Prediction of k-records from a general class of distributions under balanced type loss functions

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

a School of Mathematical Sciences, Ferdowsi University of Mahshhad, P. O. Box 91775-1159, Mashhad, Iran

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

Abstract

We study the problem of predicting future k-records based on k-record data for a large class of distributions, which includes several well-known distributions such as: Exponential, Weibull (one parameter), Pareto, Burr type XII, among others. With both Bayesian and non-Bayesian approaches being investigated here, we pay more attention to Bayesian predictors under balanced type loss functions as introduced by Jafari Jozani, Marchand and Parsian (2006b). The results are presented under the balanced versions of some well-known loss functions, namely squared error loss (SEL), Varian’s linear-exponential (LINEX) loss and absolute error loss (AEL) or L_{1} loss functions. Some of the previous results in the literatures such as Ahmadi et al. (2005), and Raqab et al. (2007) can be achieved as special cases of our results.

Keywords: Absolute value error loss; Balanced loss function; Bayes prediction; Conditional median prediction; Maximum likelihood prediction; LINEX loss; Record values. AMS Subject Classification Primary 62G30; Secondary 62F15.

1 Introduction

Consider a sequence {X_{i}, i ≥ 1} of iid absolutely continuous random variables distributed according to the cumulative distribution function (cdf) F (x; θ) and probability density function (pdf) f(x; θ), where θ is an unknown parameter. An observation X_{j} will be 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. Records of iid random variables and their properties have been extensively studied in the literature. Interest in records has increased steadily over the years since Chandler (1952) formulated the theory of records. See Arnold et al. (1998), Nevzorov (2001) and the references therein for more details on applications of records. An upper k-record process is defined in terms of the k-th largest X yet seen. For a formal definition, we consider the definition in Arnold et al. (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 partially by a grant of Statistical Research and Training Center, Tehran, Iran 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. Using the joint density of usual records, the joint pdf of the first n, k-records R_{(k)} = (R_{1(k)}, ..., R_{n(k)}) is given by

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

where r_{(k)} = (r_{1(k)}, ..., r_{n(k)}) and F̄ ≡ 1 − F, see, Arnold et al. (1998). The problem of prediction of future usual (k=1) records based on observed record data has been extensively studied by several statisticians in view of classical and Bayesian framework, see Ahsanullah (1980), Awad and Raqab (2000), Dunsmore (1983), Ali Mousa et al. (2000), Jaheen (2003, 2004), Madi and Raqab (2004), Ahmadi and Doostparast (2006) and Doostparast and Ahmadi (2006) among others. Ahmadi et al. (2005) developed Bayesian inference and prediction based on k-record values under a LINEX loss function. The record statistics are of interest and importance, these statistics are applied in estimating strength of materials, predicting natural disasters, sports achievements, etc. Consider a technical systems or subsystems with k-out-of-n structure. A k-out-of-n system breaks down at the time of the (n−k+1)-th component failure. So, in reliability analysis, the life length of a k-out-of-n system is the (n−k +1)-th order statistic in a sample of size n. Consequently, the n-th upper k-record value can be regarded as the life length of a k-out-of-T_{n,k} system. Several application of k-record values can be found in the literature, for instance, see the examples cited in Kamps (1995) or Danielak and Raqab (2004b) in reliability theory. Suppose that a technical system is subject to shocks, e.g. peaks of voltage. These shocks may be modeled as realizations of records. If not record values themselves, but second or third largest values are of special interest, then the model of k-record values is adequate. When record values themselves are viewed as outliers, then second or third largest values are of special interest. Insurance claims in some non-life insurance can be used as an example. So predicting future k-record values is an important problem. Several statisticians have investigated upper bounds for the expectation of future k-records, see Klimczak (2006), Klimczak and Rychlik (2004), Raqab (2004), Danielak and Raqab (2004a, b) among others. One of the motivations here is the relative paucity of work concerning a Bayesian approach to predicting future k-records. A second important feature of this paper is the proposed use of balanced type loss functions, which to us represent an interesting tool for decision making, or specifically here for predictions. In decision theory, the loss function usually focusses on precision of estimation. However, goodness of fit is also a very important criterion. Zellner (1994), introduced the notion of a balanced loss function in the context of a general linear model to reflect both goodness of fit and precision of estimation. Often loss functions reflecting one or the other of these two criteria, but not both have been employed in statistical inferences. For example, least squares estimation reflects goodness of fit considerations whereas the use of quadratic or LINEX loss functions involve a sole emphasis on precision of estimation. Following Jafari Jozani, Marchand and Parsian (2006b) we introduce a general class of balanced type loss functions of the form of

L_{ρ,ω,d_{0}}(y, d) = ωρ(d_{0}, d) + (1 − ω)ρ(y, d), (2)

where ρ(y, d) is an arbitrary loss function in predicting y by d, while d_{0} is a chosen a priori “target” predictor of y, obtained for instance from the criterion of maximum likelihood predictor, least square or unbiasedness among others. Our objective is to predict a future k-record based on a sample of observed k-record data from Bayesian point of view for a general class of distributions under balanced type loss functions as in (2). As first modelled by Jafari Jozani, Marchand and Parsian (2006a) for a squared error ρ in (2), the choice of the target d_{0} in (2) is arbitrary. This constitutes a particularly appealing feature with the Bayesian developments of Section 4 being applicable for any target predictor d_{0}, such as those presented in Section 3. In this paper, we shall use balanced squared error loss (BSEL), and balanced absolute error loss (BAEL) or balanced L_{1} loss functions as symmetric BLFs and balanced LINEX loss function as an asymmetric BLF to derive the Bayes predictions of the future k-records in a general class of distributions based on a sample of k-record values observed.

2 Preliminaries

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

F (x; θ) = 1 − [Ḡ(x)]^{α(θ)}, −∞ ≤ c < x < d ≤ ∞, (3)

where α(θ) > 0 and G ≡ 1 − Ḡ is an arbitrary continuous distribution function, free of unknown parameters with G(c) = 0 and G(d) = 1. The family in C_{1} is well-known in the lifetime experiments as proportional hazard models (see for example Lawless, 2003), which includes several well-known lifetime distributions such as: Exponential, Pareto, Lomax, Burr type XII, Resnick, Weibull (one parameter) among others. Also, the family in C_{1} is a subclass of a regular one parameter exponential family of distributions. Let g(x) = \frac{d}{dx}G(x) be the pdf corresponding to G(x), then

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

Suppose that R_{1(k)}, R_{2(k)}, ... is the sequence of k-records coming from (3) with α(θ) = θ, then the following can be obtained easily ( see for instance Arnold et al., 1998):

• The joint pdf of R_{(k)} = (R_{1(k)}, ..., R_{n(k)}) is given by

f_{1,...,n}(r_{1(k)}, ..., r_{n(k)}; θ) = (kθ)^{n}\{Ḡ(r_{n(k)})\}^{kθ} ∏_{i=1}^{n} \frac{g(r_{i(k)})}{Ḡ(r_{i(k)})}. (5)

• The marginal pdf of the nth k-records, R_{n(k)}, is given by

f_{n}(r_{n(k)}; θ) = \frac{(kθ)^{n}}{(n − 1)!}g(r_{n(k)})\{Ḡ(r_{n(k)})\}^{kθ−1}\{− log Ḡ(r_{n(k)})\}^{n−1}. (6)

• The conditional pdf of R_{s(k)} given R_{n(k)} = r_{n(k)}, s > n is

f_{s|n}(r_{s(k)}|r_{n(k)}, θ) = \frac{(kθ)^{s−n}}{(s − n − 1)!} (log \frac{Ḡ(r_{n(k)})}{Ḡ(r_{s(k)})})^{s−n−1} \frac{g(r_{s(k)})}{Ḡ(r_{s(k)})} (\frac{Ḡ(r_{s(k)})}{Ḡ(r_{n(k)})})^{kθ} . (7)

The following results are an immediate consequences of (5) and (6):

• R_{n(k)} is a complete sufficient statistic for θ among the first n k-records.

• − log Ḡ(R_{n(k)}) has a Gamma distribution with parameters n and kθ.

• The maximum likelihood estimator (MLE) of θ is θ̂^{mle} = \frac{−n}{k log Ḡ(R_{n(k)})}.

• The UMVUE of θ is θ̂_{U} = \frac{1−n}{k log Ḡ(R_{n(k)})}.

Suppose that we observed the first n upper k-record values R_{(k)} = r_{(k)} from a distribution, with cdf and pdf given, respectively, by (3) and (4) with α(θ) = θ. Let R̂_{ω,s(k)} = ψ(R_{(k)}) be the point predictor of R_{ω,s(k)}, then we consider the balanced type predictive loss function as follows:

L_{ρ,ω,R_{s(k)}^{0}}(R_{s(k)}, R̂_{ω,s(k)}) = ωρ(R_{s(k)}^{0}, R̂_{ω,s(k)}) + (1 − ω)ρ(R_{s(k)}, R̂_{ω,s(k)}), (8)

where ω ∈ [0, 1) and ρ(R_{s(k)}, R̂_{ω,s(k)}) is an arbitrary loss function in predicting R_{s(k)} by R̂_{ω,s(k)}, while R_{s(k)}^{0} is a chosen a priori “target” predictor of R_{s(k)}, obtained for instance from the criterion of maximum likelihood predictor, best linear unbiased predictor, best linear invariant predictor, or conditional median predictor among others. In the sequel, we first derive the target predictor R_{s(k)}^{0} of R_{s(k)} based on observed records via two methods namely, conditional median and maximum likelihood.

3 Non-Bayesian Prediction

There are several non-Bayesian methods for predicting future k-records. Here we present two schemes, maximum likelihood and conditional median prediction, when the parent distribution is as in (4) with α(θ) = θ.

3.1 Conditional median prediction

The median of the distribution of R_{s(k)} given R_{n(k)} = r_{n(k)}, whose density is given in (7), is called the conditional median predictor (CMP) (see, Raqab and Nagaraja, 1995). The CMP, given in (9) below, depends on θ but we can derive a plausible predictor by using a plug-in estimate of θ. We have

∫_{r_{n(k)}}^{R_{s(k)}^{0}} f_{s|n}(r_{s(k)}|r_{n(k)}, θ)dr_{s(k)} = ∫_{R_{s(k)}^{0}}^{d} f_{s|n}(r_{s(k)}|r_{n(k)}, θ)dr_{s(k)}.

Now, suppose that we observed R_{n(k)} = r_{n(k)} then by taking t = log (\frac{Ḡ(r_{n(k)})}{Ḡ(r_{s(k)})}) the above identity can be rewritten as follows

∫_{0}^{log( \frac{Ḡ(r_{n(k)})}{Ḡ(R{0 \atop s(k)})} )} t^{s−n−1}e^{−kθt}dt = ∫_{log( \frac{Ḡ(r_{n(k)})}{Ḡ(R{0 \atop s(k)})} )}^{∞} t^{s−n−1}e^{−kθt}dt.

From the above identity we conclude that

R_{s(k)}^{0} = Ḡ^{−1} (Ḡ(r_{n(k)})e^{− \frac{Med(W)}{2kθ}} ) , (9)

where W has chi-square distribution with 2(s − n) degrees of freedom and Med(W) stands for median of W. Substituting the MLE and unbiased estimator of θ in (9) we get

R_{M,s(k)}^{0} = Ḡ^{−1} ([Ḡ(r_{n(k)})]^{(1+ \frac{Med(W)}{2n})}) (10)

and

R_{U,s(k)}^{0} = Ḡ^{−1} ([Ḡ(r_{n(k)})]^{(1+ \frac{Med(W)}{2(n−1)} )}) , (11)

respectively. For s = n + 1, using the fact that Med(χ_{(2)}^{2}) = log 4, (10) and (11) reduce to

R_{M,n+1(k)}^{0} = Ḡ^{−1} ([Ḡ(r_{n(k)})]^{(1+ \frac{log 4}{2n})}) and R_{U,n+1(k)}^{0} = Ḡ^{−1} ([Ḡ(r_{n(k)})]^{(1+ \frac{log 4}{2(n−1)} )}) .

Example 1 (i) (Exponential distribution): Taking Ḡ(x) = e^{−x}, 0 < x < ∞, in (3), X has Exponential distribution, and using (10) and (11) the target predictor (CMP) of R_{s(k)} on the basis of R_{n(k)} = r_{n(k)} are

R_{M,s(k)}^{0} = (1 + \frac{Med(W)}{2n}) R_{n(k)} and R_{U,s(k)}^{0} = (1 + \frac{Med(W)}{2n − 2}) R_{n(k)}.

For the predicting next record, s = n + 1, we get

R_{M,n+1(k)}^{0} = (1 + \frac{log 4}{2n})R_{n(k)} and R_{U,n+1(k)}^{0} = (1 + \frac{log 4}{2(n − 1)})R_{n(k)}.

(ii) (Pareto distribution): Taking Ḡ(x) = \frac{η}{x}, x > η > 0, with known η, in (3), X has Pareto distribution, and using (10) and (11) the target predictor (CMP) of R_{s(k)} on the basis of R_{n(k)} = r_{n(k)} are

R_{M,s(k)}^{0} = η (\frac{R_{n(k)}}{η} )^{−(1+ \frac{Med(W)}{2n})} , and R_{U,s(k)}^{0} = η (\frac{R_{n(k)}}{η} )^{−(1+ \frac{Med(W)}{2(n−1)} )} .

For the predicting next record, s = n + 1, we get

R_{M,n+1(k)}^{0} = η (\frac{R_{n(k)}}{η} )^{−(1+ \frac{log 4}{2n})} and R_{U,n+1(k)}^{0} = η (\frac{R_{n(k)}}{η} )^{−(1+ \frac{log 4}{2(n−1)} )} .

For k = 1 the results of Raqab et al. (2007) are obtained as an special case.

3.2 Maximum likelihood prediction

Assume that we observed r_{(k)} = (r_{1(k)}, ..., r_{n(k)}) from a population with unknown parameter θ. We intend to predict R_{s(k)}, s > n based on r_{(k)}. The predictive likelihood function of Y = R_{s(k)}, and θ is given by (see Basak and Balakrishnan, 2003)

L(r_{(k)}, r_{s(k)}, θ) = f(r_{(k)}|θ)h(r_{s(k)}|r_{(k)}, θ)
= f(r_{(k)}|θ)f_{s|n}(r_{s(k)}|r_{n(k)}, θ),

where the second equality is obtained by Markovian property of k-records. If there exists R_{L,s(k)}^{0} = R̂_{s(k)}^{mle} and θ̂_{L} = θ̂^{mle} such that

L(r_{(k)}, R_{L,s(k)}^{0}, θ̂_{L}) = sup_{r_{s(k)},θ} L(r_{(k)}, r_{s(k)}, θ),

then θ̂_{L} and R_{L,s(k)}^{0} are the predictive maximum likelihood estimator (PMLE) for θ and the maximum likelihood predictor (MLP) of R_{s(k)}, respectively. Now suppose that k-records are obtained from our proposed model (3) with α(θ) = θ, then using (5) and (7) the predictive log-likelihood function is given by

l(r_{(k)}, r_{s(k)}, θ) ∝ (s − n − 1) log (log (\frac{Ḡ(r_{s(k)})}{Ḡ(r_{n(k)})})) + log g(r_{s(k)}) + (kθ − 1) log Ḡ(r_{s(k)}) + s log θ.

Substituting θ by θ̂^{mle} in l(r_{(k)}, r_{s(k)}, θ) the MLP of R_{s(k)} can be obtained from the following equation:

\frac{g(R_{L,s(k)}^{0})}{Ḡ(R_{L,s(k)}^{0})} \{1 + \frac{s}{log Ḡ(R_{L,s(k)}^{0})} − \frac{s − n − 1}{log Ḡ(R_{L,s(k)}^{0}) − log Ḡ(r_{n(k)})}\} = \frac{g′(R_{L,s(k)}^{0})}{g(R_{L,s(k)}^{0})}, (12)

Example 2 (continued) (i)(Exponential distribution:) For the case of exponential distribution the equation (12) reduces to:

1 + \frac{s}{R_{L,s(k)}^{0}} − \frac{s − n − 1}{r_{n(k)} − R_{L,s(k)}^{0}} = 1.

Hence the MLP of R_{s(k)} is

R_{L,s(k)}^{0} = \frac{s}{n + 1}R_{n(k)}.

(ii) (Pareto distribution): Taking Ḡ(x) = \frac{η}{x}, x > η > 0, with known η, in (3) with α(θ) = θ, the maximum likelihood predictor of R_{s(k)} is given as a solution of the following equation in R_{L,s(k)}^{0}:

\frac{s − n − 1}{log r_{n(k)} − log R_{L,s(k)}^{0}} − \frac{s}{log η − log R_{L,s(k)}^{0}} = 3.

For k = 1 the results of Raqab et al. (2007) are obtained as a special case.

4 Bayesian prediction

A Bayesian approach may be adopted in order to derive the necessary predictive distribution; see Dunsmore (1983). Assuming that the parameter θ is a realization of a random variable with prior pdf π(θ). We use Gamma(α, β)-distribution as a conjugate prior for the parameter θ with the following density function,

π(θ; α, β) = \frac{β^{α}}{Γ(α)} θ^{α−1}e^{−βθ}, θ > 0, (13)

where α and β are positive constant. It follows from (5) and (13) that the posterior pdf of the parameter θ is given by

π(θ|r_{1(k)}, ..., r_{n(k)}) = \frac{(β − k log Ḡ(r_{n(k)}))^{n+α}}{Γ(n + α)} θ^{n+α−1}e^{−θ(β−k log Ḡ(r_{n(k)})}, θ > 0, (14)

where Γ(.) is the complete gamma function. Assume that we have observed the first n upper k-records r_{(k)} = (r_{1(k)}, ..., r_{n(k)}) from (3) with α(θ) = θ, and that, based on such a sample, prediction, either point or interval, is needed for s-th upper k-record, s > n. Now, let Y = R_{s(k)} be the s-th upper k-record value, s > n. The Bayes predictive density function of Y given r_{(k)} is given by (see, Arnold et al. 1998, p. 162)

f(y|r_{(k)}) = ∫_{Θ} f(y|r_{(k)}, θ)π(θ|r_{(k)})dθ. (15)

Hence, we find the Bayes predictive density function as follows

f(r_{s(k)}|r_{n(k)}) = ∫_{0}^{∞} f(r_{s(k)}|r_{n(k)}, θ)π(θ|r_{n(k)})dθ
= \frac{Γ(s + α)}{Γ(s − n)Γ(n + α)} (\frac{β + kt_{n(k)}}{β + kt_{s(k)}} )^{n+α} \frac{k^{s−n}(t_{s(k)} − t_{n(k)})^{s−n−1}g(r_{s(k)})}{(β + kt_{s(k)})^{s−n}Ḡ(r_{s(k)})},

where t_{i(k)} = − log Ḡ(r_{i(k)}), i = n, s and r_{n(k)} < r_{s(k)} < d. Let U = \frac{β−k log Ḡ(R_{n(k)})}{β−k log \overline{G}(R_{s(k)})}, then it is easy to verify that U given R_{n(k)} = r_{n(k)} has a Beta(n + α, s − n)- distribution which is independent of R_{s(k)}. Thus we can use U to construct a Bayesian prediction interval for R_{s(k)}. Let b_{γ} be the γth percentile of a Beta(n+α, s−n)-distribution, then the 100(1−γ)% prediction interval for R_{s(k)} is given by (L_{1}, L_{2}), where

L_{1} = Ḡ^{−1} \{exp [\frac{1}{k} (β − \frac{β − k log Ḡ(r_{n(k)})}{b_{1− \frac{γ}{2}} (n + α, s − n)})]\} , (16)

and

L_{2} = Ḡ^{−1} \{exp [\frac{1}{k} (β − \frac{β − k log Ḡ(r_{n(k)})}{b_{\frac{γ}{2}} (n + α, s − n)})]\} (17)

Remark 1 We get the results of Ahmadi et al. (2005) by taking Ḡ(x) = e^{−x} (Exponential distribution) in (16), and (17), and the results of Raqab et al. (2007) by choosing Ḡ(x) = \frac{η}{x}, x > η > 0, with known η (Pareto distribution) and k = 1 in (16), and (17), as special cases, respectively.

Here we consider the Bayesian point prediction of R_{s(k)} under BLF, L_{ρ,ω,R_{s(k)}^{0}}, as in (8). When ω = 0, we simply use L_{0} instead of L_{ρ,0,R_{s(k)}^{0}} unless we want to emphasize the role of ρ. The following lemma and proof are quite similar to Lemma 1 in Jafari Jozani, Marchand and Parsian (2006b), and establishes a connection between Bayesian prediction under BLF (8) for the cases ω > 0 and ω = 0.

Lemma 1 For predicting R_{s(k)} with target predictor R_{s(k)}^{0} under balanced loss function L_{ρ,ω,R_{s(k)}^{0}} as in (8) and for a prior π(θ), the Bayes predictor R̂_{ω,s(k)} corresponds to the Bayes solution R_{s(k)}^{∗} with respect to f^{∗}(r_{s(k)}|r_{n(k)}) under L_{0}; where

f^{∗}(r_{s(k)}|r_{n(k)}) = ωI_{{r_{s(k)}^{0}}}(r_{s(k)}) + (1 − ω)f(r_{s(k)}|r_{n(k)}),

i.e., f^{∗}(r_{s(k)}|r_{n(k)}) is a mixture of a point mass at r_{s(k)}^{0} and the conditional distribution R_{s(k)} given R_{n(k)} = r_{n}(k).

Proof: Let μ_{n} and ν_{n} be dominating measures of f(r_{s(k)}|r_{n(k)}) and f^{∗}(r_{s(k)}|r_{n(k)}), respectively. With the definitions of R_{s(k)}^{∗}, R̂_{ω,s(k)} and L_{0}, and with X_{s(k)} standing for the sample space of R_{s(k)}, we have

R̂_{ω,s(k)} = argmin_{r_{ω,s(k)}} ∫_{X_{s(k)}} L_{ρ,ω,R_{s(k)}^{0}}(r_{s(k)}, r_{ω,s(k)})f(r_{s(k)}|r_{n(k)})dμ_{n}(r_{s(k)})
= argmin_{r_{ω,s(k)}} ∫_{X_{s(k)}∪{r_{s(k)}^{0}}} ρ(r_{s(k)}, r_{ω,s(k)})f^{∗}(r_{s(k)}|r_{n(k)})dν_{n}(r_{s(k)})
= argmin_{r_{ω,s(k)}} ∫_{X_{s(k)}∪{r_{s(k)}^{0}}} L_{0}(r_{s(k)}, r_{ω,s(k)})f^{∗}(r_{s(k)}|r_{n(k)})dν_{n}(r_{s(k)})
= R_{s(k)}^{∗}.
□

4.1 Bayes predictor under BSEL function

Choosing ρ(y, d) = ρ_{1}(y, d) = τ(y)(y − d)^{2} with τ(·) > 0, the balanced predictive loss function in (8) becomes

L_{ρ_{1},ω,R_{s(k)}^{0}}(R_{s(k)}, R̂_{ω,s(k)}) = ωτ(R_{s(k)}^{0})(R_{s(k)}^{0} − R̂_{ω,s(k)})^{2} + (1 − ω)τ(R_{s(k)})(R_{s(k)} − R̂_{ω,s(k)})^{2}.

Under L_{0} and the prior π, we have R̂_{0,s(k)} = \frac{E_{F} [R_{s(k)}τ(R_{s(k)})|r_{n(k)}]}{E_{F} [τ(R_{s(k)})|r_{n(k)}]} (subject to the finiteness conditions E_{F} [R_{s(k)}^{i}τ(R_{s(k)})|r_{n(k)}] < ∞; i = 0, 1; for all r_{n(k)}). Thus, following Lemma 1, the Bayes point prediction of R_{s(k)} under L_{ρ_{1},ω,R_{s(k)}^{0}} is given by

R̂_{ω,s(k)} = \frac{E_{F^{∗}}[R_{s(k)}τ(R_{s(k)})|r_{n(k)}]}{E_{F^{∗}}[τ(R_{s(k)})|r_{n(k)}]}

= \frac{∫_{X_{s(k)}∪\{r_{s(k)}^{0}\}} r_{s(k)}τ(r_{s(k)})f^{∗}(r_{s(k)}|r_{n(k)})dν_{n}(r_{s(k)})}{∫_{X_{s(k)}∪\{r_{s(k)}^{0}\}} τ(R_{s(k)})f^{∗}(r_{s(k)}|r_{n(k)})dν_{n}(r_{s(k)})}

= \frac{ωr_{s(k)}^{0}τ(r_{s(k)}^{0}) + (1 − ω)E_{F} [R_{s(k)}τ(R_{s(k)})|r_{n(k)}]}{ωτ(r_{s(k)}^{0}) + (1 − ω)E_{F} [τ(R_{s(k)})|r_{n(k)}]}.

Taking τ(·) = 1 the predictive loss function L_{ρ_{1},ω,R_{s(k)}^{0}} reduces to the balanced predictive SEL function. In this case the Bayes predictor of R_{s(k)} simplifies as

R̂_{ω,s(k)} = ωR_{s(k)}^{0} + (1 − ω)E[R_{s(k)}|R_{n(k)}]  (18)
= ωR_{s(k)}^{0} + (1 − ω)R̂_{0,s(k)}.

Remark 2 It may be noted that the Bayes predictor of R_{s(k)} under balanced predictive SEL function is a convex linear combination of the target predictor R_{s(k)}^{0} and the Bayes predictor of R_{s(k)} under the usual predictive SEL function. Also, it is observed that when ω = 1, R̂_{ω,s(k)} reduces to the classical predictor R_{s(k)}^{0}, whereas, for w = 0, R̂_{ω,s(k)} reduces to R̂_{0,s(k)} which is the Bayes predictor for R_{s(k)} under SEL function. When F ∈ C_{1}, we have

R̂_{0,s(k)} = E[R_{s(k)}|r_{n(k)}]
= E_{U} [Ḡ^{−1} (Ḡ(r_{n(k)})^{\frac{1}{U}} e^{− \frac{β}{k} \frac{1−U}{U}} ) |r_{n(k)}] ,

where the random variable U = \frac{β−k log Ḡ(R_{n(k)})}{β−k log Ḡ(R_{s(k)})} given R_{n(k)} = r_{n(k)} has a Beta(n+α, s−n)- distribution. Thus we may rewrite (18) as

R̂_{ω,s(k)} = ωR_{s(k)}^{0} + (1 − ω)E_{U} [Ḡ^{−1} (Ḡ(r_{n(k)})^{\frac{1}{U}} e^{− \frac{β}{k} \frac{1−U}{U}} ) |R_{n(k)}] . (19) Example 3 (continued) (Exponential distribution:) For the case of exponential distribution the equation (19) reduces to

R̂_{ω,s(k)} = ωR_{s(k)}^{0} + (1 − ω) \{(R_{n(k)} + \frac{β}{k} ) E [\frac{1}{U} |R_{n(k)}] − \frac{β}{k} \}
= ωR_{s(k)}^{0} + (1 − ω) \{\frac{s + α − 1}{n + α − 1} (R_{n(k)} + \frac{β}{k} ) − \frac{β}{k} \} .

Taking R_{s(k)}^{0} = R_{L,s(k)}^{0} = \frac{s}{n+1}R_{n(k)} as the MLP of R_{s(k)}, then for s = n + 1 we get

R̂_{ω,n+1(k)} = ωR_{n(k)} + (1 − ω) \{\frac{n + α}{n + α − 1} (R_{n(k)} + \frac{β}{k} ) − \frac{β}{k} \} .

When k = 1 and ω = 0, we find the results of Jaheen (2004) as an special case.

4.2 Bayes predictor under BAEL function

Following Jafari Jozani, Marchand and Parsian (2006b), taking ρ(y, d) = ρ_{2}(y, d) = |y − d|, we get balanced absolute error loss (BAEL) function as

L_{ρ_{2},ω,R_{s(k)}^{0}}(R_{s(k)}, R̂_{ω,s(k)}) = ω|R_{s(k)}^{0} − R̂_{ω,s(k)}| + (1 − ω)|R_{s(k)} − R̂_{ω,s(k)}|.

Since the Bayes prediction of R_{s(k)} under L_{ρ_{2},0,R_{s(k)}^{0}} is the median of the Bayes predictive density function f(r_{s(k)}|r_{n(k)}) = ∫ f(r_{s(k)}|r_{n(k)}, θ)π(θ)dθ, following Lemma 1, the Bayes prediction of R_{s(k)} under BAEL, L_{ρ_{2},ω,R_{s(k)}^{0}}, is a median of

f^{∗}(r_{s(k)}|r_{n(k)}) = ωI_{{r_{s(k)}^{0}}}(r_{s(k)}) + (1 − ω)f(r_{s(k)}|r_{n(k)}),

for all r_{n(k)}. It is easy to realize that when ω ≥ \frac{1}{2}, R̂_{ω,s(k)} = R_{s(k)}^{0}, i.e. the point mass of R_{s(k)}^{0} has large enough probability to force the median to be R_{s(k)}^{0}. For ω < \frac{1}{2}, the cdf corresponding to f^{∗}(r_{s(k)}|r_{n(k)}) is given by

F^{∗}(r_{s(k)}|r_{n(k)}) = \{ \begin{matrix}(1 − ω)P (R_{s(k)} ≤ r_{s(k)}|r_{n(k)}), & r_{s(k)} < r_{s(k)}^{0}, \\ ω + (1 − ω)P (R_{s(k)} ≤ r_{s(k)}|r_{n(k)}), & r_{s(k)} ≥ r_{s(k)}^{0}. \end{matrix}

Taking ζ(r_{n(k)}) = (1 − ω)P (R_{s(k)} ≤ r_{s(k)}|r_{n(k)}) and F_{s|n}^{−1} as the inverse cumulative Bayes predictive density function, straightforward calculations lead to

R̂_{ω,s(k)} = \{ \begin{matrix}F_{s|n}^{−1}(\frac{1−2ω}{2(1−ω)}), & 0 ≤ ζ(r_{n(k)}) < \frac{1}{2} − ω, \\ R_{s(k)}^{0}, & \frac{1}{2} − ω < ζ(r_{n(k)}) < \frac{1}{2}, \\ F_{s|n}^{−1}(\frac{1}{2(1−ω)}), & \frac{1}{2} ≤ ζ(r_{n(k)}) < 1. \end{matrix}  (20)

Note that the above result is general with respect to the choice of the target predictor, the model and the prior distribution. Also in predicting R_{n+1(k)}, when the target predictor is chosen to be the maximum likelihood predictor of R_{n+1(k)}, i.e. R_{n+1(k)}^{0} = R_{n(k)}, it is easy to see that ζ(r_{n(k)}) = 0 and so R̂_{ω,n+1(k)} = r_{n(k)}. Let B(·, n + α, s − n) be the cdf of a Beta(n + α, s − n)-distribution. For F ∈ C_{1}, we can easily show that

ζ(r_{n(k)}) = 1 − B (\frac{β − log g(r_{n}(k))}{β − k log g(r_{s(k)}^{0})}, n + α, s − n) ,

and

F (r_{s(k)}|r_{n(k)}) = 1 − B (\frac{β − log g(r_{n}(k))}{β − k log g(r_{s(k)})}, n + α, s − n) .

Choosing B^{−1}(·, n + α, s − n) as the inverse cdf of a Beta(n + α, s − n)-distribution, then (20) leads us to the following Bayes prediction of R_{s(k)} under BAEL,

R̂_{ω,s(k)} = \{ \begin{matrix}B^{−1}(\frac{1}{2(1−ω)}, n + α, s − n), & \frac{1}{2} + ω ≤ B(\frac{β−log g(r_{n}(k))}{β−k log g(r_{s(k)}^{0})}, n + α, s − n) < 1, \\ R_{s(k)}^{0}, & \frac{1}{2} ≤ B(\frac{β−log g(r_{n}(k))}{β−k log g(r_{s(k)}^{0})}, n + α, s − n) < \frac{1}{2} + ω, \\ B^{−1}(\frac{1−2ω}{2(1−ω)}, n + α, s − n), & 0 ≤ B(\frac{β−log g(r_{n}(k))}{β−k log g(r_{s(k)}^{0})}, n + α, s − n) < \frac{1}{2}. \end{matrix}

4.3 Bayes predictor under balanced LINEX loss function

The choice of ρ(y, d) = ρ_{a}(y, d) = e^{a(y−d)} − a(y − d) − 1, a ≠ 0, in (8), leads to balanced LINEX loss function L_{ρ_{a},ω,R_{s(k)}^{0}} as follows

ω{e^{a(R_{s(k)}^{0}−R̂_{ω,s(k)})} − a(R_{s(k)}^{0} − R̂_{ω,s(k)}) − 1} + (1 − ω){e^{a(R_{s(k)}−R̂_{ω,s(k)})} − a(R_{s(k)} − R̂_{ω,s(k)}) − 1}.

The choice of such an asymmetric ρ_{a} is especially appropriate when over-prediction and underprediction of the same magnitude have different economic consequences. For more details on LINEX loss see the review paper of Parsian and Kirmani (2002). The unique Bayes prediction of R_{s(k)} under L_{ρ_{a},0,R_{s(k)}^{0}} with respect to the prior distribution π(θ) is given by R̂_{0,s(k)} = −\frac{1}{a} log E_{F} [e^{−aR_{s(k)}}|r_{n(k)}]. Following Lemma 1, the unique Bayes predictor of R_{s(k)} under L_{ρ_{a},ω,R_{s(k)}^{0}} is given as follows

R̂_{s(k)} = −\frac{1}{a} log E_{F^{∗}}[e^{−aR_{s(k)}}|R_{n(k)}]
= −\frac{1}{a} log(ωe^{−aR_{s(k)}^{0}} + (1 − ω)E_{F} [e^{−aR_{s(k)}}|R_{n(k)}])
= −\frac{1}{a} log(ωe^{−aR_{s(k)}^{0}} + (1 − ω)e^{−aR̂_{0,s(k)}})

and for F ∈ C_{1} reduces to

R̂_{ω,s(k)} = −\frac{1}{a} log (ωe^{−aR_{s(k)}^{0}} + (1 − ω)E [exp\{Ḡ^{−1} (Ḡ(R_{n(k)})^{\frac{1}{U}} e^{− \frac{β}{k} \frac{1−U}{U}} )\}|R_{n(k)}]) ,  (21)

where U|r_{n(k)} ∼ Beta(n + α, s − n).

Example 4 (continued) (Exponential distribution:) For the case of exponential distribution, since

e^{−aR̂_{0,s(k)}} = E [e^{−a( \frac{rn(k)}{U} + \frac{β}{k} \frac{1−U}{U} )}|r_{n(k)}]
= \frac{e^{− \frac{aβ}{k}}}{B(n + α, s − n)}Ψ_{s}(a, α, β, k, r_{n(k)}),

where

B(n + α, s − n) = \frac{Γ(n + α)Γ(s − n)}{Γ(s + α)},

and

Ψ_{s}(a, α, β, k, r_{n(k)}) = ∫_{0}^{1} exp (−a(r_{n(k)} + \frac{β}{k})\frac{1}{u}) u^{n+α−1}(1 − u)^{s−n−1}du,

the Bayes prediction of R_{s(k)} obtained from F ∈ C_{1} under balanced LINEX loss function is given by

R̂_{ω,s(k)} = −\frac{1}{a} log (ωe^{−ar_{s(k)}^{0}} + \frac{(1 − ω)e^{− \frac{aβ}{k}}}{B(n + α, s − n)}Ψ_{s}(a, α, β, k, r_{n(k)})) .

For k = 1 and ω = 0 the results of Ahmadi et al. (2005) are obtained as an special case.

5 Conclusion

• The balanced type loss function, L_{ρ,ω,R_{s(k)}^{0}} as in (8), is a mixture of two loss functions, ρ(R̂_{ω,s(k)}, R_{s(k)}^{0}) and ρ(R̂_{ω,s(k)}, R_{s(k)}). Therefore it may be more appropriate than either of them in predicting the future records. This loss, which depends on the observed value of R_{s(k)}^{0}, reflects a desire of closeness of R̂_{ω,s(k)} both to R_{s(k)} and R_{s(k)}^{0}. In this paper, under non-Bayesian and Bayesian frameworks, we have studied the prediction of future k-records based on observed records which come from a general class of continuous distributions. The Bayes predictors of the s-th future k-record are obtained under BLF. We presented the results for the balanced type version of some well-known loss functions namely SEL, L_{1} and LINEX loss.

• Taking a sequence of iid random variables X_{1}, X_{2}, ⋯, from the class C_{2} of continuous distribution function F (x; θ) with

F (x; θ) = [G(x)]^{θ}, −∞ ≤ c < x < d ≤ ∞, θ > 0

where G(x) is an arbitrary continuous distribution function free of unknown parameter and G(c) = 0, G(d) = 1. It is easy to show that the results of this paper are also hold true for lower k-record data from C_{2} with some modifications. The above family of distributions is well known in the lifetime experiments as proportional reverse hazard models (see Lawless, 2003) and also a subclass of one parameter exponential family.

Acknowledgements

The authors are grateful to the two anonymous referees for making many helpful comments and suggestions on an earlier version of this paper.

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