Statistical inference based on record data from Pareto model

Transcription

1 Statistics, Vol. 41, No., April 7, Statistical inference based on record data from Pareto model MOHAMMAD Z. RAQAB, J. AHMADI* ** and M. DOOSTPARAST Department of Mathematics, University of Jordan, Amman 1194, Jordan Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box , Mashhad, Iran (Received 11 April 5; revised 11 August 5; in final form 6 October 6) In this article, based on record values from the two-parameter Pareto distribution, maximum likelihood and Bayes estimators for the two unknown parameters are obtained. The Bayes estimates are obtained on the basis of the squared error loss and linear-exponential loss functions. The admissibility of some estimators is discussed. The problem of predicting the future record values, either point or interval prediction, from the Pareto distribution, based on the past record values observed, is considered from a Bayesian approach. Also, the maximum likelihood prediction of the future records and other classical methods are used for obtaining prediction intervals for the future records. Numerical computations are given for empirical comparison purposes. Keywords: Admissibility; Bayes estimation; Bayes prediction; Conditional median prediction; Linearexponential loss; Maximum likelihood prediction; Squared error loss 1. Introduction Let {X i,i 1} be a sequence of independent and identically distributed (iid) continuous random variables, each distributed according to cumulative distribution function (cdf) F(t; θ) and probability density function (pdf) f(t; θ), where θ is a vector of parameters. An observation X j will be called an upper record value if its value exceeds that of all previous observations. Thus, X j is an upper record if X j >X i for every i<j. An analogous definition can be given for lower record values. Then the record times sequence {T n,n 1} is defined in the following manner: T 1 = 1, with probability 1, and, for n, T n = min{j : j>t n 1,X j >X Tn 1 }. The sequence of upper record values is defined by R n = X Tn,n= 1,, 3,... *Corresponding author. ahmadi@math.um.ac.ir **The second author is a member of Statistics Center of Excellence, Ferdowsi University of Mashhad Statistics ISSN print/issn online 7 Taylor & Francis DOI: 1.18/

2 16 M. Z. Raqab et al. Suppose, we observe the first m upper record values R 1 = r 1,R = r,...,r m = r m from the cdf F(x; θ) and the pdf f(x; θ). Then, the joint pdf of the first m upper record values (for more details see ref. [1]) is given by where f(r; θ) = m 1 i=1 h(r i ; θ)f(r m ; θ), <r 1 <r < <r m < +, (1) r = (r 1,r,...,r m ), h(r i ; θ) = f(r i; θ) 1 F(r i ; θ), and θ may be a vector, where is the parameter space. Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of the observations. These statistics are of interest and importance in several applications. Like extreme order statistics, record values are applied in estimating strength of materials, predicting natural disasters and sport achievements, etc. For more details and applications of record values, see Arnold et al. [1], Ahmadi [], Gulati and Padgett [3]. Some work has been done on statistical inference based on record values; see, for instance, refs. [1, 3 7] and references therein. Inference problems about θ in equation (1) can be dealt with using Bayesian analysis. The idea is that since the posterior distribution supposedly contains all the available information about θ (both sample and prior information), any inference concerning θ should be made solely through this distribution. An important element of Bayesian analysis is the selection of a loss function, L(θ, δ), where δ is a decision rule based on the data. One disadvantage when using squared error loss (SEL) is that it penalizes for overestimation and underestimation. Overestimation of a parameter can lead to more severe or less severe consequences than underestimation, or vice versa. Subsequently, the use of an asymmetrical loss function, which associates greater importance to overestimation or underestimation, can be considered for the estimation of the parameter. A useful alternative to the SEL is the convex but asymmetric loss function L(θ, δ) = e v(δ θ) v(δ θ) 1, () where θ is a univariate parameter and v =. This loss function, called the linear-exponential (LINEX) loss function was proposed by Varian [8]. The sign of v represents the direction of penalty and its magnitude represents the degree of symmetry. For v = 1, loss is quite asymmetric about zero with overestimation being more costly than underestimation. In general, when v >, this loss increases almost linearly for negative error δ θ, and almost exponential for positive error. Therefore, overestimation is a more serious mistake than underestimation. When v<, the LINEX increases are interchanged, where underestimation is more serious than overestimation. The magnitude of v reflects the degree of asymmetry, so the proposed loss function allows for an asymmetric penalty. The loss is strictly convex and for small, positive values of v, i.e. v j for j 3, the loss function is almost symmetric and not far from a SEL function. Indeed, on expanding e v(δ θ) 1 + v(δ θ)+ v (δ θ), L(θ, δ) v (δ θ), a SEL function. Thus, for small values of v, optimal estimates and prediction are not far different from those obtained with a SEL function. Writing M θ X (t) = E θ X [e tθ ] for the moment-generating function of the posterior distribution of θ, it is easily seen that the value

3 of δ(x) that minimizes E θ X [L(θ, δ(x)] in equation () is Statistical inference from Pareto model 17 δ B (X) = 1 v log M θ X( v), (3) provided, M θ X ( ) exists and is finite. Having observed a sequence of record values in a sample of size n or only a number of record values, what should be the magnitude of future record values? Several authors have studied this problem. For an exponential distribution, Ahsanullah [9] obtained the best linear unbiased predictor (BLUP) and best linear invariant predictor (BLIP) of R s based on R i, 1 i m, for m<s, using standard least squares theory. Dunsmore [1] provided tolerance regions as well as a Bayesian predictive distribution of R s R m. Awad and Raqab [11] conducted a comparison study between several non-bayesian predictions of future records from the oneparameter exponential distribution. AL-Hussaini and Jaheen [1] presented prediction bounds for Burr type XII failure model. AL-Hussaini and Ahmad [13] derived Bayesian prediction intervals of future records from Pareto distribution. Ahmadi et al. [14] developed Bayesian inference and prediction based on k-record values under the LINEX loss function. A random variable X is said to be a two-parameter Pareto distribution, which we shall write P (α, β), if its cdf is ( ) β α F(x; α, β) = 1, x β>, α >, (4) x and hence its pdf is given by f(x; α, β) = αβα, x β>, α >. (5) xα+1 This distribution is called Pareto type I distribution (see, for example, ref. [15]). It has been used often to model naturally occurring phenomena in which the distributions of random variables of interest has long tails; for example, see ref. [16]. The rest of the paper is organized as follows: section presents the maximum likelihood and Bayes estimators based on the SEL and LINEX loss functions for the two unknown parameters. In section 3, the problem of predicting the future record values from the Pareto distribution is considered. On the basis of observed records, the maximum likelihood prediction of future record values as well as other non-bayesian methods have been used for obtaining prediction intervals for future records. The Bayesian approach is also used to establish point and interval predictions of future records. Numerical computations for the risk of estimators as well as an illustrative example are included in section 4.. Estimation of the parameters In this section, we shall be concerned with estimation of the two unknown parameters α and β of the Pareto model based on record values. Suppose, we observed the first m upper record values R 1 = r 1,R = r,...,r m = r m from a Pareto distribution, with cdf and pdf given, respectively, by equations (4) and (5). Notice that from equations (1), (4) and (5), it is easy to verify that the joint density function is given by αm β α f(r α, β) = rm α m i=1 r, β r 1 <r < <r m. (6) i It may be noticed that (R 1,R m ) is a jointly sufficient statistic.

4 18 M. Z. Raqab et al..1 Maximum likelihood estimation By equation (6), it can be easily shown that the log-likelihood function is given by l(α, β) = m log α + α log β α log r m m log r i. (7) Then, by equation (7), it can be shown that the maximum likelihood estimators of β and α are given by ˆβ M = R 1, (8) and m ˆα M =, log R m log R 1 (9) respectively. It is evident that by equation (6), ˆβ is distributed as P (α, β). Therefore, from equation (8), we get E( ˆβ M ) = βα α 1, α > 1, and the mean square error (MSE) of ˆβ M is β MSE( ˆβ M ) = (α 1)(α ), α >. It can be shown that the expression in the denominator of ˆα M is distributed as gamma distribution with parameter m and 1. Thus, we have E(ˆα M ) = mα m, m >, m + 6 MSE( ˆα M ) = (m )(m 3), m > 3.. Bayes estimation Under the assumption that both of the parameters α and β are unknown, we may consider the joint prior density function for α and β which was introduced by Arnold and Press [17], π(α, β) α a β 1 exp{ α(log c b log β)}, α >, <β<d, (1) where a, b, c, d are positive constants and d b <c. Such a prior specifies π(α) as a gamma distribution with paprameters a and log c b log d. Hence, the joint posterior distribution of α and β is given by π(α, β r) f(r α, β)π(α, β), (11) where f(r α, β) is the joint distribution function given by equation (6). By substituting equations (6) and (1) in equation (11), the joint posterior density can be written as π(α, β r) = (b + 1)[I(r m,m)] m+a e { αi(rm,β)}, α >, <β<m, (1) Ɣ(m + a)βα m a where I(x,y) = log x + log and Ɣ( ) is the complete gamma function. c y b+1, i=1 M = min{r 1,d},

5 Statistical inference from Pareto model 19 It can be shown, from equation (1), that α r has the gamma distribution with parameters (m + a,i(r m,m)). This implies that the Bayes estimator of α under SEL, as the mean of the posterior distribution of α r,isgivenby ˆα BS = m + a I(R m,m). (13) We also can obtain the Bayes estimator for β as follows: E(β r) = M βπ(α, β r) dβ dα = (b + 1)[I(r m,m)] m+a Ɣ(m + a) = (b + 1)[I(r m,m)] m+a M Ɣ(m + a) = (b + 1)[I(r m,m)] m+a M Ɣ(m + a) M α m+a exp{ αi(r m,β)} dβ dα α m+a { α(b + 1) + 1 exp α log r } mc dα M b+1 α m+a α(b + 1) + 1 exp{ αi(r m,m)} dα M t m+a = exp{ t} dt. Ɣ(m + a) t + I(r m, M)/(b + 1) Thus, ( M ˆβ BS = Ɣ(m + a) m + a, I(R ) m,m), (14) b + 1 where t x (x,y) = t + y e t dt. (15) A partial tabulation of ψ(x,y) = (y/ɣ(x)) (x 1,y)is provided by Arnold and Press [17]. Remark 1 The ˆα BS and ˆβ BS are the unique Bayes estimators of α and β, respectively, and hence are admissible. From equation (1), it can be shown that ( ) E(e αv I(rm,M) m+a r) =. I(r m,m)+ v Adopting the LINEX loss function () and using (3), we obtain the Bayes estimator of α as ( ) log 1 +. (16) ˆα BL = m + a v v I(R m,m) Obviously, ˆα BL is the unique Bayes estimator for α and hence is admissible. Remark It may be noticed that ˆα BL tends to ˆα BS as v. When r 1 <d and a, c 1, b, then the ˆα BS tends to ˆα M. Proceeding similarly, we have ˆβ BL = 1 { v log (m + a)[i(r m,m)] m+a (I(r m, M), v(r m c) 1/b, 1 } b,m+ a + 1), (17)

6 11 M. Z. Raqab et al. where (w,x,y,z) = ˆβ BL is the unique Bayes estimator, and so is admissible. w t z exp { xe yt} dt. 3. Prediction of future record values Predicting future record values is a problem of considerable interest. Point and interval prediction of future records using past data has dominated scientific research from record-breaking data. This problem was studied by several statisticians (see, refs. [9 11, 14, 18 1] among others). 3.1 Non-Bayesian prediction approach Suppose that we observe the first m record values from a population with pdf f(x; θ). Our aim is to predict Y = R s,s>m, having observed records r = (r 1,...,r m ). The predictive likelihood function of Y = R s, and θ is given by Basak and Balakrishnan [19] L(y, θ; r) = m j=1 Using equations (4), (5) and (18), we obtain h(r j ; θ) [H(y; θ) H(r m; θ)] s m 1 f(y; θ). (18) Ɣ(s m) α s β α L(y,α,β,r) = Ɣ(s m) [log y log r m ] s m 1 m j=1 r, y > r j y α+1 m > >r 1 β. So, the predictive maximum likelihood estimator (PMLE) of β is given by Also, the log predictive likelihood is given by ˆβ PM = R 1. l = s log α + α log ˆβ + (s m 1) log(log y log r m ) log Ɣ(s m) m log r i (α + 1) log y. i=1 So, the log-likelihood equations are given by and Hence, the PMLE for α is ˆα PM = 1 m log R m log R 1 l α = s α + log ˆβ log y =, l y = s m 1 y(log y log r m ) α + 1 =. y [ ( ) m log R m log R 1 4s log R m log R 1 ] 1/.

7 Statistical inference from Pareto model 111 Thus, the maximum likelihood prediction (MLP) of Y = R s is { } s Ŷ MLP = R 1 exp. (19) ˆα PM Now, let us consider the conditional pdf of Y = R s given R 1 = r 1,...,R m = r m. It is known that the conditional pdf of R s given R 1 = r 1,R = r,...,r m = r m (by Markov property) is given by αs m f(y r; α, β) = Ɣ(s m) ( log y r m ) s m 1 ( rm y ) α 1 y, y > r m. () The median of Y given R m is called the conditional median predictor (CMP) []. The density function in equation () does not depend on β. So we consider the following two cases. Case I α known. If Ŷ CMP (1) = g(r m ; α) is a function of r m, then P(g(R m ; α) Y r m,α)= 1/. From equation (), we have g(rm ;α) r m Setting log y/r m = t, we obtain log g(rm ;α)/r m α s m [log y log r m ] s m 1 r α m y α+1 Ɣ(s m) dy = 1. α s m Ɣ(s m) t s m 1 exp{ αt} dt = 1. Thus, Ŷ CMP (1) = R m exp{med(w)}, where W is a gamma random variable with shape parameter (s m) and scale parameter 1/α, say, G(s m, α). For a special case, when s = m + 1, W is an exponential random variable with scale parameter α, and so Ŷ CMP (1) = 1/α R m. (1) Setting Z 1 = log(y/r m ) in equation (19), we can easily observe that αz 1 r m χ (s m). Thus, the exact (1 γ)1% prediction interval for Y = R s is derived to be ( { } { }) χ (s m),γ / χ (s m),1 γ/ I A = R m exp,r m exp, α α where χr,p is 1pth percentile from the χ distribution with degrees of freedom r. For a special case, when s = m + 1, [ I A = (R m 1 γ ] 1/α [ γ ] ) 1/α,Rm. ()

8 11 M. Z. Raqab et al. Case II α unknown. When α is unknown, the MLE of α is obtained to be Ŷ CMP () = R m exp{med(w )}, where W G(s m, ˆα M ). For a special case, when s = m + 1, W is an exponential random variable with scale parameter ˆα M, and then Ŷ CMP () = 1/ ˆα M R m. (3) Substituting for α by its MLE (for example), we obtain an approximate prediction interval as ( { } { }) χ Iˆ (s m),γ / χ (s m),1 γ/ A = R m exp,r m exp. ˆα M ˆα M For a special case, when s = m + 1, ˆ I A = [ (R m 1 γ ] 1/ ˆαM [ γ ] ) 1/ ˆαM,Rm. (4) One can also use the pivotal approach as a procedure to obtain another prediction interval. By substituting the MLE of α in equation (), we obtain the approximate pdf of Y given R m = r m, ( ) f(y r ˆ m, ˆα M ) = ms m log(y/rm ) s m 1 ( ) { 1 exp m log(y/r } m). (5) Ɣ(s m) log(r m /r 1 ) y log(r m /r 1 ) log(r m /r 1 ) The conditional pdf in equation (5) is unimodal function of the statistic Z = log Y log r m log r m log r 1. It can be shown that Z G(s m, m), so that mz χ(s m). Then, the (1 γ)1% prediction interval for Y = R s is given by ( { log(rm /R 1 ) R m exp m χ (s m),γ / } { }) log(rm /R 1 ),R m exp χ(s m),1 γ/, m or ( { } { }) 1 1 I P = R m exp χ(s m),γ /,R m exp χ(s m),1 γ/. ˆα M ˆα M It may be noted that I P = Iˆ A. Another related prediction interval can be obtained as follows. Since the pdf in equation (5) is unimodal, the (1 γ)1% highest conditional density (HCD) prediction interval for R s is given by ( log w1 log r m I H =, log w ) log r m, log r m log r 1 log r m log r 1 where w 1 and w are the simultaneous solution of w w 1 ˆ f(y r m ) dy = 1 γ, and ˆ f(w 1 r m ) = ˆ f(w r m ).

9 Statistical inference from Pareto model 113 These equations lead to the following equations: ( G s m, m; log w ) ( log r m G s m, m; log w ) 1 log r m = 1 γ, (6) log r m log r 1 log r m log r 1 and ( ) log w1 log r s m 1 ( ) (r1 /r m )+1 m w1 =, (7) log w log r m w where t G(a,b,t)= ba x a 1 e bx dx, Ɣ(a) is the incomplete gamma function. Remark 3 It may be noted that for s = m + 1, equation (7) yields w 1 = w and then no prediction interval can be established because equation (5) is not unimodal. In this case, we obtain the approximate pdf of Y given R m = r m, ( ) ˆαM f(y r ˆ rm 1 m, ˆα M ) =ˆα M y y, y > r m. Hence, it can be shown that, the (1 γ)1% HCD prediction interval for Y = R m+1 is given by I H = ( ) R m,r m γ 1/ ˆα M. (8) 3. Bayes prediction approach In this section, we consider the problem of prediction, either point or interval, of future records based on a Bayesian approach. Assume that we have observed the first m upper records R 1 = r 1,...,R m = r m from the P (α, β). On the basis of such a sample, prediction, either point or interval, is needed for sth upper record, 1 <m<s. Now, let Y = R s be the sth upper record value, 1 <m<s. The Bayes predictive density function of Y given r is given by h(y r) = f (y r, θ)π(θ r) dθ. (9) Combining the posterior density, given by equation (1), and the conditional density, given by equation (), and integrating out the parameters α and β, one may get a Bayesian predictive density function of Y = R s, given the past m records, in the form h(y r) = M f (y α, β, r)π(α, β r) dβ dα = [log y log r m] s m 1 yb(m + a,s m) [I(r m,m)] m+a, (3) [I(y,M)] s+a where B(, ) is the complete beta function and I(x,y) is defined in section.. From equation (3), under SEL, we obtain the Bayes point predictor of the sth upper record (s m + 1) as E(Y r) = r m yh(y r) dy = I(r m,m) m+a B(m + a,s m) 1 [ log u] s m 1 [ log u + I(r m,m)] s+a r m u du.

10 114 M. Z. Raqab et al. Hence, R m J s+a 1 s m 1 1 Ŷ B = J k [J log z] k s a dz, (31) B(m + a,s m) k= z where J = I(R m,m). For a special case, when s = m + 1, the Bayesian predictive function of Y = R m+1,given the past m records is Hence, h(y r) = Ŷ B = I(r m,m) m+a+1 yb(m + a,1)i (y, M) m+a+1 = (m + a)i(r m,m) m+a+1 yi (y, M) m+a+1, y > r m. R mj m+a 1 [J log z] m a 1 dz B(m + a,1) z 1 = (m + a)r m J m+a z dz. [J log z] m+a+1 Let t(y) = I(r m, M)/I (Y, M). It can be shown that t(y)has a Beta distribution with parameters m + a and s m, which is independent of Y. Thus, t(y)is a pivotal quantity and one can use t(y)for constructing a Bayesian prediction interval for R s. Let b γ be the γ th percentile of a Beta(m + α, s m)-distribution. It can be shown that the (1 γ)1% prediction interval for Y = R s is given by I B = ( R m exp [ I(R m,m) As a special case, when s = m + 1, we have ( M b+1 I B = exp c ( )] [ ( )]) 1 1 1, R m exp I(R m,m) 1. b 1 γ/ b γ/ { [ I(r m,m) 1 γ ] 1/m+a }, Mb+1 exp c { [ γ ] }) 1/m+a I(r m,m). (3) In the context of the highest conditional posterior density (HCPD) method, we use equation (3) to solve the following equations: w h(y r) dy = 1 γ, w 1 and h(w 1 r) = h(w r). Taking the transformation u = log y log r m,wehave I(r m,m) m s log w log r m log w 1 log r m u s m 1 du = 1 γ. [1 + u/i (r m,m)] s+a Then, using u u μ 1 uμ dx = (1 + δx) ν μ F 1 (ν, μ; 1 + μ; δx),

11 Statistical inference from Pareto model 115 (cf. ref. [3] 3.194), the limits of the prediction interval can be found by solving { [ I(r m,m) m s log w ] s m ( F 1 s + a,s m; s m + 1; log r ) m/w s m I(r m,m) r m [ log w ] s m ( 1 F 1 s + a,s m; s m + 1; log r ) } m/w 1 = 1 γ, r m I(r m,m) and ( ) log w log r s m 1 ( ) m I(w,M) s+a w =. log w 1 log r m I(w 1,M) w 1 For a special case, when s = m + 1, it can be shown that the (1 γ)1% HCPD prediction interval for Y = R m+1 is given by ( [ c ] ) γ 1/(m+a) I HCPD = R m, M R b+1 m. (33) 4. Numerical computation In this section, we will report the results of a simulation study for comparing the risk of estimators. Also, a numerical example is given to illustrate the results of prediction. 4.1 Monte Carlo simulation To assess and compare the risk of estimators obtained in section, record samples of different sizes are generated as follows: (i) For a given vector of parameters (a,b,c,d), we generate α and β from the joint prior density (1). (ii) Using (α, β), obtained in (i), we generate m = (5, 1, 15) upper record values from the P (α, β) with pdf (5). (iii) The Bayes estimators (Bayes estimator ( ) ˆ BS based on SEL and for given values of v, Bayes estimators ( ) ˆ BL based on LINEX loss function) are computed from equations (13), (14), (16) and (17). (iv) We repeated the above steps n = 1 times and calculated the estimated risk (ER) given by ER(δ) = 1 n (δ i θ i ), (34) n i=1 and ER(δ) = 1 n { e v(δ i θ i ) v(δ i θ i ) 1 }, (35) n i=1 under SEL and LINEX loss function, respectively. The computation results are summarized in the tables 1 and. Empirical evidence as can be seen from tables 1 and is that: The ER( ) is decreasing in m for all cases. ER( ˆ ( ) BL ) is increasing in v.

12 116 M. Z. Raqab et al. Table 1. The values of ER(δ) in equations (34) and (35) for Bayes estimators of α. ER( ˆα BL ) ER( ˆα BL ) ER( ˆα BL ) ER( ˆα BL ) a b c d m ER( ˆα BS ) v =.1 v =.5 v = 1 v = Table. The values of ER(δ) in equations (34) and (35) for Bayes estimators of β. ER( ˆβ BL ) ER( ˆβ BL ) ER( ˆβ BL ) ER( ˆβ BL ) a b c d m ER( ˆβ BS ) v =.1 v =.5 v = 1 v = The estimated risks of the Bayes estimators, which are obtained on the basis of LINEX loss function, are smaller than the corresponding estimated risks of the estimates, which are obtained on the basis of SEL for v =.1,.5 and Numerical example In order to illustrate the usefulness of the prediction procedures obtained in section 3, the lower and upper 95% prediction limits bounds for the next record are considered in the sample of size m = 1 observed records, when both the parameters α and β are unknown and prior information about α and β suggested that a = 3,b = 1,c = 4 and d = for the joint density given in equation (1). An upper record sample of size 1 is generated from the P(α = ,β = 1.596) by equation (5) and written in order as: , , 8.738, ,.148,.3658, 5.345, , 3.38, Using our results in section 3, we have: (i) Point prediction (for the next record, R m+1 ). From equations (1), (3) and (31), we have ˆR 11 = 4.86, , , respectively. (ii) Interval prediction (for the next record, R m+1 ). From equations (), (4), (8), (3) and (33) and using above data, lower and upper 95% prediction bounds for R 11, the next record value, are presented in table 3.

13 Statistical inference from Pareto model 117 Table 3. Upper and lower prediction bounds. Lower limit Upper limit I A Iˆ A = I P I H I B I HCPD Conclusions In this article, under non-bayesian and Bayesian frameworks, we have studied the estimation as well as prediction for the two-parameter Pareto type I distribution based on record values observed. (1) The mathematical package Maple 8 is used to evaluate the numerical results. () In the Bayes theory, the prior parameters are assumed to be known. If the prior parameters are unknown, the empirical Bayes approach may be used to estimate such parameters (see, for example, ref. [4]). (3) Different values of the prior parameters a,b,c, and d have been considered but did not change the results. Acknowledgements The authors would like to thank the referees for their comments and suggestions which have led to an improved version of the manuscript. References [1] Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N., 1998, Records (New York: John Wiley). [] Ahmadi, J.,, Record values, theory and applications. Ph.D Thesis, Ferdowsi University of Mashhad, Mashhad, Iran. [3] Gulati, S. and Padgett, W.J., 3, Parametric and Nonparametric Inference from Record-Breaking Data, Lecture Notes in Statistics, Vol. 17 (New York: Springer-Verlag). [4] Ahmadi, J. and Arghami, N.R., 1, On the Fisher information in record values. Metrika, 53, [5] Ahmadi, J. and Arghami, N.R., 3, Nonparametric confidence and tolerance intervals from record values data. Statistical Papers, 44, [6] Ahmadi, J. and Arghami, N.R., 3, Comparing the Fisher information in record values and iid observations. Statistics, 37, [7] Ahmadi, J. and Balakrishnan, N., 4, Confidence intervals for quantiles in terms of record range. Statistics and Probability Letters, 68, [8] Varian, H.R.A., 1975, Bayesian approach to real estate assessment. In: S.E. Finberg and A. Zellner (Eds) Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savege (Amesterdam: North-Holland), pp [9] Ahsanullah, M., 198, Linear prediction of record values for the two parameter exponential distribution. Annals of the Institute of Statistical Mathematics, 3, [1] Dunsmore, I.R., 1983, The future occurrence of records. Annals of the Institute of Statistical Mathematics, 35, [11] Awad, A.M. and Raqab, M.Z.,, Prediction intervals for the future record values from exponential distribution: comparative study. Journal of Statistical Computation and Simulation, 65, [1] AL-Hussaini, E.K. and Jaheen, Z.F., 1995, Bayesian prediction bounds for the Burr type XII faliure model. Communications in Statistics Theory and Methods, 4(7), [13] AL-Hussaini, E.K. and Ahmad, A.A., 3, On Bayesian interval prediction of future records. Test, 1,

14 118 M. Z. Raqab et al. [14] Ahmadi, J., Doostparast, M. and Parsian, A., 5, Estimation and prediction in a two exponential distribution based on k-record values under LINEX loss function. Communications in Statistics Theory and Methods, 34(4), [15] Johnson, N.L., Kotz, S. and Balakrishnan, N., 1995, Continuous Univariate Distributions, Vol. (nd edn) (New York: John Wiley). [16] Arnold, B.C., 1983, Pareto Distributions (Maryland: International Co-operative Publishing House, Fairland). [17] Arnold, B.C. and Press, S.J., 1989, Bayesian estimation and prediction for Pareto data. Journal of American Statistical Association, 84, [18] Berred, A.M., 1998, Prediction of record values. Communication in Statistics Theory and Methods, 7(9), 1 4. [19] Basak, P. and Balakrishnan, N., 3, Maximum likelihood prediction of future record statistic. Mathematical and statistical methods in reliability. In B.H. Lindquist, and K.A. Doksum (Eds) Series on Quality, Reliability and Engineering Statistics, Vol. 7 (Singapore: World Scientific Publishing), pp [] Madi, M.T. and Raqab, M.Z., 4, Bayesian prediction of temperature records using the Pareto model. Envirometrics, 15, 1 1. [1] Ahmadi, J. and Doostparast, M., 6, Bayesian estimation and prediction for some life distributions based on record values. Statistical Papers, 47, [] Raqab, M.Z. and Nagaraja, H.N., 1995, On some predictors of future order statistics. Metron, 53, nos. 1, [3] Gradshteyn, I.S. and Ryzhik, I.M., 1994, In: A. Jeffrey (Ed.) Table of Integrals Series and Products (5th edn) (San Diego, USA: Academic Press). [4] Maritz, J.L. and Lwin, T., 1989, Empirical Bayes Methods (nd edn) (London: Chapman & Hall).