Optimal Unbiased Estimates of P {X < Y } for Some Families of Distributions

Transcription

1 Metodološki zvezki, Vol., No., 04, -9 Optimal Unbiased Estimates of P {X < } for Some Families of Distributions Marko Obradović, Milan Jovanović, Bojana Milošević Abstract In reliability theory, one of the main problems is estimating parameter R P {X < }. In this paper we shall present UMVUEs for R in different cases i.e. for different distributions of X and. Some of them are already existing and some are original. Introduction In reliability theory the main parameter is the reliability of a system, and its estimation is one of the main goals. The system fails if the applied stress X is greater than strength, so R is a measure of system performance. In most cases this parameter is given as R P {X < }, although for some discrete cases the expression R P {X } is also considered. The problem was first introduced by Birnbaum 956. Since then numerous papers have been published. Most of results are presented in Kotz et al., 003. The vast majority of papers presuppose independence of stress and strength variables, as well as that they come from the same family of, in most cases continuous, distributions. There exists a wide range of applications in engineering, military, medicine and psychology. The unbiasedness of an estimator is a desired property especially when dealing with relatively small sample sizes, where we cannot count on asymptotic unbiasedness. Since in many cases most popular estimators are biased, it is often important to find the unique minimum variance unbiased estimator UMVUE.. UMVUE of R Let X X,..., X n and,..., n be the samples from the distributions of random variables X and. Then, using the following theorem we can construct UMVUEs. Theorem If V X, is any unbiased estimator of parameter θ and T is a complete sufficient statistic for θ, then EV X, T is the UMVUE of θ. Faculty of Mathematics, University of Belgrade, Studenski trg 6, Belgrade, Serbia; addresses: marcone@matf.bg.ac.rs, mjovanovic@matf.bg.ac.rs, bojana@matf.bg.ac.rs

2 Marko Obradović, Milan Jovanović and Bojana Milošević This theorem is the combination of Rao-Blackwell and Lehmann-Scheffé theorems. Their proofs could be found in Hogg et al., 005. However, in continuous case, the use of this theorem might not be technically convenient. Therefore, the following theorems were proposed to help deriving the UMVUEs Kotz et al., 003. Theorem Let θ 0 Θ be an arbitrary value of θ and let T be a complete sufficient statistic for θ. Denote by g θ0 t and g θ0 t X x,..., X k x k, y,..., k y k the pdf of T and the conditional pdf of T for given X j x j, j y j, j,..., k, respectively. Then the UMV UE of joint pdf f θ x,..., x k, y,..., y k is of the form fx,..., x k, y,..., y k k f θ0 x j, y j g θ 0 t X x,..., X k x k, y,..., k y k. g θ0 t Theorem 3 The UMVUE of R is Ix < y fx, ydxdy, where f is given in theorem for k. Existing results In this section we present a brief summary of existing results obtained for UMVUEs of R for some distributions. Exponential distribution Let X and be independent exponentially distributed random variables with densities f X x; e x, x 0, f y; β βe βy, y 0, where and β are unknown positive parameters. The complete sufficient statistics for and β are n X j and T n j. The UMVUE of R was derived by Tong 974; 977, and it is given by n i0 n i0 i Γn Γn Γn i Γn +i+ i Γn Γn Γn +iγn i T i+, if i, if >.

3 Optimal Unbiased Estimates of P {X < } 3 Normal distribution Let X and be normally distributed independent random variables with densities f X x; µ, σ e x µ σ, x R, πσ f y; µ, σ e y µ σ, y R, πσ where µ, σ, µ and σ are unknown parameters. The complete sufficient statistics for µ, σ, µ, σ are X, S X, Ȳ, S. The UMVUE of R was derived by Downtown 973 and it is given by [ B, n B, n ] Ω u n 4 v n 4 dudv, where Ba, b is the beta function and { Ω u, v [, ] [, ] u S Xn +v S } n +Ȳ X > 0. n n Gamma distribution Let X and be independent gamma distributed random variables with densities f X x;, σ x Γ σ e x σ, x 0, f y;, σ y Γ σ e y σ, y 0, where and are known integer values and σ and σ are unknown positive parameters. The complete sufficient statistics for σ and σ are n X j and T n j. The UMVUE of R was derived by Constantine et al. 986, and it is given by n n n k n k B + +k,n B,n B,n k +k +k, B + +k,n B,n B,n k +k if T +k, if >.

4 4 Marko Obradović, Milan Jovanović and Bojana Milošević Gompertz distribution Let X and be independent Gompertz distributed random variables with densities f X x; c, λ λ e cx e λ ecx c, x > 0, f y; c, λ λ e cy e λ ecy c, y > 0, where c is a known positive value and λ and λ are unknown positive parameters. The complete sufficient statisitcs for λ and λ are W X c n e cx j, W c n e c j. The UMVUE of R was derived by Saracoglu et al. 009 and it is given by n k Γn Γn n k Γn Γn Γn kγn +k Generalized Pareto distribution Γn +kγn k WX W k, if WX < W W W X k, if WX W. Let X and be independent random variables from generalized Pareto distribution with densities f X x;, λ λ + λx +, x > 0, f y;, λ λ + λy +, y > 0, where λ is known positive value and and are unknown positive parameters. The complete sufficient statistics for parameters and are n ln + X j and T n ln + j. The UMVUE of R was derived by Rezaei et al. 00, and it is given by Poisson distribution n k Γn Γn n k Γn Γn Γn kγn +k Γn +kγn k TX k, if TX < T k, if TX T. Let X and be independent Poisson distributed random variables with mass functions P {X x; λ } e λ λ x, x 0,,..., x! P { y; λ } e λ λ y, y 0,,..., y! where λ and λ are unknown positive parameters. The complete sufficient statistics for λ and λ are n X j and T n j.

5 Optimal Unbiased Estimates of P {X < } 5 The UMVUE of R was derived by Belyaev and Lumelskii 988 and it is given by min{,t } TX n R x x n T y. x y x0 Negative binomial distribution n Let X and be indepent random variables from negative binomial distributions with mass functions m + x P {X x; m, p } p x x p m, x 0,,..., m + y P { y; m, p } p y y p m, y 0,,..., where m and m are known integer values and p and p are unknown probabilities. The complete sufficient statistics for p and p are n X j and T n j. The UMVUE of R was derived by Ivshin and Lumelskii 995 and it is given by y0 n T min{,t } x0 3 New results T yx+ m +x TX x+m n x T x x m n + m +y y+m n y T y y m n +T. T In this section we shall derive the UMVUE of R for some new distributions. The first model is where stress and strength both have Weibull distribution with known but different shape parameter and unknown rate parameters. As a special case we present the model where stress has exponential and strength has Rayleigh distribution. An example with real data for Weibull model is presented. In the second model, both stress and strength have logarithmic distribution with unknown parameters. 3. Weibull model Let X and be independent random variables from Weibull distribution with densities f X x;, σ σ x e σ x, x 0, f y;, σ σ y e σ y, y 0. The Weibull distribution is one of the most used distribution in modeling life data. Many researchers have studied the reliability of Weibull model. Most of them did not consider unbiased estimators e.g. Kundu and Gupta, 006, and recently the case with common known shape parameter has been studied in Amiri et al., 03. We consider the case where shape parameters and are known positive values, while rate parameters σ and σ are unknown positive parameters.

6 6 Marko Obradović, Milan Jovanović and Bojana Milošević The complete sufficient statistics for parameters σ and σ are n T n X j j. Since X and have exponential distributions with rate parameters σ and σ, both statistics have gamma distribution, i.e. has Γn, σ Γn, σ. Similarly, for k minn, n, k T k X,..., X k, and and T has X j has Γn k, σ and j has Γn k, σ. Using this and transformation of random variables n jk+ X j to X,..., X k, we get, for σ, gt X X x,..., X k x k t X k x j n k e t X Γn k k x j I{tX k x j }. Using theorem, we get that fx,..., x k k t k X k x j n k Γn x j t n X Γn I{t X k k x j }. For k we obtain that fx n x t X x n I{t t X n X x }. Analogously we get that fy n y t y n I{t t n y }. Denote M min{t X, t }. Using the independence of samples and the theorem 3, we obtain 0 M 0 M 0 0 I{x < y} fx fydxdy n n x t X x n dx t n X tn t n x t X x n t x n dx. t n X tn x y t y n dy

7 Optimal Unbiased Estimates of P {X < } 7 Now applying the binomial formula we obtain that the UMVUE of R is n n r+s n n n s T X r++ s r s T s, if T r0 s0 X T n n r+s n n n r+ 3. T r++, if T s r s X > T. r0 s0 3.. Exponential-Rayleigh model T r+ X As a special case of Weibull model we have a model where X has exponential and has Rayleigh distribution with densities f X x; e x, x 0, f y; β β ye β y, y 0, where and β are unknown positive parameters. The complete sufficient statistics for and β are n X j and T n j. The UMVUE of R is n r0 s0 n r0 n n s0 3.. Numerical example r+s n r++s r+s n r++s n r n r n s n s T X s, if T r+, if > T. Here we present an example with real data. We wanted to compare daily wind speeds in Rotterdam and Eindhoven. We obtained two samples of 30 randomly chosen daily wind speeds in 0. m/s from the period of April st 00 to April st 04 taken from the website of Royal Netherlands Meteorological Institute. The first sample is from Rotterdam and the second one is from Eindhoven: Rotterdam X: 48, 5, 7, 8, 40, 6, 84, 9, 35, 3, 55, 9, 45, 5, 47, 66, 38, 3, 39, 8, 50, 36, 5, 74, 53, 85, 8, 58, 8, 48. Eindhoven : 44, 5, 43, 35, 0, 59, 5, 38, 6, 5, 37, 6, 35, 7, 34, 7, 40, 37, 33, 7, 5, 50, 33, 5, 5,, 34, 39, 3, 60. It is well known that wind speed follows Weibull distribution. To check this we used Kolmogorov-Smirnov test. Since this test requires that the parameters may not be estimated from the testing sample, we estimated them beforehand using some other larger samples from the same populations. We got that X follows Weibull distribution with shape parameter.8 and rate parameter σ /47 Kolmogorov-Smirnov test statistics is 0.57 and the p-value is greater than 0., while follows Weibull distribution with shape parameter.6 and rate parameter σ /4 Kolmogorov-Smirnov test statistics is 0.58 and the p-value is greater than 0.. Finally, using 3. we estimated the probability that the daily wind speed is lower in Rotterdam than in Eindhoven to be r 0.3.

8 8 Marko Obradović, Milan Jovanović and Bojana Milošević 3. Logarithmic distribution Let X and be independent random variables from logarithmic distribution with mass functions p x P {X x; p}, x,,... ln p x P { y; q} ln q q y, y,,..., y where p and q are unknown probabilities. The logarithmic distribution has application in biology and ecology. It is often used for modeling data linked to the number of species. The complete sufficient statistics for p and q are n X j and T n j. The sum of n independent random variables with logarithmic distributions with the same parameter p has Stirling distribution of the first kind SDF Kn, p Johnson et al., 005, so has SDF Kn, p and T has SDF Kn, q with the following mass functions P { x; n, p} P {T y; n, q} where sx, n is Stirling number of the first kind. An unbiased estimator for R is I{X < }. Since n! sx, n p x x! ln p n, x n, n +,..., n! sy, n q y y! ln q n, y n, n +,..., EI{X < } t X, T t P {X <, t X, T t } P { t X, T t } M t n + P {X x}p { y}p { n X k t X x}p { n l t y} x M x yx+ t n + yx+ k P { t X }P {T t } t X!t! st X x, n st y, n n n t X x!t y!xy st X, n st, n, where M min{t X n +, t n }, using theorem we get that the UMVUE of R is min{ n +,T n } x 4 Conclusion T n + yx+ l!t! s x, n st y, n n n x!t y!xy s, n st, n. In this paper we considered the unbiased estimation of the probability P {X < } when X and are two independent random variables. Some known results of UMVUEs for R for some distributions were listed. Two new cases were presented, namely Weibull model with known but different shape parameters and unknown rate parameters and Logarithmic model with unknown parameters. An example using real data was provided.

9 Optimal Unbiased Estimates of P {X < } 9 References [] Amiri, N., Azimi, R., aghmaei, F. and Babanezhad, M. 03: Estimation of Stress-Strength Parameter for Two-Parameter Weibull Distribution. International Journal of Advanced Statistics and Probability,, 4-8. [] Belyaev,. and Lumelskii,. 988: Multidimensional Poisson Walks. Journal of Mathematical Sciences, 40, [3] Birnbaum, Z.W. 956: On a Use of Mann-Whitney Statistics. Proc. Third Berkeley Symp. in Math. Statist. Probab.,, 3-7. Berkeley, CA: University of California Press. [4] Constantine, K., Carson, M. and Tse, S-K. 986: Estimators of P < X in the gamma case. Communications in Statistics - Simulations and Computations, 5, [5] Downtown, F. 973: On the Estimation of P r < X in the Normal Case. Technometrics, 5, [6] Hogg, R.V., McKean, J.W. and Craig, A.T. 005: Introduction to Mathematical Statistics, Sixth Edition New Jersey: Pearson Prentice Hall. [7] Ivshin, V.V. and Lumelskii, a.p. 995: Statistical Estimation Problems in Stress- Strength Models. Perm, Russia: Perm University Press. [8] Johnson, N.L., Kemp, A.W. and Kotz, S. 005: Univariate Discrete Distributions. New Jersey: John Wiley & Sons. [9] Kotz, S., Lumelskii,. and Pensky, M. 003: The Stress-Strength Model and its Generalizations. Singapore: World Scientific Press. [0] Kundu, D. and Gupta, R.D. 006: Estimation of P [ < X] for Weibull Distributions. IEEE Transactions on Reliability, 55. [] Rezaei, S., Tahmasbi, R. and Mahmoodi, M. 00: Estimation of P [X < ] for Generalized Pareto Distribution. Journal of Statistical Planning and Inference, 40, [] Saracoglu, B., Kaya, M.F. and Abd-Elfattah, A.M. 009: Comparison of Estimators for Stress-Strength Reliability in the Gompertz Case. Hacettepe Journal of Mathematics and Statistics, 383, [3] Tong, H. 974: A Note on the Estimation of P < X in the Exponential Case. Technometrics, 6, 65. Errata: Technometrics, 7, 395. [4] Tong, H. 977: On the Estimation of P < X for Exponential Families. IEEE Transactions on Reliability, 6, [5] Website of Royal Netherlands Meteorological Institute, downloaded from on April th 04.