentially. The statistical study of record values started with Chandler (1952, he formulated the theory of record values as a model for successive extremes in a sequence of independently and identically random variables. Feller (19 gave some examples of record values with respect to gambling problems. Resnick (1973 discussed the asymptotic theory of records. Theory of record values and its distributional properties have been extensively studied in the literature, for example, see, Resnick (1987, Ahsanullah (1995, Nagaraja (1988, Balakrishnan and Ahsanullah (199, Raqab et al (2008, Soliman et al. (2010, Amin (2012 and Nadar et al (2013. Let {X n, n 1} be a sequence of independent and identically distributed random variables with cumulative distribution function (cdf F (x and the corresponding (pdf f(x. Set Y n = max(x 1, X 2..., X n, n 1. We say X i is a record value of {X n } if Y j > Y j 1. Thus the record values are the relative maxima of the sequence. By definition, X 1 is a record value. The random variables U(0 = 1 and U(m = min{j : j > U(m 1, X j > X U(m 1 }, are called the upper record times, and the sequence {U(m, m > 0} is called the sequence of upper record times and the sequences R m = X U(m is called the sequences of upper record values. Let R 0, R 1,..., R m be the first (m+1 th upper record values from the distribution function F (x; θ and pdf f(x; θ, then the density function of the upper record values, was found in Ahsanullah (1995 as follows 1 [ f Rm (r m ; θ = ln F (rm ; θ ] m f(rm ; θ, < r m <, m = 0, 1, 2..., Γ(m + 1 (1.1 while the likelihood function based on the first (m + 1 th upper record values R 0, R 1,..., R m is given by L(θ = m 1 I=0 f(r i ; θ F (r i ; θ f(r m; θ < r 0 < r 1 <... < r m <. (1.2 Adepoju and Chukwu (2015 defined the Kumaraswamy exponential distribution as follows F (x, λ, a, b = 1 [ 1 (1 e λx a] b, x > 0 (1.3

3 Stress-strength reliability for Kumaraswamy exponential distribution 1 where λ > 0 is a scale parameter and the other positive parameters, a and b are shape parameters. The corresponding pdf and hazard rate function are and f(x, λ, a, b = abλe λx (1 e λx a 1 [ 1 (1 e λx a] b 1, x 0, (1. h(x, λ, a, b = abλe λx (1 e λx a 1, (1.5 [1 (1 e λx a ] respectively. In this paper considers statistical inference for R = P (Y < X when the observed data X and Y are drawn from independent Kw-E distribution based on upper record values. there are some works on R on the basis of record data. For generalized exponential distribution, Baklizi (2008a compared likelihood and Bayesian estimation of stress-strength parameter using lower record values. Also, in the one and two-parameter exponential distribution, Baklizi (2008b estimated P (Y < X based on record values.nadar and Kizilaslan (201 used maximum likelihood and Bayesian approaches to obtain the estimation of P (Y < X based on a set of upper record values from Kumaraswamy distribution. The rest of the article is organized as follows. In Section 2, we derive the maximum likelihood estimator for the reliability function R of the Kumaraswamy exponential distribution based on upper record values. In Section 3, we obtain the exact interval estimation for the reliability function R of the Kw-E distribution based on upper records when the scale parameter λ and shape parameter a are known. Simulation study are obtained for the theoretical results in Section. Finally we conclude the paper in Section 5. 2 Maximum Likelihood Estimator for the Reliability Function Let X and Y be independent random variables from (Kw-E distribution (1.3 with parameters (a, λ, b 1 and (a, λ, b 2 respectively. The stress-strength reliability function R is given by R = P (Y < X = 0 x 0 f(xf(ydydx = b 2 b 1 + b 2. (2.1 Let R 0, R 1, R 2,..., R n represent the first (n + 1 upper record values arising from a sequence {X i } of iid Kw-E distribution (1.3 with parameters (a, λ, b 1 and S 0, S 1, S 2,..., S m be the first (m + 1 upper record values arising from a sequence {Y i }of iid Kw-E distribution (1.3 with parameters (a, λ, b 2 where

4 2 Essam A. Amin (a, λ are unknown. al.(1998, The Likelihood functions are given by, see Arnold et L 1 (a, λ, b 1 r = (aλb 1 n+1 Π n e λr i [K(λ, r i ] a 1 [T (a, λ, r n ] b 1 (2.2 T (a, λ, r i and L 2 (a, λ, b 2 s = (aλb 2 m+1 Π m e λs i [K(λ, s i ] a 1 [T (a, λ, s m ] b 2, (2.3 T (a, λ, s i where r = (r 0, r 1, r 2,..., r n, s = (s 0, s 1, s 2,..., s n. Let ϕ = (a, λ, b 1, b 2, the joint likelihood and the joint log likelihood is given by L(ϕ r, s = L 1 (a, λ, b 1 r L 2 (a, λ, b 2 s (2. and [ n ln L(ϕ r, s = (n + 1 ln(aλb 1 + (m + 1 ln(aλb 2 λ r i + [ n + (a 1 ln K(λ, r i + [ n ln T (a, λ, r i + ] s j ] ln K(λ, s j + b 1 ln T (a, λ, r n ] ln T (a, λ, s j + b 2 ln T (a, λ, s m (2.5, respectively. The maximum likelihood estimators of b 1, b 2, a and λ, say b 1, b 2, â and λ respectively, can be obtained as a solution of lnl(ϕ r, s b 1 = n + 1 b 1 + ln T (a, λ, r n = 0, (2. lnl(ϕ r, s b 2 = m + 1 b 2 + ln T (a, λ, s m = 0, (2.7 lnl(ϕ r, s a = n + m + 2 a + n + n ln K(λ, r i + [K(λ, r i ] a ln K(λ, r i T (a, λ, r i + ln K(λ, s j [K(λ, s j ] a ln K(λ, s j T (a, λ, s j [K(λ, r n ] a ln K(λ, r n [K(λ, s m ] a ln K(λ, s m b 1 b 2 T (a, λ, r n T (a, λ, s m = 0, (2.8

5 Stress-strength reliability for Kumaraswamy exponential distribution 3 and lnl(ϕ r, s λ [ n = n + m + 2 r i e λr i m ] + (a 1 λ K(λ, r i + s j e λs j n r i K(λ, s j [ n ] r i e λr i [K(λ, r i ] a 1 s j e λs j [K(λ, s j ] a 1 s j + a + T (a, λ, r i T (a, λ, s j b 1a r n e λrn [K(λ, r n ] a 1 b 2a s m e λsm [K(λ, s m ] a 1 T (a, λ, r n T (a, λ, s m = 0. (2.9 from (2. and (2.7 we get b1 = (n + 1 ln T, (2.10 r n b2 = (m + 1 ln T. (2.11 s n Also â, and λ can be obtained as a solution of the following nonlinear equations [ n + m + 2 n n K( λ, r i ]â ln K( λ, ri + ln K( λ, r i + ln K( λ, s j + â T r i [ [ a K( λ, s j ]â ln K( λ, sj (n + 1 K( λ, r n ] ln K( λ, rn + T + s j T r n ln T r n [ (m + 1 K( λ, s m ]â ln K( λ, sm + T s m ln T = 0, (2.12 s m

6 Essam A. Amin and [ n ] [ n + m + 2 r i e λr i + (â 1 λ K( λ, r i + s j e λs j n r i + K( λ, s j ]â 1 ]â 1 n r i e λr i [K( λ, r i s j e λs j [K( λ, s j +â T + r i T s j ]â 1 (n + 1â r n e λr n [K( λ, r n T r n ln T r n ] s j ]â 1 (m + 1â s m e λs m [K( λ, s m s m T s m ln T = 0. (2.13 After, â and λ are obtained, b 1, and b 2 can be obtained from Equations (2.10 and (2.11, respectively. Therefore the maximum likelihood estimator of R is given as R = b 2 b1 + b 2 (2.1 3 Interval Estimation for the Reliability Function when λ and a are Known In this case we consider without loss of generality that the scale parameter λ = λ 0, and a = a 0. Therefore, the R 0, R 1, R 2,..., R n be a set of upper records from Kw-E distribution with parameters (a 0, λ 0, b 1 and S 0, S 1, S 2,..., S m be independent set of upper record from Kw-E distribution with parameters (a 0, λ 0, b 2, then the maximum likelihood estimators for b 1, and b 2, say b 11, and b 21, respectively, are given from the following equations b11 = b21 = (n + 1 ln T (a 0, λ 0, r n, (3.1 (m + 1 ln T (a 0, λ 0, s n, (3.2 and the maximum likelihood estimator for the reliability function is given by R 1 = b 21 b11 + b 21. (3.3 To study the the distribution of the maximum likelihood estimator for R 1, we must obtain the distribution of b 11 and b 21. Consider Z 1 = (n+1, one ln T (a 0,λ 0,r n

7 Stress-strength reliability for Kumaraswamy exponential distribution 5 can show that the probability distribution of Z 1 is recognized as the inverted gamma distribution with parameters (n+1 and (n+1b 1. From the properties of the inverted gama we show that 2(n+1b 1 Z 1 has a chi-square distribution with 2(n + 1 degrees of freedom. Similarly Z 2 = (m+1 ln T (a 0,λ 0,s m has inverted gamma distribution with parameters (m + 1 and (m + 1b 2 and 2(m+1b 2 Z 2 has a chisquare distribution with 2(m + 1 degrees of freedom. Now since the random variables Z 1 and Z 2 are independent, then one can show that Z 1 Z 2 b 1 b 2 F 2(m+1,2(n+1 (3. Therefore, after some simple argument we can show that the distribution of 1 R 1 is, where F 1+ b 1 2(m+1,2(n+1 is F-distribution with 2(m + 1 b2 F 2(m+1,2(n+1 and 2(m + 1 degrees of freedom. Based on the sampling distribution of R 1, and after some transformation 100(1 α% confidence interval for the stress strength reliability function for the Kw-E distribution based on upper record values is (L, U where. [ ] 1 [ and U = L = 1 + Z 1 Z 2 F (1 α 2,2(m+1,2(n+1 Simulation Study 1 + Z 1 Z 2 F ( α 2,2(m+1,2(n+1 ] 1 (3.5 In this section our purpose to render numerical analysis for average mean square error AMSE of MLE and confidence intervals (C.I for reliability function R. The software package MathCAD (2001 is used for the simulation study. The following steps are considered 1- Generate N = sets of uniform (0,1 random variables, then use the usual transformation technique to get the corresponding sets of X-samples from the Kw-E distribution with parameters a, λ, b Choose from each vector the first (n + 1, n= 2,, upper record values r 0, r 1, r 2,..., r n. 3- By the same manner, the set of Y - samples from the Kw-E distribution and the first (m + 1, m= 2,, upper record values s 0, s 1, s 2,..., s n for different values of a, λ and b 2.are selected. - The validity of the estimate of R is discussed by measures the average mean square error of the estimate and the average confidence interval of the estimate. 5- Tables 1 to contains the following i Different combinations of the sample sizes and the distribution parameters. ii Simulation results for the maximum likelihood estimators R and R 1 for the

8 Essam A. Amin reliability function R iii The average mean square error (AMSE for the MLE of R and R 1 iv The 95% and 99% confidence interval for the reliability function R and the average probability interval length Table (1: MLE s and confidence interval for the reliability function when λ = 0.5, a = 0.5, b 1 = 1.5 and b 2 = 0.5 n m R R1 95% C.I 99% C.I [0.07,0.59] [0.03,0.71] ( ( [0.093,0.02] [0.03,0.717] ( ( [0.12,0.25] [0.08,0.733] ( ( [0.02,0.9] [0.038,0.595] ( ( [0.089,0.511] [0.01,0.1] ( ( [0.121,0.55] [0.088,0.5] ( ( [0.055,0.35] [0.035,0.527] ( ( [0.079,0.38] [0.055,0.525] ( ( [0.107,0.58] [0.079,0.52] ( ( * The values between brackets are the AMSE

9 Stress-strength reliability for Kumaraswamy exponential distribution 7 Table (2: MLE s and confidence interval for the reliability function when λ = 0.5, a = 0.5, b 1 = 0.5 and b 2 = 0.5 n m R R1 95% C.I 99% C.I [0.18,0.818] [0.119,0.88] ( ( [0.233,0.818] [0.17,0.882] ( ( [0.307,0.89] [0.23,0.90] ( ( [0.178,0.72] [0.115,0.83] ( ( [0.237,0.77] [0.172,0.83] ( ( [0.30,0.799] [0.23,0.855] ( ( [0.15,0.702] [0.1,0.773] ( ( [0.205,0.702] [0.18, 0.77] ( ( [0.257,0.725] [0.198,0.788] ( ( * The values between brackets are the AMSE

10 8 Essam A. Amin Table (3: MLE s and confidence interval for the reliability function when λ = 0.5, a = 0.5, b 1 = 0.5 and b 2 = 0.5 n m R R1 95% C.I 99% C.I [0.181,0.813] [0.11,0.88] ( ( [0.238,0.822] [0.17,0.885] ( ( [0.291,0.839] [0.221,0.897] ( ( [0.178,0.72] [0.115,0.83] ( ( [0.237,0.77] [0.172,0.83] ( ( [0.292,0.79] [0.225,0.88] ( ( [0.158,0.70] [0.101,0.775] ( ( [0.213,0.712] [0.155, 0.778] ( ( [0.25,0.73] [0.205,0.79] ( ( * The values between brackets are the AMSE

11 Stress-strength reliability for Kumaraswamy exponential distribution 9 Table (: MLE s and confidence interval for the reliability function when λ = 0.5, a = 0.5, b 1 = 1.5 and b 2 = 0.5 n m R R1 95% C.I 99% C.I [0.071,0.02] [0.03,0.719] ( ( [0.095,0.08] [0.05,0.721] ( ( [0.12,0.1] [0.089,0.7] ( ( [0.08,0.591] [0.02,0.21] ( ( [0.09,0.52] [0.05,0.2] ( ( [0.121,0.552] [0.089,0.] ( ( [0.058,0.] [0.03,0.532] ( ( [0.079,0.38] [0.055,0.525] ( ( [0.107,0.58] [0.079,0.52] ( ( * The values between brackets are the AMSE 5 Conclusion In this paper the estimation problem of the stress-strength reliability R = P (Y < X when X and Y are independent Kw-E distribution based on upper record values are discussed. The MLE and the confidence interval for reliability function are obtained. We have presented a simulation study, and observed the following results: 1- For fixed values n the AMSE is decreases when m increases, also for fixed m the AMSE decreases when n increases. 2- The length of the confidence interval is decreases when n or m increases. References [1] K. A. Adepoju and O. I. Chukwu, Maximum Likelihood Estimation of the Kumaraswamy Exponential Distribution with Applications, Journal of Modern Applied Statistical Methods, 1 (2015, no. 1, [2] G. Adimari and M. Chiogna, Partially parametric interval estimation of

12 70 Essam A. Amin Pr(Y > X, Computational Statistics and Data Analysis, 51 (200, [3] M. Ahsanullah, Record Statistics, Nova Science Publishers Inc., Huntington, New York, [] E. A. Amin, Bayesian and Non-Bayesian Estimation from Type I Generalized Logistic Distribution Based On Lower Record Values, Journal of Applied Sciences Research, 8 (2012, no. 1, [5] B.C. Arnold, N. Balakrishnan and H. N. Nagaraja, Records, Wiley, New York, [] A. Baklizi, Likelihood and Bayesian estimation of Pr(X < Y using lower record values from the generalized exponential distribution, Computational Statistics and Data Analysis, 52 (2008, [7] A. Baklizi, Estimation of Pr(X < Y using record values in the one and two parameter exponential distribution, Communication in Statistics-Theory and Methods, 37 (2008, [8] N. Balakrishnan and M. Ahsanullah, Recurrence relations for single and product moments of record values from generalized Pareto distribution, Communication in Statistics-Theory and Methods, 23 (199, no. 10, [9] K. N. Chandler, The distribution and frequency of record values, Journal of Royal Statistics Society, Series B, 1 (1952, [10] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, John Wiley and Sons, New York, 19. [11] R. D. Gupta and R. C. Gupta, Estimation of Pr(a x >> b y in the multivariate normal case, Statistics, 21 (1990, no. 1, [12] J. Klein and M. Moeschberger, Survival Analysis: Techniques for Censored and Truncated Data, Second ed., Springer-Verlag, Berlin, [13] S. Kotz, Y. Lumelskii and M. Pensky, The Stress-Stength Model and Its Generalizations, World Scientific, Singapore, [1] H. N.Nagaraja, Record values and related statistics. A review, Communication in Statistics- Theory and Methods, 7 (1988,

13 Stress-strength reliability for Kumaraswamy exponential distribution 71 [15] M. Nadar, A. Papadopoulos and F. Kizilaslan, Statistical analysis for Kumaraswamy s distribution based on record data, Statistical Papers, 5 (2013, no. 2, [1] M. Nadar and F. Kizilaslan, Classical and Bayesian estimation of P(X < Y using upper record values from Kumaraswamy s distribution, Statistical Papers, 55 (201, no. 3, [17] M. Z. Raqab, M. T. Madi and D. Kundu, Estimation of P(Y < X for the Three-Parameter Generalized Exponential Distribution, Communication in Statistics- Theory and Methods, 37 (2008, no. 18, [18] S. L. Resnick, Record values and maxima, Annals of probability, 1 (1973, [19] S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer, New York, [20] A. Soliman, E. A. Amin and A. A. Abd-El Aziz, Estimation and prediction from inverse Rayleigh distribution based on lower record values, Applied Mathematical Sciences, (2010, no. 2, Received: March 11, 2017; Published: March 2, 2017