STATISTICA, anno LXXVIII, n. 1, 2018 BAYESIAN INFERENCE AND PREDICTION FOR NORMAL DISTRIBUTION BASED ON RECORDS Akbar Asgharzadeh 1 Department of Statistics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran Reza Valiollahi Department of Statistics, Statistics and Computer Science, Semnan University, Semnan, Iran Adeleh Fallah Department of Statistics, Payame Noor University, P. O. Box 19395-3697, Tehran, Iran Saralees Nadarajah School of Mathematics, University of Manchester, Manchester M13 9PL, UK 1. INTRODUCTION Let X1,X2,... be independent and identical random variables with the cumulative dis- tribution function (cdf) F (x) and the probability density function (pdf) f (x). Define Yn = max X1,...,Xn f g for n 1. Then, Xj is an upper record value of this sequence if Xj > Yj 1, j > 1. ≥ − Generally, if we define the sequence U(n), n 1 as f ≥ g ¦ © U 1 1, U n min j : j U n 1 ,X X ( ) = ( ) = > ( ) j > U(n 1) − − ¦ © for n 2, then XU n , n 1 is a sequence of upper record values. The sequence ≥ ( ) ≥ U(n), n 1 represents the record times. f Chandler≥ g (1952) defined the theory of records as a model for successive extremes in a sequence of independent and identical random variables. Record data arise in many real life problems, such as in destructive stress testing, weather, hydrology, economics and sporting and athletic events. For more details and applications, see Ahsanullah (1995), Arnold et al. (1998) and Nevzorov (2000). 1 Corresponding Author. E-mail: [email protected] 16 A. Asgharzadeh et al. In the frequentist set up, the estimation and prediction problems for normal dis- tribution based on record data have been discussed by several authors. Balakrishnan and Chan (1998) obtained the best linear unbiased estimators (BLUEs) of the normal location and scale parameters, µ and σ, based on the first few upper record values. Us- ing these BLUEs, they then developed a prediction interval for a future record value. Chacko and Mary (2013) discussed classical estimation and prediction for the normal distribution based on k-records. Sajeevkumar and Irshad (2014) estimated the location parameter of distributions with known coefficient of variation by record values. The main aim of this paper is to consider estimation and prediction for normal dis- tribution based on record data in the Bayesian set up. To the best of our knowledge, this problem has not been studied before in the literature. We compute Bayes estimators of µ and σ under squared error and Linex loss functions. It is observed that Bayes estimators can not be obtained in closed forms. We use the importance sampling procedure to gen- erate samples from the posterior distributions and then compute the Bayes estimators. We then compare Bayes estimators with the maximum likelihood estimators (MLEs) and BLUEs by Monte Carlo simulations. We observe that Bayes estimators work quite well. Bayesian prediction of future records based on the first few upper records is also discussed. We use the importance sampling method to estimate the predictive distribu- tion and then compute the Bayesian predictors. The contents of this paper are organized as follows. In Section 2, we provide a brief review of frequentist estimators and predictors. In Section 3, the Bayes estimators of µ and σ are obtained using squared error and Linex loss functions. In Section 4, we discuss Bayesian prediction for future records based on the first few upper records. In Section 5, a real data set is analyzed for illustrative purposes. Monte Carlo simulations are performed to compare the proposed estimators and predictors in Section 6. Finally, we conclude the paper in Section 7. 2. FREQUENTIST ESTIMATION AND PREDICTION:A REVIEW Suppose that we observe the first n upper record values X x ,X x ,...,X U(1) = 1 U(2) = 2 U(n) 2 = xn from the normal N µ,σ distribution. For notational simplicity, we will write X for X . The likelihood function is given (see Arnold et al., 1998) by i U(i) n 1 Y− f (xi ;µ,σ) L(µ,σ x) = f (xn;µ,σ) , j 1 F (xi ;µ,σ) i=1 − 2 where f (xn;µ,σ) and F (xn;µ,σ) denote, respectively, the pdf and cdf of the N µ,σ distribution. Bayesian Inference and Prediction for Normal Distribution Based on Records 17 The likelihood function can be rewritten as 0 1 xi µ 1 n 1 φ − 1 xn µ Y− σ σ L(µ,σ x) = φ − @ A σ σ xi µ j i=1 1 Φ − − σ n n n 1 1 1 Y xi µ Y− xi µ − = φ − 1 Φ − , (1) σ σ σ i=1 i=1 − where φ( ) and Φ( ) denote, respectively, the pdf and cdf of a standard normal distribu- tion. · · The log-likelihood function is n n 1 X xi µ X− xi µ L = ln L(µ,σ x) = n lnσ + lnφ − ln 1 Φ − . (2) σ σ j − i=1 − i=1 − From (2), we obtain the likelihood equations for µ and σ as xi µ n n 1 − @ L 1 X 1 X− φ σ = (xi µ) = 0 (3) @ µ σ2 σ xi µ i=1 − − i=1 1 Φ − − σ and x n n 1 i µ @ L n 1 X 1 X (xi µ)φ σ− 2 − − = + (xi µ) = 0. (4) @ σ σ σ3 σ2 xi µ − i=1 − − i=1 1 Φ − − σ The equations (3) and (4) can be solved analytically to obtain µbML and σbML, the MLEs of µ and σ. Following the generalized least-squares approach, the BLUEs of µ and σ can be de- rived as (see Balakrishnan and Cohen, 1991) Xn Xn µbBLU = ai Xi , σbBLU = bi Xi , (5) i=1 i=1 where 1 1 1 1 α0β− α10β− α0β− 1α0β− a = − 2 (α β 1α)(1 β 11) (α β 11) 0 − 0 − − 0 − and 1 1 1 1 10β− 1α0β− 10β− α10β− b = − 2 , (α β 1α)(1 β 11) (α β 11) 0 − 0 − − 0 − 18 A. Asgharzadeh et al. where α0 = (α1,α2,...,αn) is the moment vector with αi = E (Xi ) and β = βi,j , 1 i j n is the covariance matrix with βi,j = Cov Xi ,Xj , and 10 = (1,1,...,1)1 n. ≤ ≤ ≤ × The variances of these BLUEs are given by 1 α0β− α 2 2 Var(µbBLU ) = 2 σ = V1σ (α β 1α)(1 β 11) (α β 11) 0 − 0 − − 0 − and 1 10β− 1 2 2 Var(σbBLU ) = 2 σ = V2σ . (α β 1α)(1 β 11) (α β 11) 0 − 0 − − 0 − The coefficients a, b and the values of V1 and V2 can be found in Balakrishnan and Chan (1998, Tables 3 to 5). See also Arnold et al. (1998, Table 5.3.1, pages 139 and 140). From Xn Xn those tables, we can see that ai = 1 and bi = 0. i=1 i=1 By using the BLUEs, one can construct confidence intervals (CIs) for the location and scale parameters, µ and σ, through pivotal quantities given by µbBLU µ σbBLU σ R1 = p− , R2 = p− . (6) σbBLU V1 σbBLU V2 For constructing such CIs, we require the percentage points of R1 and R2, which can be computed by using the BLUEs µbBLU and σbBLU via Monte Carlo simulations. Table 1 gives the percentage points of R1 and R2 based on ten thousand replications and different choices of n. The following algorithm was used to determine the percentage points: 1. set a value for n; 2. simulate X ,X ,...,X from a standard normal distribution; U(1) U(2) U(n) 3. compute µbBLU and σbBLU from (5); 4. then compute R1 and R2 from (6) by taking µ = 0 and σ = 1; 5. repeat steps 2 to 4 ten thousand times, obtaining ten thousand estimates for R1 and ten thousand estimates for R2; 6. compute the percentage points of R1 as the quantiles of the empirical distribution of the ten thousand estimates of R1; 7. similarly, compute the percentage points of R2 as the quantiles of the empirical distribution of the ten thousand estimates of R2. Bayesian Inference and Prediction for Normal Distribution Based on Records 19 Let R and R denote the percentage points at determined through simulation for 1(α) 2(α) α the pivotal quantities R1 and R2, respectively. Then, p p R V , R V µbBLU σbBLU 1(1 α=2) 1 µbBLU σbBLU 1(α=2) 1 − − − and p p R V , R V σbBLU σbBLU 2(1 α=2) 2 σbBLU σbBLU 2(α=2) 2 − − − form 100(1 α) percent CIs for µ and σ based on the pivotal quantities R1 and R2, respectively.− Note that if we define Y X as the next upper record value, then we can = U(n+1) predict this value by using the best linear unbiased prediction (BLUP) method. The BLUP of the next upper record value can be derived as (see Balakrishnan and Chan, 1998) Y . ÒBLU P = µbBLU + σbBLU αn+1 TABLE 1 Simulated percentage points of R1 and R2. Percentage points of R1. n 1% 2.5% 5% 10% 90% 95% 97.5% 99% 2 -4.318 -2.765 -1.766 -1.165 2.975 4.926 7.187 9.945 3 -2.240 -1.633 -1.292 -1.003 2.587 3.970 5.662 8.125 4 -1.706 -1.430 -1.223 -0.997 2.253 3.506 4.906 6.795 5 -1.704 -1.443 -1.240 -0.999 2.038 3.107 4.193 5.974 6 -1.592 -1.375 -1.185 -0.944 2.140 3.136 4.218 5.572 7 -1.598 -1.390 -1.224 -1.030 1.825 2.776 3.778 4.930 8 -1.653 -1.441 -1.254 -1.033 1.876 2.717 3.542 4.709 9 -1.664 -1.442 -1.281 -1.048 1.767 2.548 3.384 4.368 10 -1.631 -1.453 -1.264 -1.045 1.740 2.439 3.219 4.222 Percentage points of R2.