Florida International University FIU Digital Commons FIU Electronic Theses and Dissertations University Graduate School 9-3-211 Inference about Reliability Parameter with Underlying Gamma and Exponential Distribution Zeyi Wang Florida International University, wzyznn@gmail.com DOI: 1.25148/etd.FI111224 Follow this and additional works at: http://digitalcommons.fiu.edu/etd Recommended Citation Wang, Zeyi, "Inference about Reliability Parameter with Underlying Gamma and Exponential Distribution" (211). FIU Electronic Theses and Dissertations. 475. http://digitalcommons.fiu.edu/etd/475 This work is brought to you for free and open access by the University Graduate School at FIU Digital Commons. It has been accepted for inclusion in FIU Electronic Theses and Dissertations by an authorized administrator of FIU Digital Commons. For more information, please contact dcc@fiu.edu.
FLORIDA INTERNATIONAL UNIVERSITY Miami, Florida INFERENCE ABOUT RELIABILITY PARAMETER WITH UNDERLYING GAMMA AND EXPONENTIAL DISTRIBUTION A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in STATISTICS by Zeyi Wang 211
To: Dean Kenneth Furton College of Arts and Sciences This thesis, written by Zeyi Wang, and entitled Inference about Reliability Parameter with underlying Gamma and Exponential Distribution, having been approved in respect to style and intellectual content, is referred to you for judgment. We have read this thesis and recommend that it be approved. Florence George Kai Huang Jie Mi, Major Professor Date of Defense: September 3, 211 The thesis of Zeyi Wang is approved. Dean Kenneth Furton College of Arts and Sciences Dean Lakshmi N. Reddi University Graduate School Florida International University, 211 ii
ACKNOWLEDGMENTS I would like to express my gratitude to all those who gave me the possibility to complete this thesis. I would like to thank Dr. Florence George and Dr. Kai Huang for their time to serve on my committee. I really appreciate their suggestions and encouragement. I am deeply indebted to my supervisor professor Dr. Jie Mi who helped me in all the time of research for and writing of this thesis. It is a pleasure to express my sincere gratitude for his patience, motivation, enthusiasm, and immense knowledge. His encouragement, guidance and support from the initial to the final level enabled me to complete the thesis. I would like to give a special thanks to all other professors in the Department of Mathematics and Statistics who supported and encouraged me throughout my years in FIU. Finally, I wish to thank my family and friends for their unflagging spiritual support and their absolute confidence in me. iii
ABSTRACT OF THE THESIS INFERENCE ABOUT RELIABILITY PARAMETER WITH UNDERLYING GAMMA AND EXPONENTIAL DISTRIBUTION by Zeyi Wang Florida International University, 211 Miami, Florida Professor Jie Mi, Major Professor The statistical inference about the reliability parameter R involving independent gamma stress and exponential strength is considered. Assuming the shape parameter of gamma is a known arbitrary real number and the scale parameters of gamma and exponential are unknown, the UMVUE and MLE of R are obtained. A pivot is proposed. Some inference about R derived from this pivot is presented. It will be shown that the pivot can be used for testing hypothesis and constructing confidence interval. A procedure of constructing the confidence interval for R is derived. The performances of the UMVUE and MLE are compared numerically based on extensive Monte Carlo simulation. Simulation studies indicate that the performance of the two estimators is about the same. The MLE is preferred because of the simplicity of its computation. Keywords: Stress-Strength, Exponential Distribution, Gamma Distribution, UMVUE, MLE, MSE, Pivot, Bias. iv
TABLE OF CONTENTS CHAPTER PAGE 1 Introduction................................................................... 1 2 Point Estimation of Reliability Parameter...................................... 3 2.1 Uniformly Minimum Variance Unbiased Estimator (UMVUE).............. 4 2.2 Maximum Likelihood Estimator (MLE)................................... 1 3 Inference about R Based on Pivot.............................................11 3.1 Pivotal Quantity..........................................................11 3.2 Using Pivot to test two-sided and one-sided hypothesis.................... 13 3.3 Confidence interval of R.................................................. 15 4 Numerical Study..............................................................16 4.1 Example 1................................................................ 16 4.2 Example 2................................................................ 17 REFERENCES................................................................. 19 APPENDIX.................................................................... 21 v
LIST OF TABLES TABLE.....................................................................PAGE 1 For γ = 1.25, N = 1, R =.3.............................................22 2 For γ = 1.25, N = 1, R =.6.............................................22 3 For γ = 1.25, N = 1, R =.9.............................................23 4 For γ = 4.5, N = 1, R =.3.............................................. 23 5 For γ = 4.5, N = 1, R =.6.............................................. 24 6 For γ = 4.5, N = 1, R =.9.............................................. 24 vi
LIST OF FIGURES FIGURE................................................................... PAGE 1 UMVUE and MLE γ = 1.25, m = n = 5, N = 1.......................... 25 2 Mean Error of UMVUE and MLE γ = 1.25, m = n = 5, N = 1........... 25 3 MSE of UMVUE γ = 1.25, m = n = 5, 1, 2, 3, 4 N = 1................26 4 MSE of MLE γ = 1.25, m = n = 5, 1, 2, 3, 4 N = 1................... 26 5 Comparison between MSE of UMUVE and MLE γ = 1.25, m = n = 5.........27 6 Comparison between MSE of UMUVE and MLE γ = 1.25, m = n = 1........27 7 UMVUE and MLE γ = 4.5, m = n = 5, N = 1........................... 28 8 Mean Error of UMVUE and MLE γ = 4.5, m = n = 5, N = 1............ 28 9 MSE of UMVUE γ = 4.5, m = n = 5, 1, 2, 3, 4 N = 1................. 29 1 MSE of MLE γ = 4.5, m = n = 5, 1, 2, 3, 4 N = 1................... 29 11 Comparison between MSE of UMUVE and MLE γ = 4.5, m = n = 5.........3 12 Comparison between MSE of UMUVE and MLE γ = 4.5, m = n = 1........3 vii
1. Introduction Extensive research has been conducted on the stress-strength model. This model involves two independent random variables X and Y, and the parameter of interest is the probability R P (X Y ). This function has attracted a great deal of attention in the literature because of its wide applications. Naturally, from the engineering point of view X and Y can represent the strength of a structure and the stress imposed on it. With this interpretation the probablility R is often called the reliability parameter which will be used in this article. In addition to its applications in engineering, stress-strength model is also applied in many other fields. For example, in a medical application let X represent the response for a control group and Y represent the response for a treatment group (Simonoff, Hochberg, and Reiser (1986); Hauck, Hyslop, and Anderson (2)). In this case the probability P (X Y ) measures the effect of the treatment. The probability R P (X Y ) can also be used for bioequivalence assessment (Wellek (1993)). It is frequently used to assess the effectiveness of diagnostic markers in distinguishing between diseased and healthy individuals (Reiser (2)). In biology, this probability is useful in estimating heritability of a generic trait (Schwarz and Wearen (1959)). An extensive review of this model is presented in Kotz, Lumelsdii, and Pensky (23). The most recent work on the topic was published by Saracoglu and Kaya (27), Krishnamoorthy, Mukherjee, and Guo (27), Baklizi (28a, b), Eryilmaz (28a, b, c; 21), Kundu and Raqab (29), Rezaei, Tahmasbi, and Mahmoodi (21) and the references therein. 1
In the literature, various distributions of X and Y are considered which include exponential, normal, Weibull, etc. Krishnamoorthy et al. (27) gave the test and interval estimation procedures derived from the generalized variable approach for two-parameter exponential case when both the location and scale parameters are unknown. Baklizi (28a, b) derived estimators and confidence intervals by considering both one and two-parameter exponential cases with record values and lower record values. Constantine, Karson, and Tse (1986) considered the case when both X and Y have gamma distributions. It was assumed there that the two scale parameters are unknown but the two shape parameters are known integers. Under these assumptions they derived UMVUE and MLE of R and obtained exact interval estimation of R. In this article we will study the same model while X has exponential distribution and Y has gamma distribution. It is assumed that the shape parameter of the gamma distribution is known, however it can be any positive real number and is not restricted to be an integer. The thesis is organized as follows. The MLE and UMVUE of R are derived in Section 2. A pivotal quantity is presented in Section 3. Some inference about R derived from this pivot will also be shown in Section 3. It has been discovered that the pivot can be used for testing hypothesis, and the rejection region are obtained accordingly. A procedure of constructing the confidence intervals for R is derived. In section 4, I present results of numerical studies based on Monte Carlo simulations. It is observed numerically that even though the MLE of R is biased, it is superior to the UMVUE for its MSE is about the same as that of the UMVUE, and is actually smaller most the time, and furthermore its computation is much easier. 2
2. Point Estimation of Reliability Parameter Let the random strength X have exponential distribution with probability density function f X (x) = λe λx, x > and the random stress Y have gamma distribution with probability density function f Y (y) = τ γ Γ(γ) yγ 1 e τy, y > where the shape parameter γ is known but the scale parameters λ and τ are unknown. Our quantity of interest is the reliability parameter R defined as R = P (X > Y ). Suppose that X = {X 1,..., X m } is a simple random sample from the strength population and Y = {Y 1,..., Y n } is a simple random sample from the stress population, where m and n do not have to be the same. In this section we will first derive the uniformly minimum variance unbiased estimator (UMVUE) of R. Then we will derive the maximum likelihood estimator (MLE) of R. 3
Theorem 1. The UMVUE of R based on sample {X 1,..., X m } and {Y 1,..., Y n } is given by R = 1 v/u ( us/v (m 1)(1 s) m 2 (m 1)(1 s) m 2 ( us/v ( B 1 γ,(n 1)γ ( B 1 γ,(n 1)γ ) ) t γ 1 (1 t) (n 1)γ 1 dt ds, if u = m x i n y j = v i=1 j=1 ) )t γ 1 (1 t) (n 1)γ 1 dt ds + (1 v u )m 1, if u = m x i > n y j = v. i=1 j=1 Proof. Obviously the indicator function 1, if X 1 > Y 1, I(X 1 > Y 1 ) =, if X 1 Y 1. is an unbiased estimator of R. Moreover, (U, V ) is the sufficient and complete statis- tic, where U = m X i and V = i=1 n j=1 Y j. Hence, according to the Blackwell-Rao and Lehmann-Scheffe Theorem E ( I(X 1 > Y 1 ) U, V ) = P (X1 > Y 1 U, V ) is the unique UMVUE of R. In order to find P (X 1 > Y 1 U, V ) we need to derive the conditional distribution of X 1 given U = u and the conditional distribution of Y 1 given V = v. By using the routine way of multivariate change of variables it can be shown that the conditional probability density function (pdf) of X 1 given U = u is f X1 U(x u) = m 1 u (1 x u )m 2, < x < u < ; and the conditional pdf of Y 1 given V = v is f Y1 V (y v) = Γ(nγ) Γ(γ)Γ ( (n 1)γ ) yγ 1 v (1 y γ v )(n 1)γ 1 4
= 1 B ( γ, (n 1)γ ) yγ 1 v γ (1 y v )(n 1)γ 1, < y < v <. Here B(a, b) is the beta function, i.e.,. Thus, the desired UMVUE R is given by B(a, b) = Γ(a)Γ(b) Γ(a + b) R = P (X 1 > Y 1 U = u, V = v) = I(x > y)f X1 U(x u)f Y1 V (y v)dxdy = = = = u u I(y < v) B ( γ, (n 1)γ ) yγ 1 v γ (1 y v )(n 1)γ 1 (1 x u )m 2 dxdy (m 1)I(x < u) (1 x u u )m 2 I( y < min{x, v} ) B ( γ, (n 1)γ ) dxdy m 1 u (1 x u )m 2 ( m 1 u (1 x ( u )m 2 dx. min{x, v} min{x, v} 1 B ( γ, (n 1)γ ) yγ 1 v γ (m 1)I(x < u)i(y < x) u yγ 1 v (1 y γ v )(n 1)γ 1 (1 y ) v )(n 1)γ 1 dy dx 1 B ( γ, (n 1)γ ) (y v )γ 1 (1 y v )(n 1)γ 1 d y ) v (1) In the following let us consider two cases: Case 1: u v. In this case x u implies x u v and min(x, v) = x. Let t = y/v and 5
s = x/u, then from (1) R = = = u u 1 m 1 u (1 x ( x u )m 2 m 1 u (1 x ( x/v u )m 2 1 B ( γ, (n 1)γ )(y v )γ 1 (1 y v )(n 1)γ 1 d y ) dx v 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ) dx ( us/v ) (m 1)(1 s) m 2 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ds. (2) Case 2: u > v. R = v We express the integral in (1) as the sum of two other integrals as follows u + v m 1 u (1 x ( u )m 2 min{x, v} 1 B ( γ, (n 1)γ ) (y v )γ 1 (1 y v )(n 1)γ 1 d y v ) dx Note that on the interval (, v) we have x v and so min{x, v} = x, and on the interval (v, u) it holds that x v and consequently min{x, v} = v. From these observations we see that R = = v + v + m 1 u (1 x ( x u )m 2 u v m 1 u (1 x u )m 2 m 1 u (1 x u )m 2 u v m 1 u (1 x u )m 2 ( v 1 B ( γ, (n 1)γ ) (y v )γ 1 (1 y v )(n 1)γ 1 d y ) dx v ( x/v ( 1 1 B ( γ, (n 1)γ ) (y v )γ 1 (1 y v )(n 1)γ 1 d y ) dx v 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ) dx ) 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt dx 6
= = = v/u ( us/v ) (m 1)(1 s) m 2 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ds + u v m 1 u (1 x u )m 2 dx v/u ( us/v ) (m 1)(1 s) m 2 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ds + 1 v/u (m 1)(1 s) m 2 ds v/u ( us/v ) (m 1)(1 s) m 2 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ds +(1 v u )m 1. (3) Finally, from (2) and (3) we obtain the desired result. The expression of R given in Theorem 1 is complicated because the known shape parameter γ can be any positive real number. However, in the case when γ is an integer, then simple expression for R can be obtained as follows. Theorem 2. Suppose that the shape parameter γ is a known integer. Then it follows that R = ( m 1 B γ,(n 1)γ ) (n 1)γ 1 j= ( (n 1)γ 1 j ) ( 1) (n 1)γ 1 j B ( nγ j 1 nγ j, m 1 ( u v )nγ j 1, if u = m x i n y j = v ( m 1 B γ,(n 1)γ ) m 2 i= (n 1)γ 1 j= ( m 2 i )( (n 1)γ 1 j ) i=1 j=1 ) ( 1) (n 1)γ j+m i 3 (nγ j 1)(nγ j+m i 2) ( v u )m i 1 + (1 v u )m 1, if u = m x i > n y j = v i=1 j=1 7
Proof. For the case of u = m X i n Y j = v, according to Theorem 1 the UMVUE R is given as i=1 j=1 R = = = = = = 1 1 ( us/v (m 1)(1 s) m 2 ) 1 B ( γ, (n 1)γ ) tγ 1 (1 t) (n 1)γ 1 dt ds ( us/v (m 1)(1 s) m 2 1 B ( γ, (n 1)γ ) tγ 1 ) ( t) (n 1)γ 1 j dt ds 1 (m 1)(1 s) m 2 B ( γ, (n 1)γ ) ( us/v 1 ) t nγ j 2 dt ds (m 1)(1 s) m 2 B ( γ, (n 1)γ ) ( us v )nγ j 1 ds m 1 B ( γ, (n 1)γ ) 1 (n 1)γ 1 j= s nγ j 1 (1 s) m 2 ds m 1 B ( γ, (n 1)γ ) (n 1)γ 1 j= (n 1)γ 1 j= (n 1)γ 1 j= ( (n 1)γ 1 j (n 1)γ 1 j= ) ( 1) (n 1)γ 1 j ( 1) (n 1)γ 1 j ( (n 1)γ 1 j ( ) (n 1)γ 1 ( ) (n 1)γ 1 ( 1) (n 1)γ 1 j j nγ j 1 (u v )nγ j 1 ) j 1 nγ j 1 ( ) (n 1)γ 1 ( 1) (n 1)γ 1 j B ( nγ j, m 1 ) j nγ j 1 ( u v )nγ j 1 On the other hand, if u > v, then R = v/u ( us/v ) (m 1)(1 s) m 2 1 B(γ, (n 1)γ) tγ 1 (1 t) (n 1)γ 1 dt dt +(1 v u )m 1 8
= = = = m 1 B ( γ, (n 1)γ ) v/u (n 1)γ 1 j= s nγ j 1 (1 s) m 2 ds + (1 v u )m 1 m 1 B ( γ, (n 1)γ ) m 2 i= ( m 2 i m 1 B ( γ, (n 1)γ ) m 2 i= ( ) m 2 i (n 1)γ 1 j= ) ( 1) m 2 i (n 1)γ 1 j= m 2 m 1 B ( γ, (n 1)γ ) ( ) (n 1)γ 1 ( 1) (n 1)γ j 1 j nγ j 1 (u v )nγ j 1 ( ) (n 1)γ 1 ( 1) (n 1)γ j 1 j nγ j 1 (u v )nγ j 1 v/u s nγ j+m i 3 ds + (1 v u )m 1 ( ) (n 1)γ 1 ( 1) (n 1)γ j 1 j nγ j 1 (u v )nγ j 1 ( 1) m 2 i nγ j + m i 2 (v u )nγ j+m i 2 + (1 v u )m 1 i= (n 1)γ 1 j= ( 1) (n 1)γ j+m i 3 ( )( ) m 2 (n 1)γ 1 (nγ j 1)(nγ j + m i 2) (v u )m i 1 + (1 v u )m 1 i j Therefore, the desired result follows. To derive the MLE ˆR of R we need to derive the closed-form formula of R. Theorem 3. The reliability parameter R is given as τ R = ( λ + τ )γ. Proof. The reliability parameter R is given by R = P (X > Y ). By conditioning on Y we have R = P (X > y)f Y (y)dy 9
= = τ γ = e λy τ γ (λ + τ) γ τ = ( λ + τ )γ τ γ Γ(γ) yγ 1 e τy dy 1 Γ(γ) yγ 1 e (λ+τ)y dy (λ + τ) γ y γ 1 e (λ+τ)y dy Γ(γ) So, the result follows. With the help of Theorem 3 the MLE of R is readily available. Theorem 4. The MLE ˆR of R is given as γ ˆR = ( X Ȳ + γ X )γ. (4) Proof. It is easy to see that the MLEs of λ and τ are ˆλ = 1 X and ˆτ = γ Ȳ respectively where X = m X i i=1 m and Ȳ = n Y j j=1 n. Because of the invariance property of MLE we immediately obtain ˆτ ˆR = ( ˆλ + ˆτ )γ = ( γ/ȳ ) γ γ X (1/ X) = ( + (γ/ȳ ) Ȳ + γ X )γ. Thus, the desired result follows. 1
3. Inference about R Based on Pivot We will keep the same notation used in the previous section. For example, the MLE of λ is ˆλ = 1/ X and the MLE of τ is ˆτ = γ/ȳ. First let us search for a pivot. Theorem 5. Define Q = λ τ γ X Ȳ = 1 γ X ρ Ȳ where ρ = τ/λ. Then Q is a pivotal quantity and has F-distribution with numerator degrees of freedom 2m and denominator degrees of freedom 2nγ, i.e., Q F (2m, 2nγ). Proof. Note that 2λX i χ 2 (2), where χ 2 (ν) denote Chi-Square distribution with ν degrees of freedom, since X i follows exponential ditribution with mean 1/λ. Thus 2λ Similarly, we see that m X i χ 2 (2m). i=1 2τ n Y j χ 2 (2nγ). j=1 Therefore, Q = 1 γ X ρ Ȳ = γλ m X i /m i=1 τ n Y j /n j=1 = 2λ m X i /2m i=1 2τ n Y j /2nγ j=1 d = F (2m, 2nγ), because of the independence of X and Ȳ. 11
On the basis of Theorem 5 the reliability parameter R can be expressed in terms of F-distribution. Theorem 6. The reliability parameter R is given by R = R(ρ) = 1 F (γ/ρ; 2, 2γ) and strictly increase in ρ. Proof. By the definition of the reliability parameter R it follows that R = P (X > Y ) = P ( X Y > 1) = P ( 2λX 2τY > λ τ ) = P ( 2λX/2 2τY/2γ > γ ρ ) = P (ξ > γ ρ ) = 1 F ( γ ; 2, 2γ) (5) ρ where random variable ξ F (2, 2γ) and F ( ; 2, 2γ) is its cumulative distribution function (CDF). From Eq.(5) it is clear that R strictly increases in ρ >. For any R (, 1) it is easy to see that there exists a unique ρ > such that R(ρ ) = R. Actually ρ is determined by the equation F (γ/ρ ; 2, 2γ) = 1 R (6) according to Theorem 6. 12
Now we can use Q for testing hypothesis about R. Theorem 7. (Two-Sided Test) Consider testing hypothesis H : R = R vs. H a : R R. Then statistic Q = γ X/ρ Ȳ can be used as test statistic with rejection region R.R. = {(X, Y) : Q F 1 α/2 (2m, 2nγ) or Q F α/2 (2m, 2nγ)} where X = (X 1,..., X m ) and Y = (Y 1,..., Y n ). Proof. Note that the null hypothesis R = R is equivalent to H : ρ = ρ. From Theorem 5, it follows that Q F (2m, 2nγ). Therefore, the size of this test is α. Following the similar argument, we can obtain the two results below. Theorem 8. The pivot Q can be used for testing hypothesis H : R R vs. H a : R > R. The associated rejection region is R.R. = {(X, Y) : Q F α (2m, 2nγ)}. Proof. It is sufficient to show that the size of this test is α, i.e., sup P R (rejecting H ) = α. (7) R R 13
Then null hypothesis R R is equivalent to H : ρ ρ by Theorem 6, where ρ is determined by Eq.(6). Hence, it suffices to show that sup P ρ (rejecting H ) = α. (8) ρ ρ We have P ρ (rejecting H ) = P ρ ( γ X ρ Ȳ F α(2m, 2nγ) ) = P ρ (γ X ρȳ ρ ρ F α(2m, 2nγ) ) = 1 F ( ρ ρ F α(2m, 2nγ); 2m, 2nγ ). Thus sup ρ ρ P ρ (rejecting H ) = sup ρ ρ {1 F ( ρ ρ F α(2m, 2nγ); 2m, 2nγ ) } =1 F ( F α (2m, 2nγ); 2m, 2nγ ) =1 (1 α) =α. This ends the proof. Theorem 9. Consider testing hypothesis H : R R vs. H a : R < R. The test that uses test statistic Q along with rejection region R.R. = {(X, Y) : Q F 1 α (2m, 2nγ)} 14
has size α. Proof. It is the same as Theorem 8. At the end of this section we derive a procedure for constructing confidence interval of R can be obtained as ( 1 F ( Ȳ X F α/2(2m, 2nγ); 2, 2γ ), 1 F ( Ȳ X F 1 α/2(2m, 2nγ); 2, 2γ )). Proof. According to Eq.(5) the reliability parameter R is given as It then follows from Theorem 5 that R = 1 F ( γ ρ ; 2, 2γ). 1 α =P ( F 1 α/2 (2m, 2nγ) < Q < F α/2 (2m, 2nγ) ) =P ( F 1 α/2 (2m, 2nγ) < 1 γ X ρ Ȳ < F α/2(2m, 2nγ) ) =P ( Ȳ X F 1 α/2(2m, 2nγ) < γ ρ < Ȳ X F α/2(2m, 2nγ) ) ( =P F ( Ȳ X F 1 α/2(2m, 2nγ); 2, 2γ ) < F ( γ ρ ; 2, 2γ) < F ( Ȳ X F α/2(2m, 2nγ); 2, 2γ )) ( =P 1 F ( Ȳ X F α/2(2m, 2nγ); 2, 2γ ) < 1 F ( γ ; 2, 2γ) < ρ 1 F ( Ȳ X F 1 α/2(2m, 2nγ); 2, 2γ )) ( =P 1 F ( Ȳ X F α/2(2m, 2nγ); 2, 2γ ) < R < 1 F ( Ȳ X F 1 α/2(2m, 2nγ); 2, 2γ )). and consequently the desired result. 15
4. Numerical study In this section we will conduct simulation studies in which the shape parameter of the gamma distribution is γ = 1.25 and γ = 4.5 in Example 1 and Example 2 respectively. Three cases of R =.3,.6 and.9 are considered. Using N = 1 replications, in Tables 1-3 we list the average of R, MSE( R) which is the estimates of MSE of R, and the counterparts of the MLE ˆR corresponding to various sample sizes m = n = 5, 6,, 4, for the case of γ = 1.25. Tables 4-6 show the results for the case of γ = 4.5. In these tables e ˆR, R = MSE( ˆR)/ MSE( R) is the relative efficiency. Example 1. In this example, shape parameter of gamma distribution γ = 1.25 is used. The scale parameters λ = 2 and τ = 1.235, 3.962, 22.742 so that the reliability parameter R = P (X > Y ) equals to.3,.6,.9 accordingly based on Theorem 6. From the table 1-3, it is obvious that the average of the 1 values of the UMVUE R is very close to the true value of R, which is certainly consistent with the unbiasedness property of R, the estimate MSE of R is only.2 even the sample sizes are only m=n=5 and it decreases to.25 as the sample sizes increase to m=n=4. As far as the MLE ˆR, similar pattern is also observed. In addition, the bias of ˆR is positive when R =.3,.6 and is negative when R =.9. Tables and Figures can be found in the Appendix. Figure 1 is the plot of the average of the observed N = 1 values of R and the average of the observed N = 1 values of ˆR versus the true value of R (, 1). 16
Because of the unbiasedness property of R, the average of R is almost a straight line from (, ) to (1, 1). The average of ˆR has only very slight deviation from the true R. For the case of m = n = 5 the solid line curve in Figure 2 shows the behavior of the difference between the average of the observed N = 1 values of R and the true value of R (, 1). The difference is of the order of 1 3. The dotted curve in Figure 2 shows the similar picture for the MLE ˆR with error of order of 1 2. And it is notable that the MLE ˆR overestimates when R is small and underestimates when R is large. For the case of γ = 1.25 Figure 3 shows the behavior of the Mean Squared Error of R for sample sizes m = n = 5, 1, 2, 3, 4. It is obvious that the MSE of R decrease while the sample sizes increase. And the decreasing speed is also decelerated with the increasing sample sizes. The behavior of the MSE of ˆR follow the same pattern as shown in Figure 4. Example 2. In this second example, similar computation is done but with shape parameter of gamma distribution γ = 4.5 and the scale parameter λ = 2 and τ = 6.52, 16.638, 84.425 according to R =.3,.6,.9 following Theorem 6. The estimates of R and MSE in the cases R =.3,.6 and.9 are listed in the Table 4-6. For m = n = 5 the average of the observed N = 1 values of R and the average of the observed N = 1 values of ˆR versus the true value of R (, 1) is given in Figure 5. The difference of R and the average of N = 1 simulated values are plotted in Figure 6. Figure 7 17
and Figure 8 show the behavior of the MSE of R and ˆR respectively. From the above two examples, it is noticed that even though ˆR is a biased estimator of R its MSE is about the same as that of R and actually is smaller than the MSE of R. Considering this and the fact that the computation of R based on Theorem 1, is more complicated than that of ˆR based on Theorem 4, the MLE ˆR is recommended for estimating the reliability parameter R. 18
REFERENCES Baklizi, A., 28a. Estimation of Pr(X < Y) using record values in the one and two parameter exponential distributions. Comm. Statist. - Theory Methods 37, 692-698. Baklizi, A., 28b. Likelihood and Bayesian estimation of Pr(X < Y) using lower record values from the generalized exponential distribution. Comput. Statist. Data Anal. 52, 3468-3473. Church, J. D., Harris, B., 197. The Estimation of Reliability from Stress- Strength Relationships. Technometrics 12, 49-54. Eryilmaz, S., 28a. A new perspective to stress-strength models. Ann. Inst. Statist. Math. DOI: 1.117/s1463-8-211-3. Eryilmaz, S., 28b. Consecutive k-out-of-n: G system in stress-strength setup. Comm. Statist. - Simul. Comput. 37, 579-589. Eryilmaz, S., 28c. Multivariate stress-strength reliability model and its evaluation for coherent structures. J.Multivar. Anal. 99, 1878-1887. Eryilmaz, S., 21. On system reliability in stress-strength setup. Statist. Probab. Lett. 8, 834-839. Hauck, W.W., Hyslop, T., Anderson, S., 2. Generalized treatment elects for clinical trials. Statist. Med. 19, 887-899. Krishnamoorthy, K., Mukherjee, S., Guo, H., 27. Inference on reliability in two-parameter exponential stress-strength model. Metrika 65, 261-273. Kundu, D., Raqab, M.Z., 29. Estimation of R = P(Y < X) for three- 19
parameter Weibull distribution. Statist.Probab.Lett. 79, 1839-1846. Kotz, S., Lumelsdii, Y., Pensky, M., 23. The Stress-Strength Model and its Generalization. World Scientific, Singapore. O Brien, P.C., 1988. Comparing two samples: Extensions of the t, and logrank tests. J. Amer. Statist. Assoc. 83, 52-61. Reiser, B., Guttman, I., 1986. Statistical Inference for Pr(Y < X): The Normal Case. Technometrics 28, 253-257. Rezaei, S., Tahmasbi, R., Mahmoodi, M., 21. Estimation of P(Y < X) for generalized Pareto distribution. J. Statist. Plann. Inference 14, 48-494. Reiser, B., 2. Measuring the effectiveness of diagnostic markers in the presence of measurement error through the use of ROC curves. Statist. Med. 19, 2115-2129. Saracoglu, B., Kaya, M.F., 27. Maximum likelihood estimation and confidence intervals of system reliability for Gomperts distribution in stress-strength models. Selcuk J.Appl.Math. 8, 25-36. Simonoff, J.S., Hochberg, Y., Reiser, B., 1986. Alternative estimation procedures for Pr(X < Y) in categorized data. Biometrics 42, 895-97. Schwarz, L., Wearden, S., 1959. A distribution-free asymptotic method of estimating, testing and setting confidence limits to heritability. Biometrics 15, 227-235. Wellek, S., 1993. Basing the analysis of comparative bioavailability trials on an individualized statistical definition of equivalence. Biometrical J. 35, 47-55. 2
APPENDIX 21
Table 1: γ = 1.25, N = 1, R =.3 n=m R UMVUE MSE( R) R =.3 ˆR MLE MSE( ˆR) e ˆR, R 5.299369975.21279.31376827.1829164.85969965 6.299523479.1776656.3878997.1568322.8832411 7.299292717.14429986.37318211.12968356.89878837 8.39584.12872568.36994261.11736233.911724289 9.299753452.11397359.35995936.149748.92138646 1.3468633.1144365.3696257.9427346.929317856 15.297289393.6532654.31154861.623776.94965628 2.2997745.4927652.3265196.474979.963888827 3.3264421.3291566.32169468.3214554.9766389 4.352611.2447211.31964775.244573.98257742 Table 2: γ = 1.25, N = 1, R =.6 n=m R UMVUE MSE( R) R =.6 ˆR MLE MSE( ˆR) e ˆR, R 5.599437387.2578832.584773681.2215898.85923578 6.598951392.21275287.58662364.1879494.88327535 7.59997666.17585521.5891764.15815761.899362678 8.667353.1525857.5911831.1396348.91141146 9.61236847.13549913.592677321.12478542.92931571 1.598439557.12276553.598967.11435631.93151813 15.618314.774377.59582417.7374462.952394235 2.61455459.587437.59748228.5594458.963326552 3.666345.3859331.597988242.376589.975788279 4.598942225.2933121.596957569.288643.983949489 22
Table 3: γ = 1.25, N = 1, R =.9 n=m R UMVUE MSE( R) R =.9 ˆR MLE MSE( ˆR) e ˆR, R 5.899956395.3996736.883237769.4917556 1.233931 6.89923759.3331532.88548325.4368 1.2759341 7.9913634.256114.88932333.36483 1.173942495 8.899956223.221919.889836889.2584197 1.164567713 9.9111754.1893262.891174761.217535 1.148829488 1.899542968.174741.89151685.1946139 1.14163929 15.9531192.172896.8953195.116784 1.8849318 2.926517.79621.896371635.85527 1.68218699 3.939881.53877.897815471.526223 1.44348352 4.9228.389978.89875443.43973 1.35885512 Table 4: γ = 4.5, N = 1, R =.3 n=m R UMVUE MSE( R) R =.3 ˆR MLE MSE( ˆR) e ˆR, R 5.296872619.2548893.28773599.1989677.78363229 6.3113926.211993.29119653.178319.8127163 7.3667283.18255329.292798374.1525513.835651899 8.298696893.1643637.29187129.1372433.85543762 9.299821243.13995597.29354456.12166139.869283352 1.29977949.12428744.29399368.1948896.88933413 15.299262754.8531467.2952518.784245.918979678 2.29988526.6128631.296777791.574882.93826761 3.29936474.4188645.297256181.41488.95851592 4.337335.31196.298688676.3287.96872876 23
Table 5: γ = 4.5, N = 1, R =.6 n=m R UMVUE MSE( R) R =.6 ˆR MLE MSE( ˆR) e ˆR, R 5.599389361.24387944.564472844.23398279.95941991 6.599688353.1963258.57183433.1912941.974485892 7.657872.16886692.575131398.16559874.98646452 8.599963775.14397779.577551599.14264567.99747739 9.59999175.12655836.579975345.12571633.993346729 1.643646.1122382.582347269.11178991.99671413 15.6478265.7443647.58832793.744973 1.817231 2.68632.5312875.59917543.533472 1.411174 3.667212.353196.594446145.3536547 1.179957 4.6846263.265547.596235883.26881 1.972441 Table 6: γ = 4.5, N = 1, R =.9 n=m R UMVUE MSE( R) R =.9 ˆR MLE MSE( ˆR) e ˆR, R 5.89939142.3219176.879422528.451926 1.398471344 6.89963717.2563739.882758182.3463913 1.351117541 7.924744.1992949.88635666.259682 1.299923891 8.9258673.169681.888467592.2135483 1.263953 9.899443294.1464135.88899514.1825548 1.246843941 1.89967785.1296352.89367763.158357 1.2198431 15.954698.77328.894525921.87951 1.13773476 2.99277.575634.89561483.63767 1.1776979 3.936277.376961.89785859.43962 1.71627494 4.96563.286674.897866175.31752 1.52595974 24
Figure 1: γ = 1.25, m = n = 5, N = 1 Figure 2: γ = 1.25, m = n = 5, N = 1 25
Figure 3: γ = 1.25, N = 1, UMVUE Figure 4: γ = 1.25, N = 1, MLE 26
Figure 5: γ = 1.25, N = 1 Figure 6: γ = 1.25, N = 1 27
Figure 7: γ = 4.5, m = n = 5, N = 1 Figure 8: γ = 4.5, m = n = 5, N = 1 28
Figure 9: γ = 4.5, N = 1, UMVUE Figure 1: γ = 4.5, N = 1, MLE 29
Figure 11: γ = 4.5, N = 1 Figure 12: γ = 4.5, N = 1 3