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

Jafar Ahmadi,^{a,}^{1}Mohammad Jafari Jozani,^{b,}^{2} Éric Marchand,^{c,}^{3} Ahmad Parsian,^{d,}^{4}^{*}

a School of Mathematical Sciences, Ferdowsi University of Mahshhad, P. O. Box 91775-1159, Mashhad, Iran bAllameh Tabatabaie University, Department of Statistics, Faculty of Economics & Statistical Research and

Training Center (SRTC), Tehran, Iran

c Université de Sherbrooke, Département de mathématiques, Sherbrooke Qc, CANADA, J1K 2R1, d School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran

Abstract

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 are developed; and in some cases the admissibility or inadmissibility of the linear estimators are 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.

Keywords: Admissibility; Bayes estimation; Balanced loss; Squared error loss; LINEX loss; k-records.

1 Introduction

1.1 Record data

Let {X_{i}, i ≥ 1} be a sequence of iid absolutely continuous random variables distributed according to the cumulative distribution function (cdf) F (·; θ) and probability density function (pdf) f(·; θ), where θ is an unknown parameter. An observation X_{j} is called an upper record value if its value exceeds all previous observations. Thus, X_{j} is an upper record if X_{j} > X_{i} 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, Balakrishnan, and Nagaraja (1998), Nevzorov (2001) and the references therein. For a formal definition of k-records, we consider the definition in Arnold, Balakrishnan and Nagaraja (1998, page 43) for the continuous case. Let T_{1,k} = k and, for n ≥ 2,

1Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged.

2Research supported by a grant of Statistical Research and Training Center, grant No. 8707.

3Research supported by NSERC of Canada 4Research supported by a grant of the Research Council of the University of Tehran *Corresponding author. Tel.:(+98) 21 6111 2624; Fax:(+98) 21 6641 2178.

E-mail: ahmadi@math.um.ac.ir (J. Ahmadi), mohammad.jafari.jozani@usherbrooke.ca (M. Jafari Jozani), eric.marchand@usherbrooke.ca(E. Marchand), ahmad_{−}p@khayam.ut.ac.ir, (A. Parsian).

T_{n,k} = min{j : j > T_{n−1,k}, X_{j} > X_{T_{n−1,k}−k+1:T_{n−1,k}}},

where X_{i:m} denotes the i-th order statistic in a sample of size m. The sequence of upper k-records are then defined by R_{n(k)} = X_{T_{n,k}−k+1:T_{n,k}} for n ≥ 1. For k = 1, note that the usual records are recovered. These sequences of k-records were introduced by 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 F_{1:k} ≡ 1 − {1 − F }^{k}. Using the joint density of usual records, the marginal density of R_{n(k)} is obtained as

f_{n,k}(r_{n(k)}; θ) = \frac{k^{n}}{(n − 1)!} [− log F̄(r_{n(k)}; θ)]^{n−1} \{F̄(r_{n(k)}; θ)\}^{k−1}f(r_{n(k)}; θ), (1)

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

f_{1,...,n}(r; θ) = k^{n}[F̄(r_{n(k)}; θ)]^{k} ∏_{i=1}^{n} \frac{f(r_{i(k)}; θ)}{\overline{F} (r_{i(k)}; θ)}, (2)

where r = (r_{1(k)}, …, r_{n(k)}) and F̄ = 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, Jaheen and Ahmad, 2002; Jaheen, 2003 and 2004; Ahmadi, Doostparast and Parsian, 2005; Ahmadi and Doostparast, 2006). The goal of this paper is to develop Bayes estimation of functions of θ 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 θ under the model X = (X_{1}, ..., X_{n}) ∼ F_{θ}, Zellner (1994) introduced a Balanced Loss Function (BLF) as follows:

\frac{ω}{n} ∑_{i=1}^{n}(X_{i} − δ)^{2} + (1 − ω)(δ − θ)^{2},

where ω ∈ [0, 1), and considered optimal estimates relative to BLF for estimation of a scalar mean, a vector mean and a vector regression coefficients. Dey, Ghosh and Strawderman (1999), as well as Jafari Jozani, Marchand and Parsian (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, Marchand and Parsian (2006b) introduced an extended class of balanced type loss functions of the form

L_{ρ,ω,δ_{0}}^{q}(γ(θ), δ) = ωq(θ)ρ(δ_{0}, δ) + (1 − ω)q(θ)ρ(γ(θ), δ), (3)

with q(·) being a suitable positive weight function, ρ(γ(θ), δ) being as arbitrary loss function in estimating γ(θ) by δ, and δ_{0} a chosen a priori “target” estimator of γ(θ), obtained for instance from the criterion of maximum likelihood, least squares or unbiasedness among others. They give a general Bayesian connection between the cases ω > 0 and ω = 0. For the case of squared error ρ, a least squares δ_{0}, γ(θ) = θ, and q(θ) = 1 in (3) is equivalent to Zellner’s (1994) BLF, and the introduction of an arbitrary ρ extends the squared error version of (3) introduced by Jafari Jozani, Marchand and Parsian (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 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 δ_{0}, the weight ω, and to some extent, with respect to the parameter being estimated.

1.3 Proportional hazard rate models

Let X_{1}, X_{2}, ... be a sequence of iid random variables from the family of continuous distribution functions with

F (x; θ) = 1 − [Ḡ(x)]^{α(θ)}, −∞ ≤ c < x < d ≤ ∞, (4)

where α(θ) > 0, G ≡ 1 − Ḡ, and G is an arbitrary continuous distribution function with G(c) = 0 and G(d) = 1. The family in (4) is 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) = \frac{d}{dx}G(x) be the corresponding pdf, then

f(x; θ) = α(θ)g(x)[Ḡ(x)]^{α(θ)−1}, −∞ ≤ c < x < d ≤ ∞. (5)

From (4), it can be easily shown that: (i) the Fisher information contained in a single observation X is I_{X}(θ) = (\frac{α′(θ)}{α(θ)})^{2}, (ii) E[− log Ḡ(X)] = 1/α(θ), and (iii) T (X) = − ∑_{i=1}^{n} log Ḡ(X_{i}) is a complete sufficient statistic for θ. In this paper we consider the estimation of γ_{1}(θ) = θ, γ_{2}(θ) = \frac{1}{θ}, the hazard rate function at t: γ_{3}(θ) = h_{θ}(t) = f(t; θ)/F̄(t; θ), and the survival function at t: γ_{4}(θ) = R_{θ}(t) = F̄(t; θ) 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 R_{1(k)} = r_{1(k)}, R_{2(k)} = r_{2(k)}, ⋯, R_{n(k)} = r_{n(k)} from a distribution with cdf given by (4) with α(θ) = θ. 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}(r; θ) = [kα(θ)]^{n}\{Ḡ(r_{n(k)})\}^{kα(θ)} ∏_{i=1}^{n} \frac{g(r_{i(k)})}{\overline{G}(r_{i(k)})}, (6)

where r = (r_{1(k)}, …, r_{n(k)}). Let U_{n(k)} = − log Ḡ(R_{n(k)}). Then from (6), U_{n(k)} is seen to be a complete and sufficient statistic for θ with a Gamma(n, kθ) distribution. Also, the maximum likelihood estimator (MLE) of θ is given by \frac{n}{k}U_{n(k)}^{−1}. With the completeness and sufficiency of U_{n(k)} based on the first n k-records for samples from (4), we obtain with Rao-Blackwell-Lehmann-Scheffe’s Theorem that,

• \frac{n−1}{k}U_{n(k)}^{−1} is the UMVUE of γ_{1}(θ) = θ,

• \frac{k}{n}U_{n(k)} is the UMVUE of γ_{2}(θ) = \frac{1}{θ},

• \frac{n−1}{k} \frac{g(t)}{\overline{G}(t)}U_{n(k)}^{−1} is the UMVUE of γ_{3}(θ) = h_{θ}(t),

• (1 + \frac{log Ḡ(t)}{k}U_{n(k)}^{−1})^{n−1} is the UMVUE of γ_{4}(θ) = R_{θ}(t).

3 Bayesian estimation under balanced SEL

In this section we obtain Bayes estimators of γ_{i}(θ); i = 1, 2, 3, 4; based on observed k-record data generated from (4) with α(θ) = θ, using balanced SEL given in (3) with ρ(γ, δ) = (δ − γ)^{2} and equal to

L_{ω,δ_{0}}^{q}(γ(θ), δ) = ωq(θ)(δ − δ_{0})^{2} + (1 − ω)q(θ)(δ − γ(θ))^{2}. (7) Using a conjugate Gamma(α, β) prior for θ with density,

π(θ) = \frac{β^{α}}{Γ(α)} θ^{α−1}e^{−βθ}, θ, α, β > 0, (8)

the posterior pdf of θ is given by

π(θ|r_{1(k)}, ..., r_{n(k)}) = \frac{(β + ku_{n(k)})^{n+α}}{Γ(n + α)} θ^{n+α−1}e^{−θ(β+ku_{n(k)})}, θ > 0, (9)

where u_{n(k)} = − log Ḡ(r_{n(k)}). The Bayes estimator of γ(θ) under L_{ω,δ_{0}}^{q}(γ(θ), δ) is given by (see Jafari Jozani, Marchand and Parsian, 2006a)

δ_{ω,γ}^{q}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{E[q(θ)γ(θ)|R_{(k)}]}{E[q(θ)|R_{(k)}]}, (10)

where R_{(k)} = (R_{1(k)}, ..., R_{n(k)}). As apparent from (10), δ_{ω,γ}^{q}(R_{(k)}) is a convex linear combination of the target estimator δ_{0}(R_{(k)}) (weight ω), and the Bayes estimator of γ(θ) under weighted SEL function. In the sequel, we pursue with the study of Bayes estimators of γ_{i}(θ), i = 1, 2, 3, 4 for q ≡ 1 in (7).

3.1 Bayes estimator of γ_{1}(θ) = θ under balanced SEL

With a Gamma prior as in (8), the Bayes estimator of γ_{1}(θ) = θ under SEL function is given by

δ_{0,γ_{1}}(R_{(k)}) = E[θ|R_{(k)}] = \frac{n + α}{β − k log Ḡ(R_{n(k)})}.

Using (10), the Bayes estimator of γ_{1}(θ) = θ under L_{ω,δ_{0}}(γ_{1}(θ), δ) with respect to a Gamma(α, β) prior is given by

δ_{ω,γ_{1}}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{n + α}{β − k log Ḡ(R_{n(k)})}.

Among others, a plausible target estimator δ_{0}(R_{(k)}) is the MLE of γ_{1}(θ), given by −n/ (k log Ḡ(R_{n(k)})). It may be noted that δ_{ω,γ_{1}}(R_{(k)}) is the unique Bayes estimator of γ_{1}(θ) = θ under L_{ω,δ_{0}}(γ_{1}(θ), δ) with finite Bayes risk and hence is admissible. Notice that as (α, β) → (0, 0),

δ_{ω,γ_{1}}(R_{(k)}) → δ_{ω,γ_{1}}^{N}(R_{(k)}) = ωδ_{0}(R_{(k)}) − (1 − ω)\frac{n}{k log Ḡ(R_{n(k)})}.

where δ_{ω,γ_{1}}^{N}(R_{(k)}) is the Bayes estimator of γ_{1}(θ) = θ under L_{ω,δ_{0}}(γ_{1}(θ), δ) with respect to the non informative prior π(θ) = \frac{1}{θ}. It is easy to see that under L_{0,δ}(γ_{1}(θ), δ), the estimator δ_{0,γ_{1}}^{N}(R_{(k)}) is dominated by −\frac{n−2}{k log Ḡ(R_{n(k)})}, provided that n > 2, and is an inadmissible estimator of γ_{1}(θ). In fact, δ_{0,γ_{1}}^{a}(R_{(k)}) = −\frac{n−2}{k log Ḡ(R_{n(k)})} is the unique admissible estimator of γ_{1}(θ) under L_{0,δ}(γ_{1}(θ), δ) in the class of all estimators of the form of −\frac{A}{k log Ḡ(R_{n(k)})} (e.g., Ghosh and Singh, 1970). Let δ_{0,γ_{1}}^{a}(R_{(k)}) = δ_{0}(R_{(k)}) + g(R_{(k)}), where g(R_{(k)}) = δ_{0,γ_{1}}^{a}(R_{(k)}) − δ_{0}(R_{(k)}). Now, following Lemma 1 of Jafari Jozani, Marchand and Parsian (2006a), we conclude that

δ_{ω,γ_{1}}^{a}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)δ_{0,γ_{1}}^{a}(R_{(k)}),

is an admissible estimator of γ_{1}(θ) under L_{ω,δ_{0}}(γ_{1}(θ), δ), and that δ_{ω,γ_{1}}^{N}(R_{(k)}) is an inadmissible estimator of γ_{1}(θ) under BLF (regardless to the choice of target estimator δ_{0}, δ_{0} ≠ δ_{ω,γ_{1}}^{N} ). In general (see Example 1 of Jafari Jozani, Marchand and Parsian, 2006a)

δ_{ω,γ_{1}}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)(A_{0}δ_{0,γ_{1}}^{a}(R_{(k)}) + B_{0}),

is inadmissible under loss L_{ω,δ_{0}}(γ_{1}(θ), δ) if

(i) A_{0} ∈ [0, \frac{n}{n−2}], B_{0} ≤ 0, or,

(ii) A_{0} > \frac{n}{n−2}, or A_{0} < 0, or A_{0} = \frac{n}{n−2}, B_{0} ≠ 0.

Alternatively, if the target estimator is chosen as δ_{0,γ_{1}}^{a}(R_{(k)}), the linear estimators A_{ω}δ_{0,γ_{1}}^{a}(R_{(k)}) + B_{ω} are inadmissible under L_{ω,δ_{0}}(γ_{1}(θ), δ) whenever one of the following conditions holds

(i) A_{ω} ∈ [ω, \frac{n−2ω}{n−2} ], B_{ω} ≤ 0,

(ii) A_{ω} > \frac{n−2ω}{n−2} , or A_{ω} < 0, or A_{ω} = \frac{n−2ω}{n−2} , B_{ω} ≠ 0.

Remark 1 The results of this section tell us that for estimating γ_{1}(θ) for a model in (4), the estimator δ_{ω,γ_{1}}^{a}(R_{(k)}) is admissible under information-weighted balanced SEL function, i.e.,

L_{ω,δ_{0}}^{I}(θ, δ) = ω \frac{(δ − δ_{0})^{2}}{θ^{2}} + (1 − ω)(\frac{δ}{θ} − 1)^{2}.

Under L_{0,δ_{0}}^{I}(θ, δ) = (\frac{δ}{θ} − 1)^{2} it is well known that δ_{0,γ_{1}}^{a}(R_{(k)}) = −\frac{n−1}{k log Ḡ(R_{n(k)})} is, among the class of linear estimators in −\frac{1}{log Ḡ(R_{n(k)})}, the unique admissible and minimax estimator of γ_{1}(θ) (e.g., Ghosh and Singh, 1970). Suppose the target estimator δ_{0} has constant risk under L_{0,δ_{0}}^{I}(θ, δ) as those proportional to −\frac{1}{k log Ḡ(R_{n(k)})} such as δ_{0,γ_{1}}^{a}, or δ_{0,γ_{1}}^{N}. Now, Theorem 1 of Jafari Jozani, Marchand and Parsian (2006a) tells us that δ_{ω,γ_{1}}^{a}(R_{(k)}) is, among the class of linear estimators in −\frac{1}{log Ḡ(R_{n(k)})}, the unique admissible and minimax estimator of γ_{1}(θ) under L_{ω,δ_{0}}^{I}(γ_{1}(θ), δ).

Example 1 (i) Taking Ḡ(x) = e^{− \frac{x}{1−x}}, 0 < x < 1, in (4) (i.e., X ∼ \frac{Y}{Y +1} with Y ∼ Exp(1)) and choosing δ_{0}(R_{(k)}) = \frac{n−2}{k} \frac{1−R_{n(k)}}{R_{n(k)}} as an admissible target estimator of θ under L_{0,δ_{0}}(γ_{1}(θ), δ), the unique Bayes (and admissible estimator) of θ under L_{ω,δ_{0}}(γ_{1}(θ), δ) is given by

δ_{ω,γ_{1}}(R_{(k)}) = \frac{(n − 2)ω}{k} \frac{1 − R_{n(k)}}{R_{n(k)}} + \frac{(1 − ω)(n + α)R_{n(k)}}{β + (β − k)R_{n(k)}} .

(ii) (Exponential distribution). For an exponential survival function in (4); i.e., Ḡ(x) = e^{−x}; and the admissible target estimator δ_{0}(R_{(k)}) = \frac{n−2}{kR_{n(k)}} of θ, the unique Bayes estimator of θ under L_{ω,δ_{0}}(γ_{1}(θ), δ) is given by

δ_{ω,γ_{1}}(R_{(k)}) = \frac{(n − 2)ω}{kR_{n(k)}} + \frac{(1 − ω)(n + α)}{β + kR_{n(k)}}.

For ω = 0 the results of Jaheen (2004) are obtained as a special case, while, for ω = 0 and k = 1, the results of Ahmadi and Doostparast (2006) are obtained as a special case.

3.2 Bayes estimator of γ_{2}(θ) = \frac{1}{θ} under balanced SEL

In this case E[γ_{2}(θ)|r_{(k)}] = \frac{1}{n+α−1}(β−k log Ḡ(r_{n(k)})), so that the (unique) Bayes estimator of γ_{2}(θ) = \frac{1}{θ} under L_{ω,δ_{0}}(γ_{2}(θ), δ) and prior density in (8) is given by

δ_{ω,γ_{2}}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{β − kU_{n(k)}}{n + α − 1} .

It is worth mentioning that, for estimating γ_{2}(θ) under L_{0,δ_{0}}(γ_{2}(θ), δ), linear estimators of the form of A_{0}U_{n(k)} + B_{0} are admissible whenever (Lehmann and Casella, 1998)

(a) A_{0} ∈ [0, \frac{1}{n+1}], B_{0} > 0, or,

(b) A_{0} = \frac{1}{n+1}, B_{0} = 0.

So, by Lemma 1 of Jafari Jozani, Marchand and Parsian (2006a), for estimating γ_{2}(θ) under loss L_{ω,δ_{0}}(γ_{2}(θ), δ), estimators of the form of ωδ_{0}(R_{(k)}) + (1 − ω)(A_{0}U_{n(k)} + B_{0}) are admissible whenever one of the conditions (a) or (b) holds. In particular, when the target estimator is chosen to be δ_{0}(R_{(k)}) = \frac{U_{n(k)}}{n+1} , we infer that linear estimators A_{ω}U_{n(k)} +B_{ω} are admissible under loss L_{ω,δ_{0}}(γ_{2}(θ), δ) if: (a) A_{ω} ∈ [\frac{ω}{n+1}, \frac{1}{n+1}], B_{ω} > 0; or (b) A_{ω} = \frac{1}{n+1}, B_{ω} = 0. Alternatively, we can also show that A_{ω}U_{n(k)} + B_{ω} is inadmissible under loss L_{ω,δ_{0}}(γ_{2}(θ), δ), with δ_{0}(R_{(k)}) = \frac{U_{n(k)}}{n+1} , if one of the following conditions holds:

1. A_{ω} ∈ [\frac{ω}{n+1}, \frac{n+1−ω}{n(n+1)}], B_{ω} ≤ 0,

2. A_{ω} < \frac{ω}{n+1}, or A_{ω} > \frac{n+1−ω}{n(n+1)}, or A_{ω} = \frac{n+1−ω}{n(n+1)}, B_{ω} ≠ 0;

(Lehmann and Casella, 1998; Jafari Jozani, Marchand and Parsian, 2006a).

Remark 2 In estimating γ_{2}(θ) under the balanced squared error loss, an alternative choice of the target estimator is the MLE, δ_{0}^{M} = \frac{k}{n}U_{n(k)}. In such a case we show as above that the estimators A_{ω}^{M}U_{n(k)} + B_{ω}^{M} are admissible under L_{ω,δ_{0}^{M}} (γ_{2}(θ), δ) whenever (a) A_{ω}^{M} ∈ [\frac{ω}{n}, \frac{ω+n}{n+1}], B_{ω}^{M} > 0; or (b) A_{ω}^{M} = \frac{ω+n}{n+1}, B_{ω} = 0, and are inadmissible whenever (i) A_{ω}^{M} ∈ [\frac{ω}{n}, \frac{1}{n}], B_{ω}^{M} ≤ 0, or (ii) A_{ω}^{M} > \frac{1}{n}, or (iii) A_{ω}^{M} < \frac{ω}{n}, or (iv) A_{ω}^{M} = \frac{1}{n}, B_{ω}^{M} ≠ 0.

Example 2 (continued) (i) Again for the model with Ḡ(x) = e^{− \frac{x}{1−x}}, 0 < x < 1, in (4), the unique Bayes estimator of γ_{2}(θ) = \frac{1}{θ} under L_{ω,δ_{0}}(γ_{2}(θ), δ) when δ_{0}(R_{(k)}) = \frac{kR_{n(k)}}{(n+1)(1−R_{n(k)})} with respect to the prior density in (8) reduces to

δ_{ω,γ_{2}}(R_{(k)}) = (\frac{kω}{n + 1} + \frac{k − (1 − ω)β}{n + α − 1} ) \frac{R_{n(k)}}{1 − R_{n(k)}} + \frac{(1 − ω)β}{n + α − 1}.

(ii) Exponential distribution: The unique Bayes (hence admissible) estimator of γ_{2}(θ) = \frac{1}{θ} under L_{ω,δ_{0}}(γ_{2}(θ), δ) when δ_{0}(R_{(k)}) = \frac{kR_{n(k)}}{(n+1)} with respect to prior distribution (8) reduces to

δ_{ω,γ_{2}}(R_{(k)}) = (\frac{kω}{n + 1} + \frac{k(1 − ω)}{n + α − 1}) R_{n(k)} + \frac{(1 − ω)β}{n + α − 1}.

For ω = 0 and k = 1, the results of Ahmadi and Doostparast (2006) are obtained as special cases.

3.3 Bayes estimator of γ_{3}(θ) = h_{θ}(t) under balanced SEL

For estimating the hazard rate function of a proportional hazard rate model as in (4) or (5) with α(θ) = θ, we have γ_{3}(θ) = h_{θ}(t) = θ \frac{g(t)}{Ḡ(t)}. Thus, the problem of estimating γ_{3}(θ) is (essentially) equivalent to the one of estimating γ_{1}(θ) = θ. Namely, under L_{ω,δ_{0}}(γ_{3}(θ), δ) and based on k-record data,

E[γ_{3}(θ)|r_{(k)}] = \frac{g(t)}{Ḡ(t)} \frac{n + α}{β − k log Ḡ(r_{n(k)})};

so that the Bayes estimator of γ_{3}(θ) under L_{ω,δ_{0}}(γ_{3}(θ), δ) with respect to prior distribution (8) is given by

δ_{ω,γ_{3}}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{g(t)}{Ḡ(t)} \frac{n + α}{β − k log Ḡ(R_{n(k)})}.

Choosing the MLE of γ_{3}(θ) as the target estimator, i.e., δ_{0}(R_{(k)}) = −\frac{n}{k log Ḡ(R_{n(k)})} \frac{g(t)}{Ḡ(t)}, the Bayes estimator δ_{ω,γ_{3}}(R_{(k)}) reduces to

δ_{ω,γ_{3}}^{M}(R_{(k)}) = \frac{g(t)}{Ḡ(t)} \{−\frac{nω}{k log Ḡ(R_{n(k)})} + \frac{(n + α)(1 − ω)}{β − k log Ḡ(R_{n(k)})}\} .

Alternatively, if we choose the target estimator as the UMVUE of γ_{3}(θ), the Bayes estimator δ_{ω,γ_{3}}(R_{(k)}) reduces to

δ_{ω,γ_{3}}^{U}(R_{(k)}) = \frac{g(t)}{Ḡ(t)} \{−\frac{(n − 1)ω}{k log Ḡ(R_{n(k)})} + \frac{(n + α)(1 − ω)}{β − k log Ḡ(R_{n(k)})}\} .

Notice that δ_{ω,γ_{3}}(R_{(k)}) is a unique Bayes estimator, has finite Bayes risk and hence is admissible. With the above equivalence, we note that R_{ω,δ_{0}}(γ_{3}(θ), δ_{ω,γ_{3}}(R_{(k)})) = \{\frac{g(t)}{Ḡ(t)}\}^{2} R_{ω,δ_{0}}(γ_{1}(θ), δ_{ω,γ_{1}}(R_{(k)})), so that, the admissibility or inadmissibility of δ_{ω,γ_{1}}(R_{(k)}) for estimating γ_{1}(θ) under L_{ω,δ_{0}}(γ_{1}(θ), δ) is equivalent to the admissibility or inadmissibility of \frac{g(t)}{Ḡ(t)}δ_{ω,γ_{1}}(R_{(k)}) for estimating γ_{3}(θ) under L_{ω, \frac{g(t)}{Ḡ(t)} δ_{0}}(γ_{3}(θ), δ).

Example 3 (Pareto distribution). Taking Ḡ(x) = \frac{η}{x}, x > η > 0, with known η, in (4), then X has Pareto distribution. With the target estimator δ_{0} chosen as the MLE of γ_{3}(θ) (given by n/(tk log(\frac{R_{n(k)}}{η})) ), then the unique Bayes estimator of γ_{3}(θ) under BLF L_{ω,δ_{0}}(γ_{3}(θ), δ) with respect to a Gamma prior in (8) is given by

δ_{ω,γ_{3}}^{M}(R_{(k)}) = \frac{1}{t} [ω \frac{n}{k log(\frac{R_{n(k)}}{η})} + (1 − ω)\frac{n + α}{β + k log(\frac{R_{n(k)}}{η})}] .

3.4 Bayes estimator of γ_{4}(θ) = F̄(t; θ) under balanced SEL

For estimating the survival function (reliability function) of a distribution as in (4) with α(θ) = θ, we have γ_{4}(θ) = [Ḡ(t)]^{θ}. Thus, under L_{ω,δ_{0}}(γ_{4}(θ), δ) based on k-record data

E[F̄(t; θ)|R_{(k)}] = E [(Ḡ(t))^{θ}|R_{(k)}] = (\frac{β + kU_{n(k)}}{β + kU_{n(k)} − log Ḡ(t)})^{n+α} .

Choosing δ_{0} = δ_{MLE} = [Ḡ(t)]^{\frac{n}{k} U_{n(k)}^{−1}}, the unique Bayes estimator of γ_{4}(θ) with respect to a Gamma prior in (8) under L_{ω,δ_{0}}(γ_{4}(θ), δ) is obtained from (10) as

δ_{ω,γ_{4}}(R_{(k)}) = ω[Ḡ(t)]^{\frac{n}{k} U_{n(k)}^{−1}} + (1 − ω) (\frac{β + kU_{n(k)}}{β + kU_{n(k)} − log Ḡ(t)})^{n+α} .

Example 4 Exponential distribution (continued): Let Ḡ(x) = e^{−x}, x > 0, in (4). The unique Bayes estimator of the survival function (reliability function), i.e., γ_{4}(θ) = e^{−θt}, under L_{ω,δ_{0}}(γ_{4}(θ), δ) with δ_{0} = δ_{MLE}, based on k-record data with respect to prior distribution in (8) becomes

δ_{ω,γ_{4}}(R_{(k)}) = ωe^{\frac{nt}{kR_{n(k)}}} + (1 − ω) (1 + \frac{t}{β + kR_{n(k)}} )^{−(n+α)} .

Remark 3 One can also develop the previous framework for weighted balanced squared error loss. With the Fisher information equal to 1/θ^{2} for a distribution in (4) it is reasonable to choose the weight function q(θ) in (7) as 1/θ^{2}. Then, for example, the Bayes estimators of γ_{2}(θ) and γ_{4}(θ) under L_{ω,δ_{0}}^{q}, with q(θ) = 1/θ^{2} and δ_{0} = δ_{MLE}, based on n upper k-record data with respect to a Gamma prior as in (8) are given by

δ_{ω,γ_{2}}^{q}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{β − k log Ḡ(R_{n(k)})}{n + α − 3},

and

δ_{ω,γ_{4}}^{q}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{n + α}{β − k log Ḡ(R_{n(k)}) − log Ḡ(t)},

provided n + α − 3 > 0 and β − k log Ḡ(R_{n(k)}) − log Ḡ(t) > 0, respectively.

4 Bayesian estimation under balanced LINEX loss

It has long been recognized that the commonly used squared error loss, and hence balanced squared error loss 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 (a ≠ 0), obtained with the choice of ρ(γ(θ), δ) = e^{a(δ−γ(θ))} − a(δ − γ(θ)) − 1, and q(θ) = 1 in (3), and given by

L_{ω,δ_{0}}^{∗}(γ(θ), δ) = ω{e^{a(δ−δ_{0})} − a(δ − δ_{0}) − 1} + (1 − ω){e^{a(δ−γ(θ))} − a(δ − γ(θ)) − 1},

where γ(θ) is the parameter, δ_{0} is a target estimator, and ω ∈ [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 γ(θ) under LINEX loss is given by δ_{0,γ}^{∗}(R_{(k)}) = −\frac{1}{a} log(E[e^{−aγ(θ)}|R_{(k)}]). Hence, from Jafari Jozani, Marchand and Parsian (2006b, Example 5) the unique Bayes estimator of γ(θ) under L_{ω,δ_{0}}^{∗}(γ(θ), δ) with respect to the prior distribution π(θ) is given by

δ_{ω,γ}^{∗}(R_{(k)}) = −\frac{1}{a} log \{ωe^{−aδ_{0}(R_{(k)})} + (1 − ω)e^{−aδ_{0,γ}^{∗}(R_{(k)})}\} . (11)

In this section, we pursue with the study of Bayesian estimation of γ_{1}(θ) = θ, γ_{2}(θ) = \frac{1}{θ}, and the survival function γ_{4}(θ) = F̄(t; θ), for models in (4) with α(θ) = θ, based on n observed k-record data under L_{ω,δ_{0}}^{∗}(γ(θ), δ) with respect to a Gamma prior as in (8). We present a similar analysis for a scale invariant balanced LINEX loss function in subsection 4.4.

4.1 Bayes estimator of γ_{1}(θ) = θ

Under LINEX loss the Bayes estimator of γ_{1}(θ) = θ with respect to the prior distribution (8) is given by

δ_{0,γ_{1}}^{∗}(R_{(k)}) = −\frac{1}{a} log (E[e^{−aγ_{1}(θ)}|R_{(k)}]) = \frac{n + α}{a} log (1 + \frac{a}{β + kU_{n(k)}} ) ,

provided a > −β − kU_{n(k)}. Thus, from (11), the Bayes estimator of γ_{1}(θ) = θ under L_{ω,δ_{0}}^{∗}(γ_{1}(θ), δ) is given by

δ_{ω,γ_{1}}^{∗}(R_{(k)}) = −\frac{1}{a} log \{ωe^{−aδ_{0}(R_{(k)})} + (1 − ω) (\frac{β + kU_{n(k)}}{β + kU_{n(k)} + a})^{n+α}\} ,

provided a > −β − kU_{n(k)}. Notice that:

• lim_{a→0} δ_{ω,γ_{1}}^{∗}(R_{(k)}) = ωδ_{0}(R_{(k)}) + (1 − ω)\frac{n+α}{β+kU_{n(k)}} = δ_{ω,γ_{1}}(R_{(k)}), where δ_{ω,γ_{1}}(R_{(k)}) is the unique Bayes estimator of γ_{1}(θ) under the balanced SEL function.

• Since e^{−aγ_{1}(θ)} is convex in θ, Jensen’s inequality leads to the inequality δ_{ω,γ_{1}}^{∗}(R_{(k)}) ≤ δ_{ω,γ_{1}}(R_{(k)}) for a > 0. The relation is more general of course but here it tells us that the Bayes estimator of γ_{1}(θ) = θ under L_{ω,δ_{0}}^{∗}(γ_{1}(θ), δ) for a > 0 is smaller (with probability one) than the corresponding estimator under balanced SEL.

• lim_{α,β→0} δ_{ω,γ_{1}}^{∗}(R_{(k)}) = −\frac{1}{a} log\{ωe^{−aδ_{0}(R_{(k)})} +(1−ω)(1+ \frac{a}{kU_{n(k)}} )^{−n}\} = δ_{ω,γ_{1}}^{∗N}(R_{(k)}), where δ_{ω,γ_{1}}^{∗N}(R_{(k)}) is the Bayes estimate of γ_{1}(θ) under the balanced LINEX loss function with respect to non informative prior π(θ) = \frac{1}{θ} (provided a > −kU_{n(k)}).

• As in section 3.3, the results here apply for estimating γ_{3}(θ) = θ \frac{g(t)}{Ḡ(t)} under balanced LINEX loss function, L_{ω,δ_{0}}^{∗}(γ_{3}(θ), δ).

Example 5 Exponential distribution (continued): The unique Bayes estimator of γ_{1}(θ) under balanced LINEX loss with an arbitrary target estimator δ_{0} with respect to a Gamma prior as in (8) is

δ_{ω,γ_{1}}^{∗}(R_{(k)}) = −\frac{1}{a} log \{ωe^{−aδ_{0}(R_{(k)})} + (1 − ω) (\frac{β + kR_{n(k)}}{β + kR_{n(k)} + a})^{n+α}\} .

For ω = 0, we obtain the results of Ahmadi, Doostparast, and Parsian (2005) as a special case.

4.2 Bayes estimator of γ_{2}(θ) = 1/θ

For a > 0, we have

E[e^{−aγ_{2}(θ)}|r_{(k)}] = \frac{(β + ku_{n(k)})^{n+α}}{Γ(n + α)} ∫_{0}^{∞} e^{−( \frac{a}{θ} +θ(β+ku_{n(k)}))}θ^{n+α−1}dθ
= \frac{(β + ku_{n(k)})^{n+α}}{Γ(n + α)} Φ_{n}(α, β, a, k, u_{n(k)}),

where

Φ_{n}(α, β, a, k, t_{n(k)}) = 2a^{n+α}\sqrt{a(β + ku_{n(k)})} K (n + α, 2\sqrt{a(β + ku_{n(k)})}) ,

and K(ν, ·) is the modified Bessel function of the third kind, see Gradshteyn and Ryzhik (2000). Thus, from (11), the Bayes estimator of γ_{2}(θ) = 1/θ under L_{ω,δ_{0}}^{∗}(γ_{4}(θ), δ) with respect to the prior distribution (8) is given by

δ_{ω,γ_{2}}^{∗}(r_{(k)}) = −\frac{1}{a} log \{ωe^{−aδ_{0}(r_{(k)})} + (1 − ω)\frac{(β + ku_{n(k)})^{n+α}}{Γ(n + α)} Φ_{n}(α, β, a, k, u_{n(k)})\} . (12)

For ω = 0, we obtain

δ_{0,γ_{2}}^{∗}(r_{(k)}) = \frac{1}{a} log Γ(n + α) − \frac{n + α}{a} log(β + ku_{n(k)}) − \frac{1}{a} log Φ_{n}(α, β, a, k, u_{n(k)}). (13)

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

\frac{∫ R(θ)e^{U(θ)}dθ}{∫ e^{U(θ)}dθ} ≈ R(θ_{0}) − \frac{1}{2} \frac{1}{∂^{2}U/∂θ^{2}} [\frac{∂^{2}R}{∂θ^{2}} − \frac{(\frac{∂R}{∂U})(\frac{∂U}{∂θ^{3}} )}{\frac{∂^{2}U}{∂θ^{2}}}]_{θ=θ_{0}} ,

where θ_{0} is the root of the equation \frac{∂U(θ)}{∂θ} = 0. To obtain an approximation of (12), notice that

\frac{Γ(n + α)}{(β + ku_{n(k)})^{n+α}} = ∫_{0}^{∞} θ^{n+α−1}e^{−θ(β+ku_{n(k)})}dθ,

and hence

E[e^{− \frac{a}{θ}} |r_{(k)}] = \frac{∫_{0}^{∞} e^{− \frac{a}{θ}} e^{(n+α−1) log θ−θ(β+ku_{n(k)})}dθ}{∫_{0}^{∞} e^{(n+α−1) log θ−θ(β+ku_{n(k)})}dθ}
≈ e^{− \frac{a}{θ_{0}}} + \frac{θ_{0}^{2}}{2(n + α − 1)} [\frac{a^{2}}{θ_{0}^{4}} − \frac{2a}{θ_{0}^{3}}] e^{− \frac{a}{θ_{0}}} ,

where θ_{0} = \frac{n+α−1}{β+ku_{n(k)}} . So, an approximation of the Bayes estimator of γ_{2}(θ) under balanced LINEX loss L_{ω,δ_{0}}^{∗}(γ_{2}(θ), δ) is given by

δ_{ω,γ_{2}}^{∗}(r_{(k)}) ≈ −\frac{1}{a} log (ωe^{−aδ_{0}(r_{(k)})} + (1 − ω)e^{− \frac{a}{θ_{0}}} \{1 + \frac{θ_{0}^{2}}{2(n + α − 1)} [\frac{a^{2}}{θ_{0}^{4}} − \frac{2a}{θ_{0}^{3}}]\}) .

Similarly, we obtain

δ_{0,γ_{2}}^{∗}(r_{(k)}) ≈ −\frac{1}{θ_{0}} − \frac{1}{a} log (1 + \frac{1}{2θ_{0}(n + α − 1)}(\frac{a^{2}}{θ_{0}} − 2a)) .

Example 6 Exponential distribution (continued): The unique Bayes estimator of γ_{2}(θ) under balanced LINEX loss, with respect to prior distribution (8) and for δ_{0} as the MLE of γ_{2}(θ), is

δ_{ω,γ_{2}}^{∗}(R_{(k)}) = −\frac{1}{a} log\{ωe^{− \frac{ak}{n} R_{n(k)}} + (1 − ω)\frac{(β + kR_{n(k)})^{n+α}}{Γ(n + α)} Φ_{n}(α, β, a, k, R_{n(k)})\}.

For ω = 0 we obtain the results of Ahmadi, Doostparast and Parsian (2005) as a special case. Also Lindley’s approximation yields

δ_{ω,γ_{2}}^{∗}(r_{(k)}) ≈ −\frac{1}{a} log (ωe^{− \frac{ak}{n} R_{n(k)}} + (1 − ω)e^{− \frac{a(β+kR_{n(k)})}{n+α−1}} \{1 − \frac{a(β + kR_{n(k)})}{2(n + α − 1)^{2}} [2 − \frac{a(β + kR_{n(k)})}{n + α − 1}]\}) ,

provided n + α − 1 > 0.

4.3 Bayes estimator of γ_{4}(θ) = F̄(t; θ)

From (9), we have

E[e^{−aγ_{4}(θ)}|r_{(k)}] = E[e^{−a[Ḡ(t)]^{θ}}|r_{(k)}]
= ∫_{0}^{∞} e^{−a[Ḡ(t)]^{θ}} \frac{(β + ku_{n(k)})^{n+α}}{Γ(n + α)} θ^{n+α−1}e^{−θ(β+ku_{n(k)})}dθ.

Following Lindley’s method, it is easy to verify that

E[e^{−aγ_{4}(θ)}|r_{(k)}] = \frac{∫_{0}^{∞} e^{−aγ_{4}(θ)}θ^{n+α−1}e^{−θ(β+ku_{n(k)})}dθ}{∫_{0}^{∞} θ^{n+α−1}e^{−θ(β+ku_{n(k)})}dθ}
≈ e^{−a(Ḡ(t))^{θ_{0}}} − \frac{θ_{0}^{2}a^{2} log^{2} Ḡ(t)}{2(n + α − 1)}e^{−a(Ḡ(t))^{θ_{0}}} ,

where θ_{0} = \frac{n+α−1}{β+ku_{n(k)}} . Thus

δ_{ω,γ_{4}}^{∗}(R_{(k)}) ≈ −\frac{1}{a} log \{ωe^{−aδ_{0}(R_{(k)})} + (1 − ω)e^{−a[Ḡ(t)]^{θ_{0}}} (1 − \frac{θ_{0}^{2}a^{2} log^{2} Ḡ(t)}{2(n + α − 1)})\} (14)

is an approximation of the Bayes estimator of γ_{4}(θ) under balanced LINEX loss L_{ω,δ_{0}}^{∗}(γ_{4}(θ), δ) with respect to prior distribution (8), and for an arbitrary target estimator δ_{0}. Example 7 Exponential distribution (continued): Expression (14) becomes

δ_{ω,γ_{4}}^{∗}(r_{(k)}) ≈ −\frac{1}{a} log \{ωe^{−aδ_{0}(r_{(k)})} + (1 − ω)e^{−ae^{(−θ_{0}R_{n(k)})}} (1 − \frac{θ_{0}^{2}a^{2}R_{n(k)}^{2}}{2(n + α − 1)})\} ,

where θ_{0} = \frac{n+α−1}{β+kR_{n(k)}} , provided that n + α − 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:

L_{ω,δ_{0}}^{SI}(γ(θ), δ) = ω \{e^{a( \frac{δ}{δ_{0}} −1)} − a(\frac{δ}{δ_{0}} − 1) − 1\} + (1 − ω) \{e^{a( \frac{δ}{γ(θ)} −1)} − a(\frac{δ}{γ(θ)} − 1) − 1\} ,

with a ≠ 0 and δ_{0} being a target estimator of γ(θ). It can be shown that the Bayes estimator of γ(θ) under L_{0,δ_{0}}^{SI}(γ(θ), δ) based on a sample of n upper k-record values is given by the following equation (e.g., Parsian and Sanjari, 1993),

E [\frac{1}{γ(θ)}e^{a \frac{δ_{0,γ} (R_{(k)})}{γ(θ)}} |R_{(k)}] = e^{a}E[\frac{1}{γ(θ)}|R_{(k)}].

Using Lemma 1 of Jafari Jozani, Marchand and Parsian (2006b), the Bayes estimator of γ(θ) under modified balanced LINEX loss function L_{ω,δ_{0}}^{SI}(γ(θ), δ) is given as a solution of the following equation with respect to δ_{ω,γ},

\frac{ω}{δ_{0}(R_{(k)})}e^{a \frac{δω,γ (R_{(k)})}{δ_{0}(R_{(k)})}} + (1 − ω)E [\frac{1}{γ(θ)}e^{a \frac{δω,γ (R_{(k)})}{γ(θ)}} |R_{(k)}] = \frac{ωe^{a}}{δ_{0}(R_{(k)})} + (1 − ω)e^{a}E[\frac{1}{γ(θ)}|R_{(k)}].

Now, for estimating γ_{2}(θ) = \frac{1}{θ} under L_{0,δ_{0}}^{SI}(γ(θ), δ), the Bayes estimator δ_{0,γ_{2}} satisfies the following equation,

E[θe^{{aθδ_{0,γ_{2}}(R_{(k)})}}|R_{(k)}] = e^{a}E[θ|R_{(k)}],

which reduces to

δ_{0,γ_{2}}(R_{(k)}) = c(α)U_{n(k)} + d(α, β),

where c(α) = [1 − e^{− \frac{a}{n+α+1}} ]/a and d(α, β) = c(α)β. Note that 0 < c(α) < c^{∗} = \frac{1}{a}(1 − e^{− \frac{a}{n+1}} ). Now, it follows from Parsian and Sanjari (1993) that the linear estimator A_{0}U_{n(k)} + B_{0} is inadmissible under L_{0,δ_{0}}^{SI}(γ(θ), δ) whenever (a) A_{0} < 0 or B_{0} < 0; or (b) A_{0} > c^{∗}, B_{0} ≥ 0; or (c) 0 ≤ A_{0} < c^{∗}, B_{0} = 0; and is admissible if A_{0} ∈ [0, c^{∗}], B_{0} ≥ 0. Unfortunately, for the case of ω > 0, it seems difficult to obtain a closed form for the Bayes estimator of γ_{2}(θ) under L_{ω,δ_{0}}^{SI}(γ(θ), δ) and one may have to rely on a numerical evaluation.

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, Marchand and Parsian (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 X_{1}, X_{2}, ... from the class of continuous distribution functions F (·; θ) with

F (x; θ) = [H(x)]^{β(θ)}, −∞ ≤ c < x < d ≤ ∞, β(θ) > 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.

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