Bayes estimation based on k -record data from a general class of distributions under balanced type loss functions

Journal of Statistical Planning and Inference 139 (2009) 1180 -- 1189

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Journal of Statistical Planning and Inference journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / j s p i

Bayes estimation based on k-record data from a general class of distributions under balanced type loss functions Jafar Ahmadia,1 , Mohammad Jafari Jozanib,2 , Éric Marchandc,3 , Ahmad Parsiand,∗,4 a

School of Mathematical Sciences, Ferdowsi University of Mahshhad, P.O. Box 91775-1159, Mashhad, Iran Department of Statistics, Faculty of Economics, Allameh Tabatababie University, & Statistical Research and Training Center (SRTC), Tehan, Iran c Département de mathématiques, Université de Sherbrooke, Sherbrooke, Qc, Canada J1K 2R1 d School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran b

A R T I C L E

I N F O

Article history: Received 11 February 2007 Received in revised form 7 July 2008 Accepted 23 July 2008 Available online 30 July 2008 Keywords: Admissibility Bayes estimation Balanced loss Squared error loss LINEX loss k-Records

A B S T R A C T

A semi-parametric class of distributions that includes several well-known lifetime distributions such as exponential, Weibull (one parameter), Pareto, Burr type XII and so on is considered in this paper. Bayes estimation of parameters of interest based on k-record data under balanced type loss functions is developed; and in some cases the admissibility or inadmissibility of the linear estimators is considered. The results are presented under the balanced versions of two well-known loss functions, namely squared error loss (SEL) and Varian's linear-exponential (LINEX) loss. Some recently published results on Bayesian estimation using record data are shown to be special cases of our results. © 2008 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Record data Let {Xi , i  1} be a sequence of iid absolutely continuous random variables distributed according to the cumulative distribution function (cdf) F(·; ) and probability density function (pdf) f (·; ), where  is an unknown parameter. An observation Xj is called an upper record value if its value exceeds all previous observations. Thus, Xj is an upper record if Xj > Xi for every i < j. Analogously, an upper k-record value is defined in terms of the k-th largest X yet seen. It is of interest to note that there are situations in which only records are observed, such as in destructive stress testing, meteorology, hydrology, seismology, and mining. For a more specific example, consider the situation of testing the breaking strength of wooden beams as described in Glick (1978). Interest in records has increased steadily over the years since Chandler's (1952) formulation. Useful surveys are given by the books of Arnold et al. (1998), Nevzorov (2001) and the references therein. For a formal definition of k-records, we consider the definition

∗

Corresponding author. Tel.: +98 21 6111 2624; fax: +98 21 6641 2178. E-mail addresses: [email protected] (J. Ahmadi), [email protected] (M. Jafari Jozani), [email protected] (É. Marchand), [email protected] (A. Parsian). 1 Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged. 2 Research supported by a grant of Statistical Research and Training Center, Grant no. 8707. 3 Research supported by NSERC of Canada. 4 Research supported by a grant of the Research Council of the University of Tehran. 0378-3758/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2008.07.008

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in Arnold et al. (1998, p. 43) for the continuous case. Let T1,k = k and, for n  2, Tn,k = min{j : j > Tn−1,k , Xj > XT

n−1,k −k+1:Tn−1,k

},

where Xi:m denotes the i-th order statistic in a sample of size m. The sequences of upper k-records are then defined by Rn(k) = XT −k+1:T for n  1. For k = 1, note that the usual records are recovered. These sequences of k-records were introduced by n,k

n,k

Dziubdziela and Kopocinski (1976) and they have found acceptance in the literature. Nagaraja (1988) pointed out that k-records with an underlying distribution function F can be viewed as ordinary record values (k = 1) based on the distribution of the minimum with cdf F1:k ≡ 1 − {1 − F}k . Using the joint density of usual records, the marginal density of Rn(k) is obtained as fn,k (rn(k) ; ) =

kn n−1 ¯ ¯ [− log F(r {F(rn(k) ; )}k−1 f (rn(k) ; ), n(k) ; )] (n − 1)!

(1)

and the joint pdf of the first n k-records is given by f

1, ...,n

k ¯ (r; ) = kn [F(r n(k) ; )]

n f (r  i(k) ; ) , F(ri(k) ; ) i=1

(2)

where r = (r1(k) , . . . , rn(k) ) and F¯ = 1 − F (see Arnold et al., 1998). The problem of estimation based on record data has been previously studied in the literature, in particular under a Bayesian framework (e.g., Ali Mousa et al., 2002; Jaheen, 2003, 2004; Ahmadi et al., 2005; Ahmadi and Doostparast, 2006). The goal of this paper is to develop Bayes estimation of functions of  based on k-record data from a general class of distributions under balanced type loss functions, which we now describe. 1.2. Balanced type loss functions To reflect both goodness of fit and precision of estimation in estimating an unknown parameter  under the model X = (X1 , . . . , Xn ) ∼ F , Zellner (1994) introduced a balanced loss function (BLF) as follows: n 

n

(Xi − )2 + (1 − )( − )2 ,

i=1

where  ∈ [0, 1), and considered optimal estimates relative to BLF for estimation of a scalar mean, a vector mean and a vector regression coefficients. Dey et al. (1999), as well as Jafari Jozani et al. (2006a) studied the notion of a BLF from the perspective of unifying a variety of results both frequentist and Bayesian. They showed in broad generality that frequentist and Bayesian results for BLF follow from and also imply related results for SEL functions. Jafari Jozani et al. (2006b) introduced an extended class of balanced type loss functions of the form L

q

,,0

((), ) = q()(0 , ) + (1 − )q()((), ),

(3)

with q(·) being a suitable positive weight function, ((), ) being as arbitrary loss function in estimating () by , and 0 a chosen a priori “target” estimator of (), obtained for instance from the criterion of maximum likelihood, least squares or unbiasedness among others. They give a general Bayesian connection between the cases  > 0 and  = 0. For the case of squared error , a least squares 0 , () = , and q() = 1 in (3) is equivalent to Zellner's (1994) BLF, and the introduction of an arbitrary  extends the squared error version of (3) introduced by Jafari Jozani et al. (2006a). In this paper, we shall use balanced squared error loss (balanced SEL) and balanced LINEX loss to illustrate Bayesian estimation of parameters of interest in a class of distributions (described in Section 1.3) based on a sample of k-record values. A companion paper (Ahmadi et al., 2008) considers a similar framework but for prediction of future k-records. Much of the analysis below is unified with respect to the choice of the target estimator 0 , the weight , and to some extent, with respect to the parameter being estimated. 1.3. Proportional hazard rate models Let X1 , X2 , . . . be a sequence of iid random variables from the family of continuous distribution functions with

() , ¯ F(x; ) = 1 − [G(x)]

−∞  c < x < d  ∞,

(4)

¯ and G is an arbitrary continuous distribution function with G(c) = 0 and G(d) = 1. The family in (4) is where () > 0, G ≡ 1 − G, well-known in lifetime experiments as proportional hazard rate models (see for example Lawless, 2003), and includes several well-known lifetime distributions such as exponential, Pareto, Lomax, Burr type XII, and so on. Let g(x) = (d/dx)G(x) be the corresponding pdf, then

()−1 , ¯ f (x; ) = ()g(x)[G(x)]

−∞  c < x < d  ∞.

(5)

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From (4), it can be easily shown that: (i) the Fisher information contained in a single observation X is IX () = ( ()/ ())2 ,  ¯ ¯ i ) is a complete sufficient statistic for . In this paper we consider the (ii) E[− log G(X)] = 1/ (), and (iii) T(X) = − ni=1 log G(X ¯ ), and the survival function at t: estimation of 1 () = , 2 () = 1/ , the hazard rate function at t: 3 () = h (t) = f (t; )/ F(t; ¯ 4 () = R (t) = F(t; ) of the parent distribution in (4) based on observed k-record data. 2. Frequentist estimation Suppose we observe the first n upper k-record values R1(k) = r1(k) , R2(k) = r2(k) , . . . , Rn(k) = rn(k) from a distribution with cdf given by (4) with () = . Using (4), (5), and (2), it is easy to verify that the joint density function of the n upper k-record values is given by f

1, ...,n

k() ¯ (r; ) = [k()]n {G(r n(k) )}

n g(r  i(k) ) i=1

G(ri(k) )

,

(6)

¯ where r = (r1(k) , . . . , rn(k) ). Let Un(k) = − log G(R n(k) ). Then from (6), Un(k) is seen to be a complete and sufficient statistic for  with

−1 a Gamma(n, k) distribution. Also, the maximum likelihood estimator (MLE) of  is given by (n/k)Un(k) . With the completeness

and sufficiency of Un(k) based on the first n k-records for samples from (4), we obtain with Rao–Blackwell–Lehmann–Scheffe's Theorem that: −1 • ((n − 1)/k)Un(k) is the UMVUE of 1 () = ,

• (k/n)Un(k) is the UMVUE of 2 () = 1/ ,

−1 • ((n − 1)/k)(g(t)/G(t))Un(k) is the UMVUE of 3 () = h (t), −1 n−1 ¯ ) is the UMVUE of 4 () = R (t). • (1 + (log G(t)/k)U n(k)

3. Bayesian estimation under balanced SEL In this section we obtain Bayes estimators of i (); i = 1, 2, 3, 4; based on observed k-record data generated from (4) with () = , using balanced SEL given in (3) with (, ) = ( − )2 and equal to L

q

,0

((), ) = q()( − 0 )2 + (1 − )q()( − ())2 .

(7)

Using a conjugate Gamma(, ) prior for  with density,

() =

 −1 −  e ,

()

, ,  > 0,

(8)

the posterior pdf of  is given by

(|r1(k) , . . . , rn(k) ) =

( + kun(k) )n+ n+−1 −(+ku ) n(k) ,  e

(n + )

 > 0,

(9)

q ¯ where un(k) = − log G(r n(k) ). The Bayes estimator of () under L, ((), ) is given by (see Jafari Jozani et al., 2006a) 0

q, (R(k) ) = 0 (R(k) ) + (1 − )

E[q()()|R(k) ] E[q()|R(k) ]

,

(10) q

where R(k) = (R1(k) , . . . , Rn(k) ). As apparent from (10), , (R(k) ) is a convex linear combination of the target estimator 0 (R(k) ) (weight ), and the Bayes estimator of () under weighted SEL function. In the sequel, we pursue with the study of Bayes estimators of i (), i = 1, 2, 3, 4 for q ≡ 1 in (7). 3.1. Bayes estimator of 1 () =  under balanced SEL With a Gamma prior as in (8), the Bayes estimator of 1 () =  under SEL function is given by

0,1 (R(k) ) = E[|R(k) ] =

n+ . ¯  − k log G(R n(k) )

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Using (10), the Bayes estimator of 1 () =  under L, (1 (), ) with respect to a Gamma(, ) prior is given by 0

,1 (R(k) ) = 0 (R(k) ) + (1 − )

n+ . ¯  − k log G(R n(k) )

¯ Among others, a plausible target estimator 0 (R(k) ) is the MLE of 1 (), given by −n/(k log G(R n(k) )). It may be noted that ,1 (R(k) ) is the unique Bayes estimator of 1 () =  under L, (1 (), ) with finite Bayes risk and hence is admissible. Notice that as 0

(, ) → (0, 0),

,1 (R(k) ) → N ,1 (R(k) ) = 0 (R(k) ) − (1 − )

n , ¯ k log G(R n(k) )

where N ,1 (R(k) ) is the Bayes estimator of 1 () =  under L,0 (1 (), ) with respect to the non-informative prior () = 1/ . It ¯ is easy to see that under L0, (1 (), ), the estimator N 0,1 (R(k) ) is dominated by −(n − 2)/(k log G(Rn(k) )), provided that n > 2, and is a ¯ an inadmissible estimator of 1 (). In fact, 0, (R(k) ) = −(n − 2)/(k log G(Rn(k) )) is the unique admissible estimator of 1 () under 1 a ¯ L0, (1 (), ) in the class of all estimators of the form of −A/(k log G(R n(k) )) (e.g., Ghosh and Singh, 1970). Let 0,1 (R(k) )= 0 (R(k) )+ a g(R(k) ), where g(R(k) ) = 0, (R(k) ) − 0 (R(k) ).Now, following Lemma 1 of Jafari Jozani et al. (2006a), we conclude that 1

a

,1 (R(k) ) = 0 (R(k) ) + (1 − )a0,1 (R(k) ) is an admissible estimator of 1 () under L, (1 (), ), and that N ,1 (R(k) ) is an inadmissible estimator of 1 () under BLF 0 N (regardless to the choice of target estimator 0 , 0 , ). In general (see Jafari Jozani et al., 2006a, Example 1) 1

,1 (R(k) ) = 0 (R(k) ) + (1 − )(A0 a0,1 (R(k) ) + B0 ) is inadmissible under loss L, (1 (), ) if 0

(i) A0 ∈ [0, n/(n − 2)], B0  0, or, (ii) A0 > n/(n − 2), or A0 < 0, or A0 = n/(n − 2), B0 0. Alternatively, if the target estimator is chosen as a0, (R(k) ), the linear estimators A a0, (R(k) ) + B are inadmissible under 1 1 L, (1 (), ) whenever one of the following conditions holds: 0

(i) A ∈ [, (n − 2)/(n − 2)], B  0, (ii) A > (n − 2)/(n − 2), or A < 0, or A = (n − 2)/(n − 2), B 0.

Remark 1. The results of this section tell us that for estimating 1 () for a model in (4), the estimator a, (R(k) ) is admissible 1 under information-weighted balanced SEL function, i.e., LI ,0 (, ) =  Under LI

0,0

( − 0 )2

2

 2  + (1 − ) −1 .



¯ (, ) = (/  − 1)2 it is well known that a0, (R(k) ) = −(n − 1)/(k log G(R n(k) )) is, among the class of linear estimators in 1

¯ −1/log G(R n(k) ), the unique admissible and minimax estimator of 1 () (e.g., Ghosh and Singh, 1970). Suppose the target estimator a N ¯ 0 has constant risk under LI0, (, ) as those proportional to −1/k log G(R n(k) ) such as 0,1 , or 0,1 . Now, Theorem 1 of Jafari 0 ¯ Jozani et al. (2006a) tells us that a, (R(k) ) is, among the class of linear estimators in −1/log G(R n(k) ), the unique admissible and 1 (1 (), ). minimax estimator of 1 () under LI ,0 ¯ Example 1. (i) Taking G(x) = e−x/(1−x) , 0 < x < 1, in (4) (i.e., X ∼ Y/(Y + 1) with Y ∼ Exp(1)) and choosing 0 (R(k) ) = ((n − 2)/k) (1 − Rn(k) )/Rn(k) as an admissible target estimator of  under L0, (1 (), ), the unique Bayes (and admissible estimator) of  under L, (1 (), ) is given by 0

,1 (R(k) ) =

(1 − )(n + )Rn(k) (n − 2) 1 − Rn(k) + . k Rn(k)  + ( − k)Rn(k)

0

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¯ (ii) Exponential distribution: For an exponential survival function in (4); i.e., G(x) = e−x ; and the admissible target estimator

0 (R(k) ) = (n − 2)/kRn(k) of , the unique Bayes estimator of  under L, (1 (), ) is given by 0

,1 (R(k) ) =

(n − 2) (1 − )(n + ) + . kRn(k)  + kRn(k)

For  = 0 the results of Jaheen (2004) are obtained as a special case, while, for  = 0 and k = 1, the results of Ahmadi and Doostparast (2006) are obtained as a special case. 3.2. Bayes estimator of 2 () = 1/  under balanced SEL ¯ In this case E[2 ()|r(k) ]=(1/(n+−1))(−k log G(r n(k) )), so that the (unique) Bayes estimator of 2 ()=1/  under L,0 (2 (), ) and prior density in (8) is given by

,2 (R(k) ) = 0 (R(k) ) + (1 − )

 − kUn(k) . n+−1

It is worth mentioning that, for estimating 2 () under L0, (2 (), ), linear estimators of the form of A0 Un(k) + B0 are admissible 0

whenever (Lehmann and Casella, 1998) (a) A0 ∈ [0, 1/(n + 1)], B0 > 0, or, (b) A0 = 1/(n + 1), B0 = 0.

So, by Lemma 1 of Jafari Jozani et al. (2006a), for estimating 2 () under loss L, (2 (), ), estimators of the form of 0 (R(k) ) + 0 (1 − )(A0 Un(k) + B0 ) are admissible whenever one of the conditions (a) or (b) holds. In particular, when the target estimator is chosen to be 0 (R(k) ) = Un(k) /(n + 1), we infer that linear estimators A Un(k) + B are admissible under loss L, (2 (), ) if: (a) 0

A ∈ [/(n + 1), 1/(n + 1)], B > 0; or (b) A = 1/(n + 1), B = 0. Alternatively, we can also show that A Un(k) + B is inadmissible under loss L, (2 (), ), with 0 (R(k) ) = Un(k) /(n + 1), if one of the following conditions holds: 0

1. A ∈ [/(n + 1), (n + 1 − )/n(n + 1)], B  0, 2. A < /(n + 1), or A > (n + 1 − )/n(n + 1), or A = (n + 1 − )/n(n + 1), B 0; (Lehmann and Casella, 1998; Jafari Jozani et al., 2006a). Remark 2. In estimating 2 () under the balanced SEL, an alternative choice of the target estimator is the MLE, M 0 = (k/n)Un(k) . M M In such a case we show as above that the estimators AM  Un(k) + B are admissible under L,M (2 (), ) whenever (a) A ∈ 0 M M M [/n, ( + n)/(n + 1)], BM  > 0; or (b) A = ( + n)/(n + 1), B = 0, and are inadmissible whenever (i) A ∈ [/n, 1/n], B  0, or M < /n, or (iv) AM = 1/n, BM 0. > 1/n, or (iii) A (ii) AM    

¯ Example 2 (Continued). (i) Again for the model with G(x) = e−x/(1−x) , 0 < x < 1, in (4), the unique Bayes estimator of 2 () = 1/  under L, (2 (), ) when 0 (R(k) ) = kRn(k) /(n + 1)(1 − Rn(k) ) with respect to the prior density in (8) reduces to 0

,2 (R(k) ) =



k − (1 − ) k + n+1 n+−1



Rn(k) 1 − Rn(k)

+

(1 − ) . n+−1

(ii) Exponential distribution: The unique Bayes (hence admissible) estimator of 2 () = 1/  under L, (2 (), ) when 0 (R(k) ) = kRn(k) /(n + 1) with respect to prior distribution (8) reduces to   k k(1 − ) (1 − ) + . ,2 (R(k) ) = Rn(k) + n+1 n+−1 n+−1

0

For  = 0 and k = 1, the results of Ahmadi and Doostparast (2006) are obtained as special cases. 3.3. Bayes estimator of 3 () = h (t) under balanced SEL For estimating the hazard rate function of a proportional hazard rate model as in (4) or (5) with () = , we have 3 () = ¯ h (t) = g(t)/ G(t). Thus, the problem of estimating 3 () is (essentially) equivalent to the one of estimating 1 () = . Namely, under L, (3 (), ) and based on k-record data, 0

E[3 ()|r(k) ] =

n+ g(t) , ¯ ¯  − k log G(r G(t) n(k) )

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so that the Bayes estimator of 3 () under L, (3 (), ) with respect to prior distribution (8) is given by 0

,3 (R(k) ) = 0 (R(k) ) + (1 − )

n+ g(t) . ¯ ¯  − k log G(R G(t) n(k) )

¯ ¯ Choosing the MLE of 3 () as the target estimator, i.e., 0 (R(k) ) = −(n/k log G(R n(k) ))g(t)/ G(t), the Bayes estimator ,3 (R(k) ) reduces to   g(t) (n + )(1 − ) n M (R ) = + . − ,3 (k) ¯ ¯ ¯ k log G(R )  − k log G(R ) G(t) n(k)

n(k)

Alternatively, if we choose the target estimator as the UMVUE of 3 (), the Bayes estimator ,3 (R(k) ) reduces to

U ,3 (R(k) ) =

g(t) ¯ G(t)

 −

(n + )(1 − ) (n − 1) + ¯ ¯ k log G(R )  − k log G(R n(k) n(k) )

 .

Notice that ,3 (R(k) ) is a unique Bayes estimator, has finite Bayes risk and hence is admissible. With the above equiva2R ¯ lence, we note that R, (3 (), ,3 (R(k) )) = {g(t)/ G(t)} ,0 (1 (), ,1 (R(k) )), so that, the admissibility or inadmissibility of 0 ¯ ,1 (R(k) ) for estimating 1 () under L, (1 (), ) is equivalent to the admissibility or inadmissibility of (g(t)/ G(t)) ,1 (R(k) ) 0

for estimating 3 () under L,(g(t)/ G(t)) ¯  (3 (), ). 0

¯ Example 3 (Pareto distribution). Taking G(x) = /x, x >  > 0, with known , in (4),then X has Pareto distribution. With the target estimator 0 chosen as the MLE of 3 () (given by n/(tk log(Rn(k) / ))), then the unique Bayes estimator of 3 () under BLF L, (3 (), ) with respect to a Gamma prior in (8) is given by 0

⎡

M ,3 (R(k) ) =

1 t

⎢ ⎢ ⎢ ⎢ ⎣

⎤

k log

n

+ (1 − ) Rn(k)



⎥ ⎥ n+

⎥ . Rn(k) ⎥ ⎦  + k log



¯ ) under balanced SEL 3.4. Bayes estimator of 4 () = F(t;

. ¯ For estimating the survival function (reliability function) of a distribution as in (4) with () = , we have 4 () = [G(t)] Thus, under L, (4 (), ) based on k-record data 0

 |R ] = ¯ ¯ )|R ] = E[(G(t)) E[F(t; (k) (k) ¯ Choosing 0 = MLE =[G(t)]



n+

 + kUn(k)

.

¯  + kU n(k) − log G(t)

(n/k)U −1

n(k) , the unique Bayes estimator of 

is obtained from (10) as ¯ ,4 (R(k) ) = [G(t)]

(n/k)U −1

n(k)

+ (1 − )

4 () with respect to a Gamma prior in (8) under L,0 (4 (), )

n+

 + kU n(k) ¯  + kUn(k) − log G(t)

.

¯ Example 4 (Exponential distribution (continued)). Let G(x) = e−x , x > 0, in (4). The unique Bayes estimator of the survival function −  t , under L, (4 (), ) with 0 = MLE , based on k-record data with respect to prior (reliability function), i.e., 4 () = e 0

distribution in (8) becomes



,4 (R(k) ) = ent/kRn(k) + (1 − ) 1 +

t  + kRn(k)

−(n+) .

Remark 3. One can also develop the previous framework for weighted balanced SEL. With the Fisher information equal to 1/ 2 for a distribution in (4) it is reasonable to choose the weight function q() in (7) as 1/ 2 . Then, for example, the Bayes estimators

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of 2 () and 4 () under L in (8) are given by

q

,0

, with q() = 1/ 2 and 0 = MLE , based on n upper k-record data with respect to a Gamma prior as

q,2 (R(k) ) = 0 (R(k) ) + (1 − )

¯  − k log G(R n(k) ) n+−3

q,4 (R(k) ) = 0 (R(k) ) + (1 − )

n+ ¯ ¯  − k log G(Rn(k) ) − log G(t)

and

¯ ¯ provided n +  − 3 > 0 and  − k log G(R n(k) ) − log G(t) > 0, respectively. 4. Bayesian estimation under balanced LINEX loss It has long been recognized that the commonly used SEL, and hence balanced SEL are inappropriate in many practical situations especially when overestimation and underestimation of the same magnitude have different consequences. A useful alternative to the balanced SEL is the asymmetric balanced LINEX loss function with shape parameter a (a0), obtained with the choice of ((), ) = ea(−()) − a( − ()) − 1, and q() = 1 in (3), and given by a(−0 ) − a( −  ) − 1} + (1 − ){ea(−()) − a( − ()) − 1}, L∗ 0 ,0 ((), ) = {e

where () is the parameter, 0 is a target estimator, and  ∈ [0, 1). A review of LINEX loss functions and their properties is given by Parsian and Kirmani (2002). Subject to finite posterior risk, the Bayes estimator of () under LINEX loss is given by ∗0, (R(k) ) = −(1/a) log(E[e−a() |R(k) ]). Hence, from Jafari Jozani et al. (2006b, Example 5) the unique Bayes estimator of () ((), ) with respect to the prior distribution () is given by under L∗ ,0 



∗

∗, (R(k) ) = − log e−a0 (R(k) ) + (1 − )e−a0, (R(k) ) . 1 a

(11)

¯ ), In this section, we pursue with the study of Bayesian estimation of 1 () = , 2 () = 1/ , and the survival function 4 () = F(t; for models in (4) with () = , based on n observed k-record data under L∗ ((), ) with respect to a Gamma prior as in (8). ,0 We present a similar analysis for a scale invariant balanced LINEX loss function in Section 4.4. 4.1. Bayes estimator of 1 () =  Under LINEX loss the Bayes estimator of 1 () =  with respect to the prior distribution (8) is given by 1 n+ log ∗0,1 (R(k) ) = − log(E[e−a1 () |R(k) ]) = a a



a 1+  + kU n(k)

provided a > −  − kU n(k) . Thus, from (11), the Bayes estimator of 1 () =  under L∗ ( (), ) is given by ,0 1 ⎧

n+ ⎫ ⎨ ⎬  + kU n(k) 1 −a0 (R(k) ) ∗ ,1 (R(k) ) = − log e + (1 − ) ⎩ ⎭ a  + kU n(k) + a provided a > −  − kU n(k) . Notice that: • lima→0 ∗, (R(k) ) = 0 (R(k) ) + (1 − )(n + )/( + kU n(k) ) = ,1 (R(k) ), where ,1 (R(k) ) is the unique Bayes estimator of 1 1 () under the balanced SEL function. • Since e−a1 () is convex in , Jensen's inequality leads to the inequality ∗, (R(k) )  ,1 (R(k) ) for a > 0. The relation is more 1 ( (), ) for a > 0 is smaller (with general of course but here it tells us that the Bayes estimator of 1 () =  under L∗ ,0 1 probability one) than the corresponding estimator under balanced SEL. −a0 (R(k) ) ∗N + (1 − )(1 + a/kU n(k) )−n } = ∗N • lim,→0 ∗, (R(k) ) = −(1/a) log{e ,1 (R(k) ), where ,1 (R(k) ) is the Bayes estimate 1 of 1 () under the balanced LINEX loss function with respect to non-informative prior () = 1/  (provided a > − kU n(k) ). ¯ • As in Section 3.3, the results here apply for estimating 3 () = g(t)/ G(t) under balanced LINEX loss function, L∗ ( (), ). ,0 3

J. Ahmadi et al. / Journal of Statistical Planning and Inference 139 (2009) 1180 -- 1189

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Example 5 (Exponential distribution (continued)). The unique Bayes estimator of 1 () under balanced LINEX loss with an arbitrary target estimator 0 with respect to a Gamma prior as in (8) is ⎧

n+ ⎫ ⎨ ⎬  + kRn(k) 1 −a0 (R(k) ) ∗ ,1 (R(k) ) = − log e + (1 − ) . ⎩ ⎭ a  + kRn(k) + a For  = 0, we obtain the results of Ahmadi et al. (2005) as a special case. 4.2. Bayes estimator of 2 () = 1/  For a > 0, we have ( + kun(k) )n+  ∞ −(a/ +(+ku )) n+−1 n(k)  e d

(n + ) 0 n+ ( + kun(k) ) = n (, , a, k, un(k) ),

(n + )

E[e−a2 () |r(k) ] =

where









n (, , a, k, tn(k) ) = 2an+ a( + kun(k) )K n + , 2 a( + kun(k) ) , and K( , ·) is the modified Bessel function of the third kind, see Gradshteyn and Ryzhik (2000). Thus, from (11), the Bayes ( (), ) with respect to the prior distribution (8) is given by estimator of 2 () = 1/  under L∗ ,0 4   ( + kun(k) )n+ 1 ∗,2 (r(k) ) = − log e−a0 (r(k) ) + (1 − ) n (, , a, k, un(k) ) . (12) a

(n + ) For  = 0, we obtain

∗0,2 (r(k) ) =

1 n+ 1 log (n + ) − log( + kun(k) ) − log n (, , a, k, un(k) ). a a a

(13)

One can use Lindley's method for obtaining an approximation of (12) or (13). Lindley (1980) derived the approximation:

⎤ ⎡   jR jU  ⎢ 2 ⎥ jU ⎢j R R()eU() d 1 1 j3 ⎥ ⎢ ⎥ ≈ R(0 ) − − ,  ⎢ ⎥ 2 j2 U/ j2 ⎣ j2 eU() d j2 U ⎦

j2

=0

where 0 is the root of the equation jU()/ j = 0. To obtain an approximation of (12), notice that  ∞

(n + ) = n+−1 e−(+kun(k) ) d, ( + kun(k) )n+ 0 and hence  ∞ −a/  (n+−1) log −(+ku ) n(k) d e e E[e−a/  |r(k) ] = 0  ∞ (n+−1) log −(+kun(k) ) d 0 e   2 2  a 2a 0 −a/  0 ≈e + − e−a/ 0 , 2(n +  − 1) 4 3 0

0

( (), ) where 0 = (n +  − 1)/( + kun(k) ). So, an approximation of the Bayes estimator of 2 () under balanced LINEX loss L∗ ,0 2 is given by    20 a2 1 2a ∗,2 (r(k) ) ≈ − log e−a0 (r(k) ) + (1 − )e−a/ 0 1 + − . a 2(n +  − 1) 4 3 0

Similarly, we obtain 1 1 ∗0,2 (r(k) ) ≈ − − log 0 a



1 1+ 20 (n +  − 1)



a2

0

− 2a

.

0

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Example 6 (Exponential distribution (continued)). The unique Bayes estimator of 2 () under balanced LINEX loss, with respect to prior distribution (8) and for 0 as the MLE of 2 (), is   ( + kRn(k) )n+ 1 ∗,2 (R(k) ) = − log e−(ak/n)Rn(k) + (1 − ) n (, , a, k, Rn(k) ) . a

(n + ) For  = 0 we obtain the results of Ahmadi et al. (2005) as a special case. Also Lindley's approximation yields    a( + kRn(k) ) a( + kRn(k) ) 1 −(ak/n)Rn(k) −a(+kRn(k) )/(n+−1) ∗ ,2 (r(k) ) ≈ − log e + (1 − )e 1− 2− a n+−1 2(n +  − 1)2 provided n +  − 1 > 0. ¯ ) 4.3. Bayes estimator of 4 () = F(t; From (9), we have 

¯

E[e−a4 () |r(k) ] = E[e−a[G(t)] |r(k) ]  ∞ n+  ( + kun(k) ) ¯ = e−a[G(t)] n+−1 e−(+kun(k) ) d.

(n + ) 0 Following Lindley's method, it is easy to verify that  ∞ −a () n+−1 −(+ku ) n(k) d 4 e  e E[e−a4 () |r(k) ] = 0  −  (  +ku ) ∞ n+−1 n(k) d e 0  ¯

0

≈ e−a(G(t))

¯ 0 2 a2 log2 G(t) ¯ e−a(G(t)) , − 0 2(n +  − 1)

where 0 = (n +  − 1)/( + kun(k) ). Thus 

∗,4 (R(k) ) ≈ − log e−a0 (R(k) ) + (1 − )e−a[G(t)] 1 a

¯

0



¯ 2 a2 log2 G(t) 1− 0 2(n +  − 1)

 (14)

( (), ) with respect to prior distribution is an approximation of the Bayes estimator of 4 () under balanced LINEX loss L∗ ,0 4 (8), and for an arbitrary target estimator 0 . Example 7 (Exponential distribution (continued)). Expression (14) becomes ⎧ ⎛ ⎞⎫ ⎨ 20 a2 R2n(k) ⎬ (−0 Rn(k) ) 1 −a0 (r(k) ) ∗ −ae ⎝1 − ⎠ , ,4 (r(k) ) ≈ − log e + (1 − )e ⎩ a 2(n +  − 1) ⎭ where 0 = (n +  − 1)/( + kRn(k) ), provided that n +  − 1 > 0. 4.4. Bayes estimation under modified balanced LINEX loss function One may wish to use the following scale invariant version of balanced LINEX loss:           LSI ((), ) =  ea(/ 0 −1) − a − 1 − 1 + (1 − ) ea(/ ()−1) − a −1 −1 , ,0 0 () with a0 and 0 being a target estimator of (). It can be shown that the Bayes estimator of () under LSI ((), ) based on a 0,0 sample of n upper k-record values is given by the following equation (e.g., Parsian and Sanjari, 1993): % & & $ $ 1 1 a0, (R(k) )/ () %% R(k) = ea E e |R(k) . E % () () Using Lemma 1 of Jafari Jozani et al. (2006b), the Bayes estimator of () under modified balanced LINEX loss function LSI ((), ) ,0 is given as a solution of the following equation with respect to , : % & & $ $  ea 1 a, (R(k) )/ () %% 1 a (R )/  (R ) R(k) = e , (k) 0 (k) + (1 − )E + (1 − )ea E e |R(k) . % 0 (R(k) ) 0 (R(k) ) () ()

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Now, for estimating 2 () = 1/  under LSI ((), ), the Bayes estimator 0, satisfies the following equation: 2 0,0 {a0, (R(k) )}

E[e

2

|R(k) ] = ea E[|R(k) ],

which reduces to

0,2 (R(k) ) = c()Un(k) + d(, ), where c() = [1 − e−a/(n++1) ]/a and d(, ) = c(). Note that 0 < c() < c∗ = 1/a(1 − e−a/n+1 ). Now, it follows from Parsian and Sanjari (1993) that the linear estimator A0 Un(k) +B0 is inadmissible under LSI ((), ) whenever (a) A0 < 0 or B0 < 0; or (b) A0 > c∗ , 0,0 B0  0; or (c) 0  A0 < c∗ , B0 = 0; and is admissible if A0 ∈ [0, c∗ ], B0  0. Unfortunately, for the case of  > 0, it seems difficult to obtain a closed form for the Bayes estimator of 2 () under LSI ((), ) and one may have to rely on a numerical evaluation. ,0 5. Concluding remarks In this paper, we showed how to develop Bayes estimation in the context of upper k-record data from a general class of distributions under some balanced type loss functions. We applied results from Jafari Jozani et al. (2006a,b), as well as from the literature on k-records. With respect to several criteria (choice of balanced type loss, choice of target estimator, underlying model and parameter of interest), the treatment is unified. Illustrations were given for the particular class of balanced squared error and LINEX losses. The admissibility or inadmissibility of some estimators was discussed. Lindley's approximation was illustrated in some cases. Finally, for a sequence of iid random variables X1 , X2 , . . . from the class of continuous distribution functions F(·; ) with F(x; ) = [H(x)]() ,

−∞  c < x < d  ∞, () > 0,

H an arbitrary continuous distribution function with H(c) = 0 and H(d) = 1, one can show that the results of this paper also hold true for lower k-record data with some minor modifications. The above family of distributions is well known in lifetime experiments, and referred to as proportional reversed hazard rate models (see Lawless, 2003). Acknowledgements The authors thank an anonymous referee for helpful comments and suggestions on an earlier version of this paper. References Ahmadi, J., Doostparast, M., 2006. Bayesian estimation and prediction for some life distributions based on record values. Statist. Papers 47, 373–392. Ahmadi, J., Doostparast, M., Parsian, A., 2005. Estimation and prediction in a two-parameter exponential distribution based on k-record values under LINEX loss function. Comm. Statist. Theory Methods 34, 795–805. Ahmadi, J., Jafari Jozani, M., Marchand, É., Parsian, A., 2008. Prediction of k-records from a general class of distributions under balanced type loss functions. Metrika, to appear. Ali Mousa, M.A.M., Jaheen, Z.F., Ahmad, A.A., 2002. Bayesian estimation, prediction and characterization for the Gumbel model based on records. Statistics 36, 65–74. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1998. Records. Wiley, New York. Chandler, K.N., 1952. The distribution and frequency of record values. J. Roy. Statist. Soc. Ser. B 14, 220–228. Dey, D., Ghosh, M., Strawderman, W.E., 1999. On estimation with balanced loss functions. Statist. Probab. Lett. 45, 97–101. Dziubdziela, W., Kopocinski, B., 1976. Limiting properties of the k-th record values. Zastosowania Mat. 15, 187–190. Ghosh, J.K., Singh, R., 1970. Estimation of the reciprocal of the scale parameter of a gamma density. Ann. Inst. Statist. Math. 22, 51–55. Glick, N., 1978. Breaking record and breaking boards. Amer. Math. Monthly 85, 2–26. Gradshteyn, I.S., Ryzhik, I.M., 2000. Tables of Integrals, Series, and Products. sixth ed. Academic Press, UK. Jafari Jozani, M., Marchand, É., Parsian, A., 2006a. On estimation with weighted balanced-type loss function. Statist. Probab. Lett. 76, 773–780. Jafari Jozani, M., Marchand, É., Parsian, A., 2006b. Bayes estimation under a general class of balanced loss functions. Rapport de recherche 36, Département de mathématiques, Université de Sherbrooke http://www.usherbrooke.ca/mathematiques/telechargement. Jaheen, Z.F., 2003. A Bayesian analysis of record statistics from the Gompertz model. Appl. Math. Comput. 145, 307–320. Jaheen, Z.F., 2004. Empirical Bayes analysis of record statistics based on LINEX and quadratic loss functions. Comput. Math. Appl. 47, 947–954. Lawless, J.L., 2003. Statistical Models and Methods for Lifetime Data. second ed. Wiley, New York. Lehmann, E.L., Casella, G., 1998. Theory of Point Estimation. Springer, New York Inc. Lindley, D.V., 1980. Approximate Bayesian methods. In: Bernardo, J., Degroot, M., Lindley, D., Smith, A. (Eds.), Bayesian Statistics, Valencia, pp. 223–245. Nagaraja, H.N., 1988. Record values and related statistics—a review. Comm. Statist. Theory Methods 17, 2223–2238. Nevzorov, V., 2001. Records: Mathematical Theory. Translation of Mathematical Monographs, vol. 194. American Mathematical Society, Providence, RI. Parsian, A., Kirmani, S.N.U.A., 2002. Estimation under LINEX loss function. In: Ullah, A., Wan, A.T.K., Chaturvedi, A. (Eds.), Handbook of Applied Econometrics and Statistical Inference. Inc., pp. 53–76. Parsian, A., Sanjari, N., 1993. On the admissibility and inadmissibility of estimators of scale parameters using asymmetric loss functions. Comm. Statist. Theory Methods 22, 2877–2901. Zellner, A., 1994. Bayesian and non-Bayesian estimation using balanced loss functions. In: Berger, J.O., Gupta, S.S. (Eds.), Statistical Decision Theory and Methods V. Springer, New York, pp. 337–390.