Statistical inference of P (X < Y ) for the Burr Type XII distribution based on records

Transcription

1 Statistical inference of P X < Y for the Burr Type XII distribution based on records Fatih Kızılaslan and Mustafa Nadar Abstract In this paper, the maximum likelihood and Bayesian approaches have been used to obtain the estimates of the stress-strength reliability = P X < Y based on upper record values for the two-parameter Burr Type XII distribution. A necessary and sufficient condition is studied for the existence and uniqueness of the maximum likelihood estimates of the parameters. When the first shape parameter of X and Y is common and unknown, the maximum likelihood ML estimate and asymptotic confidence interval of are obtained. In this case, the Bayes estimate of has been developed by using Lindley s approximation and the Markov Chain Monte Carlo MCMC method due to lack of explicit forms under the squared error SE and linear-exponential LINEX loss functions for informative prior. The MCMC method has been also used to construct the highest posterior density HPD credible interval. When the first shape parameter of X and Y is common and known, the ML, uniformly minimum variance unbiased UMVU and Bayes estimates, Bayesian and HPD credible as well as exact and approximate intervals of are obtained. The comparison of the derived estimates is carried out by using Monte Carlo simulations. Two real life data sets are analysed for the illustration purposes. Keywords: Burr Type XII distribution, Stres-strength model, ecord values, Bayes estimation. AMS Classification: 6N5, 6F1, 6F Introduction Let X 1, X,... be a sequence of continuous random variables. X k is an upper record value if its value is greater than all preceding values X 1, X,..., X k 1. By definition, X 1 is an upper record value. An analogous definition can be provided for lower record values. The theory of record values was first introduced by Chandler [17] and it has been extensively studied in the literature since then. More details Department of Mathematics, Gebze Technical University, Kocaeli, Turkey, kizilaslan@gtu.edu.tr, fkizilaslan@yahoo.com Corresponding Author. This study is a part of the first author s Doctor of Philosophy Thesis entitled Statistical inference for some distributions based on record values submitted to the Graduate School of Natural and Applied Science, Gebze Technical University, 15. Department of Mathematical Engineering, Istanbul Technical University, Istanbul, Turkey, nadar@itu.edu.tr

2 and references may be found in Ahsanullah [], Arnold et al. [5] and Nevzorov [38]. ecord values and the associated statistics are of interest in many real life applications, such as weather, sports, economics, life-tests and so on. For example, in the manufacturing industry, it might be interesting to a researcher to determine the minimum failure stress of the products sequentially, while the amount of the rainfall that is grater smaller than the previous once is of importance to climatologists and hydrologists. In some experiments, an observation is stored only if it is an upper lower record value because the measurement saving can be important especially when the sample size is very big, costly or all some portion of the data is destroyed. For specific examples, see Gulati and Padgett [3]. In the reliability context, the stress-strength model can be described as an assessment of reliability of a system in terms of random variables X representing stress experienced by the system and Y representing the strength of the system available to overcome the stress. If the stress exceeds the strength, then the system will fail. Thus = P X < Y is a reliability of a system. The main idea was introduced by Birnbaum [13] and developed by Birnbaum and McCarty [14]. A comprehensive account of this topic is presented by Kotz et al. [4]. It provides an excellent review of the development of the stress-strength up to the year 3. In the literature, many papers are available for an estimate of the reliability based on a random sample or record values. When the X and Y are independent and follow the Burr Type III, X and XII, generalized exponential, Weibull, Gompertz, Kumaraswamy and Levy distributions, the estimation of based on a random sample were studied by Mokhlis [31], Ahmad et al. [1], Awad and Gharraf [9], Kundu Gupta [5, 6], Saraçoğlu et al. [41], Nadar et al. [3] and Najarzadegan et al. [36], respectively. When the X and Y are independent and follow one and two parameters generalized exponential, Weibull, exponentiated gumbel, Kumaraswamy, one and two parameters exponential and Burr Type X distributions, the classical and Bayesian estimates of based on records were considered by Baklizi [1], Asgharzadeh et al. [7], Baklizi [11], Tavirdizade [43], Nadar and Kızılaslan [33], Baklizi [1] and Tavirdizade and Garehchobogh [44], respectively. The Burr Type XII distribution was introduced by Burr [16]. If a random variable X follows a Burr Type XII distribution, denoted by X Burrα, β, then the cumulative distribution function cdf and the probability density function pdf are given by, respectively, 1.1 F x; α, β = x α β, x >, α >, β >, 1. fx; α, β = αβx α x α β+1, x >. Here α > and β > are the two shape parameters. This distribution has been studied by the several authors; see, for example, Al-Hussaini and Jaheen [3, 4], Ghitany and Al-Awadhi [1], Nadar and Papadopoulos [35], Nadar and Kızılaslan [34] and ao et al. [4]. The main purpose of this paper is to improve the inference procedures for the stress-strength reliability based on upper record values while the measurements follow the two-parameter Burr Type XII distribution when the first shape parameters are common. When the first shape parameter α is unknown, the ML and Bayes estimates, as well as asymptotic confidence and HPD credible intervals are

3 3 derived. When α is known, different estimates, namely ML, UMVU, Bayes and empirical Bayes estimates, are obtained. The Bayes estimates of under the SE and LINEX loss functions are derived in closed forms for informative and non informative prior cases. It is also obtained by using Lindley s approximation and MCMC method. The exact and other Bayes estimates are compared in terms of estimated risk E by the Monte Carlo simulations. Also, the exact and asymptotic confidence intervals, as well as Bayesian, empirical Bayesian and HPD credible intervals are constructed for. The rest of the paper is organized as follows. In Section, a necessary and sufficient condition for the existence and uniqueness of the ML estimates of the parameters is established when α is unknown. The ML and Bayesian estimates as well as the asymptotic confidence and HPD credible intervals of are obtained. In Section 3, the ML and UMVU estimates, as well as exact and asymptotic confidence intervals are obtained for when α is known. The Bayes estimates are derived analytically and also obtained by using Lindley s approximation and MCMC method for informative and non informative prior cases. Moreover, Bayesian, empirical Bayesian and HPD credible intervals of are constructed. In Section 4, the different proposed methods have been compared by using Monte Carlo simulations and the findings are illustrated by tables and plots. Furthermore, two real data sets analysis are presented. Finally, we conclude the paper in Section 5.. Estimation of when the first shape parameter α is common In this section, we investigate the properties of = P X < Y, when the first shape parameter α is common for the distributions of X and Y. The ML estimates, its existence and uniqueness, asymptotic confidence intervals, as well as Bayes estimates and HPD credible interval for are obtained..1. MLE of. Let X Burrα, β 1 and Y Burrα, β are independent random variables. Then, the reliability = P X < Y is.1 = P X < Y = = β 1 β 1 + β. f Y yp X < Y Y = ydy The estimate of are considered based on upper record data on both variables. Let 1,..., n be a set of upper records from Burrα, β 1 and S 1,..., S m be a set of upper records from Burrα, β independently from the first sample. The likelihood functions based on records are given by, see Arnold et al. [5], n 1 fr i ; α, β 1 L 1 β 1, α r = fr n ; α, β 1 1 F r i ; α, β 1, < r 1 <... < r n <, m 1 gs j ; α, β L β, α s = gs m ; α, β 1 Gs j ; α, β, < s 1 <... < s m <, j=1 where r = r 1,..., r n, s = s 1,..., s m, f and F are the pdf and cdf of X follows Burrα, β 1, respectively and g and G are the pdf and cdf of Y follows Burrα, β,

4 4 respectively. Then, the joint likelihood function of β 1, β, α given r, s is given by. Lβ 1, β, α r, s = h 1 r; αh s; αα n+m β n 1 β m e β1t1rn;α e βtsm;α, where.3 h 1 r; α = n r α 1 i 1 + r α i, h s; α = m s α 1 j 1 + s α, j j=1.4 T 1 r n ; α = ln1 + r α n, T s m ; α = ln1 + s α m. The joint log-likelihood function is.5 lβ 1, β, α r, s = ln h 1 r; α + ln h s; α + n + m ln α + n ln β 1 +m ln β β 1 T 1 r n ; α β T s m ; α. The ML estimates of β 1, β and α are given by.6 β1 =.7 β = n T 1 r n ; α, m T s m ; α, and α is the solution of the following non-linear equation n + m n ln r i m ln s j n r α + α 1 + r α + i 1 + s α n ln r n j=1 j ln1 + rn α 1 + rn a m s α m ln s m ln1 + s α m 1 + s α =. m Therefore, α can be obtained as a solution of the non-linear equation of the form hα = α where n ha = n + m ln r i m ln s j n r α 1 + r α + i 1 + s α n ln r n.8 j=1 j ln1 + rn α 1 + rn a ] m s α 1 m ln s m ln1 + s α m 1 + s α. m Since, α is a fixed point solution of the non-linear Equation.8, its value can be obtained using an iterative scheme as: hα j = α j+1, where α j is the j th iterate of α. The iteration procedure should be stopped when α j+1 α j is sufficiently small. After α is obtained, β 1 and β can be obtained from.6 and.7, respectively. Therefore, the MLE of, say, is given as.9 = β1 β 1 + β.

5 5.. Existence and uniqueness of the ML estimates. We establish the existence and uniqueness of the ML estimates of the parameters β 1, β and α. We present the following lemma that will be used in proof of. Theorem..1. Lemma. Let [ wx = [ln1 + x] ln1 + x + ξ x x ] 1, where ξx = x lnx/1 + x. Then wx for x. Proof. For a proof, one may refer to Ghitany and Al-Awadhi [1]... Theorem. The ML estimates of the parameters β 1, β and α are unique, with β 1 = n/t 1 r n ; α, β = m/t s m ; α where α is the solution of the non-linear equation Gα n + m n ln r i m ln s j + α 1 + r α + i 1 + s α j=1 j m s α m ln s m ln1 + s α m 1 + s α =, m n r α n ln r n ln1 + rn α 1 + rn a if at least one of the r i, i = 1,..., n or s j, j = 1,..., m is less than unity. Proof. We have G lim α Gα = lim α n + m α + 1 n ln r i + 1 m j=1 ln s j n ln r n ln and Let m ln s m ln =. G 1 α; r = n α + n j=1 ln r i 1 + r α i n r α n ln r n ln1 + rn α 1 + rn a, G α; s = m m α + ln s j m s α 1 + s α m ln s m j ln1 + s α m 1 + s a. m Then, Gα = G 1 α; r + G α; s. Firstly, we consider the limit of G 1 α; r as α. i If r n is less than unity, that is r i < 1, i = 1,..., n, then G 1 ; r lim α G 1α; r = lim α = n ln r i ln r n <. n α + n ln r i 1 + r α i n ln r n/1 + r α n ln1 + r α n/r α n

6 6 ii If only r n is greater than or equal to unity, that is r n 1 and r i < 1, i = 1,..., n 1, then n 1 n G 1 ; r = lim α α + ln r i 1 + ri α = n 1 ln r i <. + ln r n 1 + rn α n rα n ln r n /1 + rn α ln1 + rn α iii If r n and some r i record values are greater than unity and some r i record values are less than unity, that is r n > 1 and r i > 1, i = p,..., t, 1 < p t < n, then G 1 ; r = lim n n α α + ln r i n ln r i 1 + ri α ri α n rα n ln r n /1 + rn α ln1 + rn α = n r i<1 r i<1 ln r i <. r i>1 When the conditions given in i-iii holds for s j, j = 1,..., m, G α; s < as α. So that, the limit of Gα = G 1 α; r + G α; s < as α when r i, i = 1,..., n and s j, j = 1,..., m satisfy any of the conditions given in i-iii. Next, we need to show the limit of Gα < as α for s j > 1, j = 1,..., m and when the conditions given i-iii holds for r i, i = 1,..., n or r i > 1, i = 1,..., n and when the conditions given i-iii holds for s j, j = 1,..., m. In particular, when s j > 1, j = 1,..., m and the conditions given i holds for r i, i = 1,..., n, we can take α large enough, such that G α; s + and G 1 α; r + G α; s < as α. Other cases can be obtained similarly. Finally, we need to show that there is no solution if all records are greater than unity, that is r i > 1, i = 1,..., n and s j > 1, j = 1,..., m. If r i > 1, i = 1,..., n, then G 1 α; r < n [ α + n ln r 1 r α ] n n 1 + r1 α 1 + rn α + as α. Similarly, G α; s + as α. Therefore, Gα + as α. Except all records are greater than unity, we obtain that lim α Gα = and lim α Gα <. By the intermediate value theorem Gα has at least one root in,. If it can be shown that Gα is decreasing, then the proof will be completed. It is easily obtained that dg 1 α; r dα = 1 α [n + = 1 α [ n n ξ r α i r α i ξ r α i r α i + + nξ rn α 1 ln1 + rn α r α n n ln1 + r α n wrα n ]. ] 1 ln1 + rn α

7 7 Similarly, dg α; s dα = 1 m ξ s α j n α + ln1 + s α m wsα m. j=1 s α j It is clear that dg 1 α; r/dα < and dg α; s/dα < by using.1 Lemma. Therefore, dgα/dα <. Finally, we will show that the ML estimates of β 1, β, α maximizes the loglikelihood function lβ 1, β, α r, s. Let Hβ 1, β, α be the Hessian matrix of lβ 1, β, α r, s at β 1, β, α. It is clear that if deth for the critical point β 1, β, α and deth 1 <, deth > and deth 3 < at β 1, β, α then it is a local maximum of lβ 1, β, α r, s, where H 1 = l β1, H = l β1 l β β 1 l β 1 β l β, H 3 = H and l = lβ 1, β, α r, s. It can be easily seen that deth 1 β 1, β ln1 + r α, α = n <, n deth β 1, β ln1 + r α, α = n ln1 + s α m >, n m and deth β 1, β, α = G α ln1 + r α n ln1 + s α m <. α n m Hence, β 1, β, α is the local maximum of lβ 1, β, α r, s. Since there is no singular point of lβ 1, β, α r, s and it has a single critical point then, it is enough to show that the absolute maximum of the function is indeed the local maximum. Assume that there exist an α in the domain in which l α > l α, where l α = l β 1, β, α r, s. Since α is the local maximum there should be some point α 1 in the neighborhood of α such that l α > l α 1. Let kα = l α l α then k α >, kα 1 < and k α =. This implies that α 1 is a local minimum of the l α, but α is the only critical point so it is a contradiction. Therefore, β 1, β, α is the absolute maximum of lβ 1, β, α r, s..3. emark. In case all records are greater than one, we can still get a unique solution of the parameters when we divide the record values, say by r n or by s m or divide r i by r n and divide s j by s m as long as the transformed observations follow from Burr Type XII..3. Asymptotic distribution and confidence intervals for. The Fisher information matrix of I Iβ 1, β, α is given by E l E l β1 β 1 β E l β 1 α I = E l β β 1 E l E l β β α = I 11 I 1 I 13 I 1 I I 3, E l E l E l I 31 I 3 I 33 α β 1 α β α

8 8 where I 11 = n/β1, I = m/β, n ln n I 13 = E 1 + n α = βn 1 αγn ψ 1n, β 1, I 3 = βm αγm ψ x ln xln1 + x a 1 1m, β, ψ 1 a, b = 1 + x b+ dx, I 33 = n + m α + + βm+1 n β1ψ i i, β 1 m α + Γi ψ m, β α, ψ a, b = Γm j=1 β j ψ j, β α Γj + βn+1 1 ψ n, β 1 α Γn x ln x ln1 + x a x b+3 dx. By the asymptotic properties of the MLE, is asymptotically normal with mean and asymptotic variance σ = 3 3 j=1 β i β j I 1 ij, where β 3 α and I 1 ij is the i, jth element of the inverse of the Iβ 1, β, α, see ao [39]. Then,.1 σ = I11 1 β + I1 1 1 β 1 β + I 1 β, where β 1 = β β 1 + β, = β β 1 β 1 + β. Therefore, an asymptotic 11 γ% confidence interval of is.11 zγ/ σ, + z γ/ σ, where z γ is the upper γth quantile of the standard normal distribution and σ is the value of σ at the MLE of the parameters. If the likelihood equations have a unique solution θ n, then θ n is consistent, asymptotically normal and efficient see Lehmann and Casella [8]. When the likelihood equations have a unique solution, the observed information matrix J m β 1, β, α/m is a consistent estimator for I m β 1, β, α/m see Appendix C in Lawless [7]. The observed information matrix Jβ 1, β, α is given by l l l where Jβ 1, β, α = J 11 = n β 1 β1 l β β 1 l α β 1 β 1 β l β l α β β 1 α l β α l α, J 1 = J 1 = rα n ln r n 1 + rn a, J = m β = J 11 J 1 J 13 J 1 J J 3, J 31 J 3 J 33, J 3 = J 3 = sα m ln s m 1 + s α, m

9 9 J 33 = n + m α + ln sm n r α i ln ri 1 + r α i + m s α j j=1 ln s j 1 + s α j ln + β 1 rn α rn 1 + rn a +β s α m 1 + s α m. Therefore, an asymptotic 11 γ% confidence interval of can be obtained following from Equation.11 by replacing I with J in Equation Bayes estimation of. Bayesian approach has a number of advantages over the conventional frequentist approach. Bayes theorem is a consistent way to modify our beliefs about the parameters given the data that actually occurred see Bolstad [15]. In this subsection, we consider the Bayes estimates of the stressstrength reliability for Burr Type XII distribution under different loss functions. In the Bayesian inference, the most commonly used loss function is the squared error SE loss, Lθ, θ = θ θ, where θ is an estimate of θ. This loss function is symmetrical and gives equal weight to overestimation as well as underestimation. It is well known that the use of symmetric loss functions may be inappropriate in many circumstances, particularly when positive and negative errors have different consequences. A useful asymmetric loss function is the linear-exponential LINEX loss, Lθ, θ = e vθ θ vθ θ 1, v, introduced by Varian [46]. The sign and magnitude of v represents the direction and degree of asymmetry, respectively. For v close to zero, the LINEX loss is approximately equal to the SE loss and therefore almost symmetric. We assume that all parameters β 1, β and α are unknown and have independent gamma prior distributions with parameters a i, b i, i = 1,, 3, respectively. The density function of a gamma random variable X with parameters a, b is fx = ba Γa xa 1 e xb, x >, a, b >. Then, the joint posterior density function of β 1, β and α is π β 1, β, α r, s = Ir, sh 1 r; αh s; αα n+m+a3 1 β n+a1 1 1 β m+a 1.1 exp { αb 3 β 1 b 1 + T 1 r n ; α β b + T s m ; α}, where [Ir, s] 1 Γn + a 1 Γm + a = h 1 r; αh s; αα n+m+a3 1 e αb3 b 1 + T 1 r n ; α n+a1 b + T s m ; α m+a dα. Then, the Bayes estimate of a given measurable function of β 1, β and α, say uβ 1, β, α under the SE loss function is.13 û B = uβ 1, β, απβ 1, β, α r, sdβ 1 dβ dα. It is not possible to compute Equation.13 analytically. Two approaches can be applied to approximate Equation.13, namely, Lindley s approximation and MCMC method.

10 Lindley s approximation. Lindley proposed a method to approximate the ratio of two integrals such as Equation.13 in [3]. This procedure are also employed to the posterior expectation of the function Uλ, for given x, is uλe Qλ dλ Euλ x = e Qλ dλ, where Qλ = lλ + ρλ, lλ is the logarithm of the likelihood function and ρλ is the logarithm of the prior density of λ. Using Lindley s approximation, Euλ x is approximately estimated by Euλ x = u + 1 u ij + u i ρ j σ ij + 1 L ijk σ ij σ kl u l i j +terms of order n or smaller, where λ = λ 1, λ,..., λ m, i, j, k, l = 1,..., m, λ is the MLE of λ, u = uλ, u i = u/ λ i, u ij = u/ λ i λ j, L ijk = 3 l/ λ i λ j λ k, ρ j = ρ/ λ j, and σ ij = i, jth element in the inverse of the matrix { L ij } all evaluated at the MLE of the parameters. For the three parameter case λ = λ 1, λ, λ 3, Lindley s approximation leads to û B = Euλ x = u + u 1 c 1 + u c + u 3 c 3 + c 4 + c [Au 1σ 11 +u σ 1 + u 3 σ 13 + Bu 1 σ 1 + u σ + u 3 σ 3 + Cu 1 σ 31 + u σ 3 + u 3 σ 33 ], evaluated at λ = λ 1, λ, λ 3, where i j k l λ c i = ρ 1 σ i1 + ρ σ i + ρ 3 σ i3, i = 1,, 3, c 4 = u 1 σ 1 + u 13 σ 13 + u 3 σ 3, c 5 = 1 u 11σ 11 + u σ + u 33 σ 33, A = σ 11 L σ 1 L 11 + σ 13 L σ 3 L 31 + σ L 1 + σ 33 L 331, B = σ 11 L 11 + σ 1 L 1 + σ 13 L 13 + σ 3 L 3 + σ L + σ 33 L 33, C = σ 11 L σ 1 L 13 + σ 13 L σ 3 L 33 + σ L 3 + σ 33 L 333. In our case, λ 1, λ, λ 3 β 1, β, α and ρ 1 = a 1 1 b 1, ρ = a 1 b, ρ 3 = a 3 1 b 3, β 1 β α L 11 = n β1, L = m β, L 13 = L 31 = rα n ln r n 1 + r a n L 33 = n + m α β 1 r α n ln rn 1 + r a n n, L 3 = L 3 = sα m ln s m 1 + s α, m ln ri m 1 + ri α s α ln s j j 1 + s α j=1 j ln sm, r α i β s α m 1 + s α m

11 11 σ ij, i, j = 1,, 3 are obtained by using L ij, i, j = 1,, 3 and L 111 = β 3 1, L = m, β 3 ln L 133 = L 331 = rn α rn ln 1 + rn a, L 33 = L 3 = s α sm m 1 + s α, m n + m n 3 3 ln L 333 = α 3 ri α 1 ri α ri m 1 + r α s α j 1 s α ln s j j i 1 + s α j=1 j ln β 1 rn1 α rn α rn ln 1 + rn α β s α m1 s α sm m 1 + s α. m Moreover, A = σ 11 L σ 33 L 331, B = σ L + σ 33 L 33 and C = σ 13 L σ 3 L 33 + σ 33 L 333. To obtain the Bayes estimate of under the SE loss function, we take uβ 1, β, α = = β 1 /β 1 + β. Then, u 3 = u 13 = u 3 = u 33 =, and u 1 = β 1 = u 11 = β 1 β β 1 + β, u = = β = β β 1 + β 3, u = β 1 β = β 1 + β 3, c 4 = u 1 σ 1, c 5 = 1 u 11σ 11 + u σ. β 1 β 1 + β, u 1 = u 1 = β 1 β β 1 + β 3, Hence, the Bayes estimate of under the SE loss function is given as BS,Lindley = + [u 1 c 1 + u c + c 4 + c 5 ] {A [u 1σ 11 + u σ 1 ] + B [u 1 σ 1 + u σ ] + C [u 1 σ 31 + u σ 3 ]}. Notice that all parameters are evaluated at β 1, β, α. For the Bayes estimate of under the LINEX loss function, we take uβ 1, β, α = e v. Then, u 3 = u 13 = u 3 = u 33 =, u 1 = vβ e v β 1 + β, u 11 = ve v vβ + β 1 β + β β 1 + β 4, u = vβ 1e v β 1 + β, u = ve v vβ1 β 1 β β1 β 1 + β 4, u 1 = ve v vβ1 β β 1 + β 4 + β 1 β β 1 + β 3 and c 4 = u 1 σ 1, c 5 = 1 u 11σ 11 + u σ. Then, the Bayes estimate of under the LINEX loss function is given as.15 BL,Lindley = 1 v ln Ee v, where Ee v = e v + [u 1c 1 + u c + c 4 + c 5] {A [u 1σ 11 + u σ 1 ] + B [u 1σ 1 + u σ ] + C [u 1σ 31 + u σ 3 ]}.

12 1 Notice that all parameters are evaluated at β 1, β, α..4.. MCMC method. In the previous subsection, the Bayes estimate of are obtained by using Lindley s approximation under the SE and the LINEX loss functions. Since the exact probability distribution of is not known, it is difficult to evaluate Bayesian credible interval of. For this reason, we use the MCMC method to compute the Bayes estimate under the SE and the LINEX loss functions as well as the HPD credible interval. We consider the MCMC method to generate samples from the posterior distributions and then compute the Bayes estimate of under the SE and the LINEX loss functions. The joint posterior density of β 1, β and α is given by Equation.1. It is easy to see that the posterior density functions of β 1, β and α are β 1 α, r, s Gamman + a 1, b 1 + T 1 r n ; α, and.17 β α, r, s Gammam + a, b + T s m ; α, { n πα β 1, β, r, s α n+m+a3 1 exp αb 3 β 1 T 1 r n ; α ln1 + ri α n m m β T s m ; α + α ln r i + ln s j ln1 + s α j. j=1 j=1 Therefore, samples of β 1 and β can be generated by using the gamma distribution. However, the posterior distribution of α cannot be reduced analytically to well known distribution, therefore it is not possible to sample directly by standard methods. If the posterior density of α is unimodal and roughly symmetric then it is often convenient to approximate it by a normal distribution see Gelman et al. []. Since the posterior density of α is log-concave density so unimodal and it is roughly symmetric by experimentation, we use the Metropolis-Hasting algorithm with the normal proposal distribution to generate a random sample from the posterior density of α. The hybrid Metropolis-Hastings and Gibbs sampling algorithm, which will be used to solve our problem, is suggested by Tierney [45]. This algorithm combines the Metropolis-Hastings with Gibbs sampling scheme under the normal proposal distribution. Step 1. Start with initial guess α. Step. Set i = 1. Step 3. Generate β i 1 from Gamman + a 1, T 1 r n ; α i 1 + b 1. Step 4. Generate β i from Gammam + a, T s m ; α i 1 + b. Step 5. Generate α i from πα β 1, β, r, s using the Metropolis-Hastings algorithm with the proposal distribution qα Nα i 1, 1 : a Let v = α i 1. b Generate w from the proposal distribution q. πw β i c Let pv, w = min 1, 1, βi, r, s qv πv, r, s qw. β i 1, βi

13 13 d Generate u from Uniform, 1. If u pv, w then accept the proposal and set α i = w; otherwise, set α i = v. Step 6. Compute the i = β i 1 /βi 1 + β i. Step 7. Set i = i + 1. Step 8. epeat Steps -7, N times, and obtain the posterior sample i, i = 1,..., N. This sample are used to compute the Bayes estimate and to construct the HPD credible interval for. The Bayes estimate of under the SE and the LINEX loss function are given as BS,MCMC = 1 N M N M i=m+1 i, BL,MCMC = 1 v ln Ee v = 1 v ln 1 N M N M i=m+1 e vi where M is the burn-in period. The HPD 11 γ% credible interval of is obtained by using the method given in Chen and Shao [18]. From MCMC, the sequence 1,..., N, are obtained and ordered as 1 <... < N. The credible intervals are constructed as j, j+[n1 γ] for j = 1,..., N [N1 γ] where [x] denotes the largest integer less than or equal to x. Then, the HPD credible interval of is that interval which has the shortest length. 3. Estimation of when the first shape parameter α is known In this section, we consider the estimation of when α is known, say α = α. Let 1,..., n be a set of upper records from Burrα, β 1 and S 1,..., S m be a set of upper records from Burrα, β independently from the first sample MLE estimation and confidence intervals of. Based on the samples described above, the MLE of, say MLE, is 3.1 MLE = β 1 nt s m ; α β 1 + β = nt s m ; α + mt 1 r n ; α, where T 1 r n ; α = ln1 + rn α, T s m ; α = ln1 + s α m. It is easy to see that β 1 ln1 + rn α χ n and β ln1 + s α m χ m. Therefore, F 1 = MLE 1 MLE is an F distributed random variable with n, m degrees of freedom. The pdf of MLE is n 1 nβ1 1 r r f MLE r = r n 1 Bm, n mβ n+m, 1 + nβ11 r mβ r.

14 14 where < r < 1. The 11 γ% exact confidence interval for can be obtained as ,, 1 + F m,n; γ 1 MLE 1 + F m,n;1 γ 1 MLE MLE MLE where F m,n; γ and F m,n;1 γ are the lower and upper γ th percentile points of a F distribution with m, n degrees of freedom. On the other hand, the approximate confidence interval of can be easily obtained by using the Fisher information matrix. The Fisher information matrix of β 1, β is I = E E l β1 l β 1 β E E l β 1 β l β = n/β 1 m/β By the asymptotic properties of the MLE, MLE is asymptotically normal with mean and asymptotic variance σ = I 1 ij β i β j where I 1 ij j=1 is the i, j th element of the inverse of the I, see ao [39]. Then 3.3 σ = 1 1 n + 1 m Therefore, an asymptotic 11 γ% confidence interval for is 3.4 MLE z γ/ σ, MLE + z γ/ σ, where z γ is the upper γth percentile points of a standard normal distribution and σ is the value of σ at the MLE of the parameters. 3.. UMVUE of. In this subsection, we obtain the UMVUE of. When the first shape parameter α is known, T 1 r n ; α, T s m ; α is a sufficient statistics for β 1, β. It can be shown that it is also a complete sufficient statistic by using Theorem 1-9 in Arnold [6]. Let us define { 1 if 1 < S φ 1, S 1 = 1. if 1 S 1 Then E φ 1, S 1 = so it is an unbiased estimator of. Let P 1 = ln1 + α 1 and P = ln1 + S α 1. The UMVUE of, say U, can be obtained by using the ao-blackwell and the Lehmann-Scheffe s Theorems, see Arnold [6],. U = E φp 1, P T 1, T = φp 1, P fp 1, p T 1, T dp 1 dp P P 1 = φp 1, P f P1 T 1 p 1 T 1 f P T p T dp 1 dp, P 1 P

15 15 where T 1, T = T 1 r n ; α, T s m ; α, fp 1, p T 1, T is the conditional pdf of P 1, P given T 1, T. Using the joint pdf of 1, n and S 1, S m and after making a simple transformation, we obtain the f P1 T 1 p 1 T 1 and f P T p T, and are given by Therefore, 3.5 U = f P1 T 1 p 1 T 1 = n 1 t 1 p 1 n, < p 1 < t 1, t n 1 1 f P T p T = m 1 t p m, < p < t. = = t m 1 f P1 T 1 p 1 T 1 f P T p T dp 1 dp P 1<P t1 t { t p 1 n 1m 1 t1 p1n p t n 1 1 n 1m 1 t1 p1n t n 1 1 t p m t m 1 t p m t m 1 F 1 1, 1 m; n; t 1 /t if t t 1 1 F 1 1, 1 n; m; t /t 1 if t < t 1, dp dp 1 if t t 1 dp 1 dp if t < t 1 where F 1.,.;.;. is Gauss hypergeometric function, see formula in Gradshteyn and yzhik [] Bayes estimation of. In this subsection, we assume that β 1 and β are unknown and have independent gamma prior distributions with parameters a i, b i, i = 1,, respectively. Then, the joint posterior density function of β 1 and β is 3.6 π β 1, β α, r, s = λδ1 1 λδ Γδ 1 Γδ βδ1 1 1 β δ 1 e β1λ1 e βλ, where λ 1 = b 1 + T 1 r n ; α, λ = b + T s m ; α, δ 1 = n + a 1, δ = m + a. We can obtain the posterior pdf of using the joint posterior density function and is given by 3.7 f r = λδ1 1 λδ Bδ 1, δ r δ1 1 1 r δ 1, < r < 1. δ1+δ rλ rλ The Bayes estimate of, say BS, under the SE loss function is BS = 1 r f rdr. After making suitable transformations and simplifications by using formula in Gradshteyn and yzhik [], we get 3.8 BS = δ1 δ 1 λ 1 δ 1+δ λ δ δ 1 λ δ 1+δ λ 1 F 1 δ 1 + δ, δ 1 + 1; δ 1 + δ + 1; 1 λ1 λ if λ 1 < λ F 1 δ 1 + δ, δ ; δ 1 + δ + 1; 1 λ λ 1 if λ λ 1.

16 16 The Bayes estimator of under the LINEX loss function, say BL, is BL = 1 v ln E e v, where E. denotes posterior expectation with respect to the posterior density of. It can be easily obtained that Ee v = = 1 { e vr f rdr λ1 λ δ1 Φ 1 δ 1, δ 1 + δ, δ 1 + δ, 1 λ1 λ, v if λ 1 < λ, λ 1, v if λ λ 1 λ λ 1 δ e v Φ 1 δ, δ 1 + δ, δ 1 + δ, 1 λ where Φ 1.,.,.,.,. is confluent hypergeometric series of two variables, see formulas and in Gradshteyn and yzhik []. Therefore, [ ] 1 v c 1 + ln Φ 1 δ 1, δ 1 + δ, δ 1 + δ, 1 λ1 λ 3.9 BL =, v if λ 1 < λ [ ], 1 v c + ln Φ 1 δ, δ 1 + δ, δ 1 + δ, 1 λ λ 1, v if λ λ 1 where c 1 = δ 1 lnλ 1 /λ and c = δ lnλ /λ 1 v. If we use the Jeffrey s non informative prior, is given by det I, then the joint prior density function is πβ 1, β 1/β 1 β. Therefore, the joint posterior density function of β 1 and β is πβ 1, β α, r, s = and the posterior pdf of is given by T 1 n m T ΓnΓm βn 1 1 β m 1 e β1t1 e βt, f r = T 1 n m T r n 1 1 r m 1, < r < 1, Bn, m rt rt n+m where T 1 = T 1 r n ; α and T = T s m ; α. The Bayes estimate of under the SE and the LINEX loss function, say BS and BL respectively, are 3.1 BS = { and 3.11 BL = T1 n T n n+m F 1 n + m, n + 1; n + m + 1; 1 T1 T if T 1 < T T T 1 m n n+m, F 1 n + m, m; n + m + 1; 1 T T 1 if T T 1 1 v 1 v [ ] c 3 + ln Φ 1 n, n + m, n + m, 1 T1 T, v [ ] c 4 + ln Φ 1 m, n + m, n + m, 1 T T 1, v if T 1 < T, if T T 1 where c 3 = n lnt 1 /T and c 4 = m lnt /T 1 v The Bayes estimates are not always derived in the closed forms. However, for our case the Bayes estimates are obtained in the closed form. These estimates can be obtained by using alternative methods such as Lindley s approximation and the MCMC method. The purpose of applying all these two methods is to see how good the approximate methods compared with the exact one. If these result are close, then it will be encouraging to use the approximate methods when the exact form can not be obtained as in the case of α unknown. These estimators will be

17 17 compared in the simulation study section. Next, we give the Bayes estimates of using Lindley s approximation and the MCMC method Lindley s approximation. The approximate Bayes estimate of under the SE and the LINEX loss functions for the informative prior case, say BS,Lindley and BL,Lindley respectively, are 3.1 BS,Lindley = and n + a 1 1 1, m + a BL,Lindley = 1 v ln [ 1 + v 1 v n + a v 1 v v ] +, m + a 1 n+a1 1 m+a 1 b +T s m;α. where = β 1, β 1 = β 1+ β b and β 1+T 1r n;α = If we use the Jeffrey s non informative prior, the approximate Bayes estimate of under the SE and the LINEX loss functions, say BS,Lindley and BL,Lindley respectively, are 3.14 BS,Lindley = and n 1 1, m BL,Lindley = 1 v ln [ 1 + v 1 v n 1 + v 1 v v ] +, m 1 where = b 1 b1+ b, b 1 = n 1 T 1r n;α and b = m 1 T s m;α MCMC method. It is clear from Equation 3.6 that the marginal posterior densities of β 1 and β are gamma distribution with the parameters δ 1, λ 1 and δ, λ, respectively. We generate a samples by using Gibss sampling from these distributions. The following algorithm are used. Step 1. Set i = 1. Step. Generate β i 1 from Gammaδ 1, λ 1. Step 3. Generate β i from Gammaδ, λ. Step 4. Compute the i = β i 1 /βi 1 + β i. Step 5. Set i = i + 1. Step 6. epeat Steps -5, N times, and obtain the posterior sample i, i = 1,..., N. This sample is used to compute the Bayes estimate and to construct the HPD credible interval for. The Bayes estimate of under the SE and the LINEX

18 18 loss functions are given as BS,MCMC = 1 N i, N BL,MCMC = 1 v ln Ee v = 1 v ln 1 N N e vi. The HPD 11 γ% credible interval of can be obtained by the method of Chen and Shao [18]. Its algorithm is given in Subsection Empirical Bayes estimation of. We obtained the Bayes estimates of using three different ways. It is clear that these estimates depend on the prior parameters. However, the Bayes estimates can be also obtained independently of the prior parameters. These prior parameters could be estimated by means of an empirical Bayes procedure, see Lindley [9] and Awad and Gharraf [9]. Let 1,..., n and S 1,..., S m be two independent random samples from Burrα, β 1 and Burrα, β, respectively. For fixed r, the function L 1 β 1 α, r of β 1 can be considered as a gamma density with parameters n + 1, T 1 r n ; α. Therefore, it is proposed to estimate the prior parameters α 1 and β 1 from the samples as n + 1 and T 1 r n ; α, respectively. Similarly, α and β could be estimated from the samples as m + 1 and T s m ; α, respectively. Hence, the empirical Bayes estimate of with respect to SE and LINEX loss functions, say EBS and EBL, respectively, could be given as 3.16 EBS = { c6 c 7 F 1 n + m +, n + ; n + m + 3; c 9 if T 1 < T c 6 c 8 F 1 n + m +, m + 1; n + m + 3; c 1 if T T 1, and 3.17 EBL = 1 v 1 v n + 1 ln T1 T + ln c 11 m + 1 ln T T 1 v + ln c 1 if T 1 < T. if T T 1 where c 6 = n + 1/n + m +, c 7 = T 1 /T n+1, c 8 = T /T 1 m+1, c 9 = 1 T 1 /T, c 1 = 1 T /T 1, c 11 = Φ 1 n + 1, n + m +, n + m +, c 9, v and c 1 = Φ 1 m + 1, n + m +, n + m +, c 1, v Bayesian credible intervals for. We know that β 1 α, r Gammaδ 1, λ 1 and β α, s Gammaδ, λ. Then, λ 1 β 1 α, r χ n+a 1 and λ β α, s χ m + a. Therefore, W = λ β α, s /m + a λ 1 β 1 α, r /n + a 1 is an F distributed random variable with m +a, n +a 1 degrees of freedom and the 11 γ% Bayesian credible interval for can be obtained as , 1 + CF m+a,n+a 1; γ 1 + CF m+a,n+a 1;1 γ

19 where C = m+ab1+t1rn;α n+a, F 1b +T s m;α m+a,n+a 1; γ and F m+a,n+a 1;1 γ are the lower and upper γ th percentile points of a F distribution with m+a, n+ a 1 degrees of freedom. Moreover, this interval can be obtained independently of these parameters by using the empirical method given in Subsection 3.4. In this case, the posterior distributions of β 1 and β have gamma distributions with parameters n + 1, T 1 r n ; α and m+1, T s m ; α, respectively and the 11 γ% Bayesian credible interval for can be obtained as C 1 F 4m+,4n+; γ, 1 + C 1 F 4m+,4n+;1 γ where C 1 = 4m+T1rn;α 4n+T, F s m;α 4m+,4n+; γ and F 4m+,4n+;1 γ are the lower and upper γ th percentile points of a F distribution with 4m +, 4n + degrees of freedom. 4. Numerical experiments In this section, firstly the Monte Carlo simulations for the comparison of the derived estimates are presented, then two real life data sets are analysed Simulation study. In this subsection, we present some numerical results to compare the performance of the different estimates for different sample sizes and different priors. The performances of the point estimates are compared by using estimated risks Es. The performances of the confidence and credible intervals are compared by using average interval lengths and coverage probabilities cps. The E of θ, when θ is estimated by θ, is given by Eθ = 1 N θi θ i, N under the SE loss function. Moreover, the E of θ under the LINEX loss function is given by Eθ = 1 N e v θ i θ i v θi θ i 1, N where N is the number of replications. All of the computations are performed by using MATLAB 1a. All the results are based on 3 replications. We consider two cases separately to draw inference on, namely when the common first shape parameter α is unknown and known. In both cases we generate the upper record values with the sample sizes; n, m = 5, 5, 8, 8, 1, 1, 1, 1, 15, 15 from the Burr Type XII distribution and different values of n and m, given in Table 1, are considered. In Table 1, the ML and Bayes estimates of and their corresponding Es are listed when α is unknown. The Bayes estimates are computed by using Lindley s approximation and MCMC method under the SE and the LINEX v = 1 and 1 loss functions for different prior parameters. In the Bayesian case, Prior 1: a 1, b 1 = 4,, a, b = 4,, a 3, b 3 = 3, 3, Prior : a 1, b 1 = 5, 1, a, b = 3, 3/, a 3, b 3 = 3, 3/ and Prior 3: a 1, b 1 = 5, 1/, a, b = 19

20 3, 3, a 3, b 3 = 3, 3/, are used for =.56,.7145 and.995, respectively. Moreover, the 95% asymptotic confidence intervals, which are computed based on Fisher information and observation matrices, and HPD credible intervals with their cps are listed. From Table 1, the Es of all estimates decrease as the sample sizes increase in all cases, as expected. The Bayes estimates under the SE and LINEX loss functions generally have smaller E than that of ML estimates. Moreover, the Es of the Bayes estimates based on Lindley s approximation are generally smaller than that of MCMC method. These estimates are close to each other as the sample sizes increase. The average lengths of the intervals decrease as the sample sizes increase. The asymptotic confidence intervals based on Fisher information and observation matrices are very similar, as expected. The average lengths of the HPD Bayesian credible intervals are smaller than that of the asymptotic confidence intervals. In the MCMC case, we run three MCMC chains with fairly different initial values and generated 1 iterations for each chain. To diminish the effect of the starting distribution, we generally discard the first half of each sequence and focus on the second half. To provide relatively independent samples for improvement of prediction accuracy, we calculate the Bayesian MCMC estimates by the means of every 5 th sampled values after discarding the first half of the chains see Gelman et al. []. The scale reduction factor estimate = V arψ/w is used to monitor convergence of MCMC simulations where ψ is the estimand of interest, V arψ = n 1W/n + B/n with the iteration number n for each chain, the between- and within- sequence variances B and W see Gelman et al. []. In our case, the scale factor value of the MCMC estimates are found below 1.1 which is an acceptable value for their convergency. In Table and 3, the ML, UMVU and Bayesian estimates of and their corresponding Es are listed when α is known α = 3. In this case, the Bayes estimates are evaluated analytically under the SE and the LINEX v = 1 and 1 loss functions for different prior parameters. Moreover, it is also computed by using Lindley s approximation and MCMC method. In the Bayesian case, Prior 4: a 1, b 1 = 6, 5/, a, b = 4,, Prior 5: a 1, b 1 = 1,, a, b = 3, 3/ and Prior 6: a 1, b 1 = 15, 5/4, a, b =, are used for =.5484,.756 and.9165, respectively. In addition, the empirical Bayes estimates are obtained. All point estimates of are listed in Table. The exact and asymptotic confidence intervals are computed from Equations 3. and 3.4. The Bayesian and empirical Bayesian credible intervals are computed from Equations 3.18 and The HPD credible interval is constructed by using the MCMC samples. All interval estimates of are listed in Table 3. From Table, the Es of all estimates decrease as the sample sizes increase in all cases, as expected. The Bayes estimates with their corresponding Es based on Lindley s approximation and MCMC method are very close to the exact values. The Es of the ML, UMVU, Bayes and empiric Bayes under the SE loss function estimates are ordered as E BS < E EBS < E MLE < E U when =.5484,.756 and E BS < E U < E MLE < E EBS when = Moreover, the Es of the Bayes estimates under the LINEX loss function have smaller than that of ML estimates. From Table 3, the average

21 1 lengths of the intervals decrease as the sample sizes increase. The average lengths of the empirical Bayesian credible intervals are smallest, but their cps are not preferable. The HPD Bayesian credible intervals are more suitable than others in terms of the average lengths and cps. In Table 4, the ML, UMVU and Bayesian estimates of and their corresponding Es are listed when α is known α = 3. In this case, the Bayes estimates are evaluated analytically and by using Lindley s approximation under the SE and the LINEX v = 1 and 1 loss functions for the non informative prior. Moreover, the exact and asymptotic confidence intervals are computed from Equations 3. and 3.4. The point and interval estimates are computed for =.5,.33,.5,, 7,.9 and.9 when β 1, β =, 6,, 4,,, 7, 3, 18, and 3,, respectively. From Table 4, the Es of all estimates decrease as the sample sizes increase in all cases, as expected. The Bayes estimates under the SE loss function with their corresponding Es are close to their response in the ML case. Moreover, the Bayes estimates with their corresponding Es based on Lindley s approximation are very close the exact values. The Es of the ML, UMVU and Bayes under the SE loss function estimates are ordered as E BS < E MLE < E U when =.5,.33,.5,.7 and E U < E MLE < E BS when =.9,.9. The Es of ML and Bayes estimates have larger values when the true value of is around.5 and it decreases as the true value of approaches the extremes. Furthermore, the average lengths of the intervals decrease as the sample sizes increase. When =.5,.9 and.9 the lengths of the asymptotic confidence intervals are smaller than that of exact confidence intervals, but for =.33,.5 and, 7 it is other way around. On the other hand, to compare the performance of the different estimates of, the graphs of MSEs and Biases are obtained for different n and m when α is unknown and known cases. When α is unknown, the graphs are plotted based on the MLE and Lindley methods in Figure 1. When α is known, the graphs are plotted based on the MLE, UMVUE and Lindley methods in Figure. For each choices of β 1, β, α or β 1, β, we use the following procedure for the comparison of the estimates. Step 1: For given β 1, β, α or β 1, β, we compute. Step : For given different n and m, we generate a sample from the Burr Type XII distributions for the strength and the stress variables. Step 3: The different estimates of are computed. Step 4: Steps -3 are repeated 3 times, the MSEs and Biases are calculated and are given by MSE s,k = N i /N and Bias s,k = N i /N. From the Figures 1 and, it is observed that the MSEs and Biases of the estimates decrease when the sample size increases, as expected. The MSEs of the Bayes estimates under the SE and LINEX v = 1, 1 loss functions are smaller than that of other estimates. Moreover, the MSE is small for the extreme values of, but it is large when is around.5 for all types of estimates. When is around.5, the MSEs of UMVUE are greater than that of MLE and when is around extreme values, the MSEs of UMVUE are smaller than that of MLE in Figure. Notice that the similar outcomes are observed in all Tables.

22 Table 1. Estimates of when α is unknown and the true values of =.56,.7145 and.995 by using the Priors 1-3. n, m MLE Bayes estimates under the LINEX Bayes under the SE v = 1 v = 1 Asymptotic C.I. Asymptotic C.I. HPD Credible BS,Lindley BS,MCMC BL,Lindley BL,MCMC BL,Lindley BL,MCMC based on Fisher based on Observ. Interval 5, ,.788.1, , / / / , , , , / / /.984 8, , , , / / / , ,.734.7, , / / / , , , , / / /.978 1, , , , / / /.98 1, , , , / / / , ,.69.36, , / / /.98 15, , , , / / / , , , , / / /.989 5, , , , / / /.99 5, , , , / / /.999 8, , , , / / /.975 8, , , , / / / , ,.899.5, , / / /.969 1, , , , / / / , , , , / / / , , , , / / / , , , , / / / , , , , / / /.9673

23 3 Table 1 continued n, m MLE Bayes estimates under the LINEX Bayes under the SE v = 1 v = 1 Asymptotic C.I. Asymptotic C.I. HPD Credible BS,Lindley BS,MCMC BL,Lindley BL,MCMC BL,Lindley BL,MCMC based on Fisher based on Observ. Interval 5, , , , / / /.975 5, , , , / / / , , , , / / / , , , , / / / , , , , / / / , , , , / / / , , , , / /.99.16/.977 1, , , , / / / , , , , / / / , , , , / / /.9847 Notes: The first row represents the average estimates and the second row represents corresponding Es for each choice of n, m. But, for the last three columns, the first row represents 95% confidence interval and the second row represents their lengths and cp s.

24 4 Table. Estimates of when α is known α = 3 and the true values of =.5484,.756 and.9165 by using the Priors 4-6. n, m MLE U BS Bayes estimates under the LINEX Bayes estimates under the SE v = 1 v = 1 BS,Lindley BS,MCMC EBS BL BL,Lindley BL,MCMC EBL BL BL,Lindley BL,MCMC 5, , , , , , , , , , , , , , , Note: The first row represents the average estimates and the second row represents corresponding Es for each choice of n, m. EBL

25 5 Table 3. Confidence intervals of when α is known α = 3 and the true values of =.5484,.756 and n, m Exact C.I. Asymptotic C.I. Bayesian Credible I. HPD Bayes C.I. Empirical Bayes C.I. 5, , , , , , / / / / / ,8.3351, , , , , / / / / /.867 1,1.357, , , , , / / / / /.83 1,1.3683, , , , , / / / / /.87 15, , ,.75.46, , , / / / / /.87 5, , , , , , / / / / /.843 8,8.549, , , , , / / / / /.818 1,1.563, , , , , / / / / /.816 1,1.5791, , , , , / /.9.78/ / / ,15.635, , , , , / / / / /.897 5, , , , , , / / / / /.857 8,8.7945, , , , , / / / / /.815 1,1.815, , , , , / / / / /.833 1,1.8199, , , , , / / / / / , , , , , , / / / / /.757 Note: The first row represents a 95% confidence interval and the second row represents their lengths and cp s.