Best Linear Unbiased and Invariant Reconstructors for the Past Records

Transcription

1 BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull Malays Math Sci Soc (2) 37(4) (2014), Best Linear Unbiased and Invariant Reconstructors for the Past Records 1 B KHATIB AND 2 JAFAR AHMADI 1,2 Departent of Statistics, Ordered and Spatial Data Center of Excellence, Ferdowsi University of Mashhad, P O Box 1159, Mashhad 91775, Iran 1 khatib b@yahooco, 2 ahadi-j@uacir Abstract The best linear unbiased reconstructor and the best linear invariant reconstructor of the past upper records based on observed upper record values for the location scale faily of distributions are derived The results are obtained in detail for two-paraeter exponential distribution A coparison study is perfored in ters of ean squared reconstruction error and Pitan s easure of closeness criteria Finally, a real exaple is given to illustrate the proposed procedures 1 Introduction 2010 Matheatics Subject Classification: 62G30 (62N99) Keywords and phrases: Best linear unbiased estiator (BLUE), best linear invariant estiator (BLIE), Pitan closeness, location-scale faily Let {X i,i > 1} be a sequence of independent and identically distributed (iid) continuous rando variables An observation X j is called an upper record value if its value exceeds that of all previous observations, i e, X j is an upper record value if X j > X i for every i < j These type of data are of great iportance in several real-life probles involving weather, industry, econoic and sport data For ore details and applications of record values, see for exaple, Arnold et al [4] Ahsanullah and Shakil [3] obtained soe characterization results of Rayleigh distribution based on conditional expectation of record values The prediction proble of the future records based on past records have been extensively studied in both frequentist and Bayesian approaches; Awad and Raqab [6] considered the prediction proble of the future nth record value based on the first ( < n) observed record values fro one-paraeter exponential distribution Raqab [17] established different point predictors and prediction intervals for future records based on observed record values fro two-paraeter exponential distribution Raqab and Balakrishnan [18] discussed the proble of prediction of the future record values in nonparaetric settings Ahadi and Doosparast [2] investigated Bayesian estiation and prediction for soe life distributions based on upper record values MirMostafaee and Ahadi [16] obtained several point Counicated by M Ataharul Isla Received: March 1, 2012; Revised: October 2, 2012

2 1018 B Khatib and J Ahadi predictors such as the best unbiased predictor, the best invariant predictor and axiu likelihood predictor of future order statistics on the basis of observed record values fro two-paraeter exponential distribution Recently, soe works have been done on the reconstruction proble Kliczak and Rychlik [14] obtained upper bounds for the expectations of increents of past order statistics and records based on observed order statistics and records, respectively Balakrishnan et al [10] investigated the proble of reconstructing past upper records fro the known values of future records in exponential and Pareto distributions Razkhah et al [19] obtained soe point and interval reconstructors for the issing order statistics in exponential distribution Asgharzadeh et al [5] studied the reconstruction of the past failure ties based on left-censored saples for the proportional reversed hazard rate odels The ain goal of this paper is to investigate the best linear unbiased reconstructor (BLUR) and the best linear invariant reconstructor (BLIR) based on observed future record data fro two-paraeter exponential distribution We also intend to present a coparison study of coon reconstructors of past records fro exponential distribution With this in ind, we consider two criteria: ean squared reconstruction error (MSRE) and Pitan s easure of closeness (PMC) In recent years, any probles involving ordered data and Pitan closeness have been investigated Pitan closeness of records to population quantiles was explored in Ahadi and Balakrishnan [1] Soe interesting results regarding the saple edian and Pitan closeness were provided by Iliopoulous and Balakrishnan [13] Based on a Type-II right censored saple fro one-paraeter exponential distribution, Balakrishnan et al [9] copared the best linear unbiased predictor (BLUP) and the best linear invariant predictor (BLIP) of the censored order statistics in the one-saple case in ters of PMC The rest of this paper is organized as follows Section 2 contains soe preliinaries In Sections 3 and 4, the BLUR and BLIR of the past upper records fro two-paraeter exponential distribution are investigated, respectively These two reconstructors and two others where obtained by Balakrishnan et al [10] are copared in ters of MSRE and PMC criteria in Section 5 A real exaple is given in Section 6 2 Preliinaries Let R 1,,R n be the first n upper record values coing fro a X-sequence of iid continuous rando variables with cuulative distribution function (cdf) F(x; θ) and probability density function (pdf) f (x;θ), where θ Θ ay be a vector of paraeters and Θ is the paraeter space Suppose we have failed to observe the first records, naely, we have observed R = (R +1,R +2,,R n ), + 1 < n In this case, the joint pdf of R can be expressed as follows (see, for exaple, Arnold et al [4, pp 10 11]) (21) ( H(r+1 ;θ) ) f R (r;θ) = f (r +1 ;θ)! n 1 i=+1 f (r i+1 ;θ) F(r i ;θ), r +1 < < r n, where F(x;θ) = 1 F(x;θ) is the survival function of X, H(x;θ) = log F(x;θ), r = (r +1,,r n ) is the observed value of R Fro (21) and using the Markovian property of record values, it can be shown that the conditional pdf of R l given the observed data set R is just the conditional pdf of R l given R +1 and is given by f Rl R(r l r;θ) = f Rl R +1 (r l r +1 ;θ)

3 (22) Best Linear Unbiased and Invariant Reconstructors for the Past Records 1019 = ( H(rl ;θ) ) l 1( H(r+1 ;θ) H(r l ;θ) ) l B(l, l + 1) ( H(r +1 ;θ) ) f (r l ;θ) F(r l ;θ), r l < r +1, where B(,) is the coplete beta function Fro (22), the unbiased reconstructor of R l, l = 1,,, is the conditional ean of R l given R +1, ie, (23) R l = E(R l R) = E(R l R +1 ), and hence it depends only on R +1 If the paraeters in (23) are unknown, they have to be estiated in this conditional expectation A rando variable X is said to have a two-paraeter exponential distribution, denoted by Exp(µ,σ), if its cdf is (24) F(x;θ) = 1 e x µ σ, x µ, σ > 0, where θ = (µ,σ), µ and σ are the location and scale paraeters, respectively The exponential distribution is a very coonly used distribution in reliability engineering and survival analysis We refer the reader to Balakrishnan and Basu [7] for soe research based on exponential distribution In what follows, we shall study the behavior of reconstructors of the past records for Exp(µ,σ) in detail The first result for the case of Exp(µ,σ) is that the unbiased reconstructor of R l based on R is given by (25) R l (µ) = µ + l + 1 (R +1 µ), it does not depend on the scale paraeter When µ is unknown, we can substitute it with its estiator based on R = (R +1,R +2,,R n ) 3 The best linear unbiased reconstructor In this section, first we focus our attention to derive the BLUR in general for the location scale failies Let X = (X +1,,X n ) be an observed ()-diensional rando vector fro the cdf F(x; µ,σ) = F 0 ((x µ)/σ), where µ and σ are the location and scale paraeters, respectively, and F 0 does not depend on µ and σ Also, let Y be an unknown past observation fro the sae distribution Reconstructing the outcoe Y based on the observed value of X is the ain goal of this section Toward this end, let us assue that a 0 = E((Y µ)/σ),a i = E((X i µ)/σ),w i = Cov((Y µ)/σ,(x i µ)/σ) and v i, j = Cov((X i µ)/σ,(x j µ)/σ), for i, j = + 1,,n Denote by 1 the vector of 1, s, a = (a +1,,a n ) T, w T = (w +1,,w n ) and V = (v i, j ),i, j = + 1,,n To obtain the BLUR of Y based on X, we will have to iniize the variance of (Y cx T ) subject to the condition of unbiasedness where c = (c +1,,c n ) is an arbitrary vector of real values Then, following along the sae lines as in the prediction proble, the BLUR of Y is given by (see, Goldberger [12] in the context of prediction) (31) Ŷ = ˆµ(X) + ˆσ(X)a 0 + w T V 1 [X T ˆµ(X)1 ˆσ(X)a], where ˆµ(X) and ˆσ(X) are the BLUEs of µ and σ on the basis of X, respectively, and V 1 is the inverse atrix of V Now, assue that the observed data are the upper record values fro the two-paraeter exponential distribution Using (31), the BLUR of R l based on R is given by (32) ˆR l = ˆµ + l ˆσ + ω T Σ 1 [R T ˆµ1 ˆσα],

4 1020 B Khatib and J Ahadi where ˆµ and ˆσ are the BLUEs of µ and σ based on R, respectively, α is the vector of eans of the corresponding record values taken fro Exp(0,1), σ 2 Σ is the covariance atrix of R and ω T = 1/σ 2 (Cov(R +1,R l ),Cov(R +2,R l ),,Cov(R n,r l )) By doing soe algebraic calculations it can be shown that ω T = (l,l,,l) and (33) ω T Σ 1 l = ( + 1,0,0,,0) Substituting (33) into (32), we get (34) ˆR l = ˆµ + l ˆσ Furtherore, the BLUEs of µ and σ on the basis of R are given by (35) ˆµ = nr +1 ( + 1)R n n ( + 1) and ˆσ = R n R +1 n ( + 1), respectively (see, Arnold et al [4]) Therefore, by substituting (35) in (34), it is deduced that (36) ˆR l = (n l)r +1 (( + 1) l)r n n ( + 1) is the BLUR of R l on the basis of the data set R The MSRE of ˆR l is coputed to be (37) MSRE( ˆR l ) = E[ ˆR l R l ] 2 = σ 2 (n l)(( + 1) l) n ( + 1) It is easy to show that MSRE( ˆR l ) is increasing in and decreasing with respect to n and l, if the other arguents are fixed Moreover, ˆR l and MSRE( ˆR l ) converge to R +1 ˆσ and σ 2 ()/(n ( + 1)), respectively, when l tends to Reark 31 It ay be noted that by substituting the BLUE of µ into the unbiased reconstructor of R l in (25), the sae result will be obtained for ˆR l as in (36), ie, R l ( ˆµ) = ˆR l 4 The best linear invariant reconstructor Sae as in previous section, first we present the results for BLIR in general for location scale failies Let X = (X +1,,X n ) and Y be as defined in Section 3 Then, follow the results of Mann [15], in the context of prediction, the BLIR of Y based on the data set X can be derived as (41) Ỹ = Ŷ ( c c 2 ) ˆσ(X), where Ŷ is the BLUR of Y, c 2 = σ 2 Var( ˆσ(X)) and c 1 = 1/σ 2 Cov( ˆσ(X),(1 w T V 1 1) ˆµ(X) + (a 0 w T V 1 a) ˆσ(X)), with V, w, a, ˆµ(X), ˆσ(X) and a 0 being as defined in Section 3 When the upper record statistics R = (R +1,R +2,,R n ) fro the Exp(µ,σ) distribution are observed, using (41), the BLIR of R l on the basis of R is as follows (42) R l = ˆR l ( V 1 1 +V 2 ) ˆσ,

5 Best Linear Unbiased and Invariant Reconstructors for the Past Records 1021 where V 2 = σ 2 Var( ˆσ) and V 1 = σ 2[ Cov( ˆσ,(1 ω T Σ 1 1) ˆµ +(l ω T Σ 1 α) ˆσ) ] in which ω,σ 1 and α are as defined in Section 3, ˆµ and ˆσ are the BLUEs of µ and σ, respectively, and ˆR l is the BLUR of R l By perforing soe algebraic calculations, we obtain ( ) n + 1 l R+1 ( + 1 l ) R n (43) R l = Moreover, it can be shown that the MSRE of R l is given by ( )( ) n + 1 l + 1 l (44) MSRE( R l ) = E[ R l R l ] 2 = σ 2, which is increasing with respect to and decreasing with respect to n and l, if the other arguents are fixed Also, R l and MSRE( R l ) converge to R +1 σ and σ 2 (1+1/(n )), respectively, when l tends to Reark 41 It ay be noted that the BLIEs of µ and σ for a two-paraeter exponential distribution are given by (see, Arnold et al [4]) (45) µ = (n + 1)R +1 ( + 1)R n and σ = R n R +1, respectively By plugging the BLIE of µ into (25), we arrive at the sae result for R l as given in (43), ie, R l ( µ) = R l 5 Coparison results In Sections 3 and 4, we obtained two reconstructors (BLUR and BLIR) for R l based on R fro a two-paraeter exponential distribution Balakrishnan et al [10] also considered the sae plan and obtained two other reconstructors for R l naely axiu likelihood reconstructor (MLR), denoted by ˆR l,m, and conditional edian reconstructor (CMR), denoted by ˆR l,c For a coparison study, let us restate their results here The MLR of R l is (51) ˆR l,m = (n l + 1)R +1 ( l)r n + 1 with (52) MSRE( ˆR l,m ) = σ 2 { (n + 1 l)( + 1 l) + 1 The CMR of R l is + } (n l)(n + 1 l) ( + 1) 2 (53) ˆR l,c = (R n R +1 )M l, + nr +1 R n with ( ) (1 MSRE( ˆR l,c ) = σ {( 2 Ml, ) l)( + 2 l) + ( 1) ( ) } (1 Ml, ) (54) () 2 ( + 1 l)( 1), where M l, stands for the edian of beta distribution with paraeters l and l + 1 In this section, we intend to copare the four reconstructors based on MSRE and PMC criteria

6 1022 B Khatib and J Ahadi 51 Coparison based on MSRE Using (37) and (44), we have (55) MSRE( R l ) = MSRE( ˆR l ) σ 2 ( ( + 1) l ) 2 (n ( + 1))() Fro (55), it is obvious that BLIR is better than BLUR in the sense of MSRE Moreover, fro (44) and (52), we have ( ) n + 1 l 2 (56) MSRE( R l ) = MSRE( ˆR l,m ) σ 2 ( ), which iplies that BLIR is better than MLR Furtherore, using (37) and (52), we get (57) ( ) MSRE( ˆR l ) = MSRE( ˆR l,m ) σ 2 n l ( + 1) 2 ( 1) () 2 (n + 2(l 1)) Using (57) it is deduced that BLUR is better than MLR when (n ) 2 > (n+ 2(l 1)) In order to coplete the coparison study, we have coputed the nuerical values of MSREs of four reconstructors for n = 10 and all selected values of l and The results are presented in Table 1 Table 1 Nuerical values of MSREs for n = ,2,3, and 4 stand for the σ 2 MSRE of ˆR l,m, ˆR l,c, ˆR l and R l, respectively For n = 10 fro Table 1, by epirical evidence, it is observed that in the sense of MSRE, BLIR is ore accurate reconstructor than others Furtherore, in the ost cases the MSRE of ˆR l,m is greater than others

7 Best Linear Unbiased and Invariant Reconstructors for the Past Records Coparison based on PMC Here, we use the PMC criterion to copare the reconstructors ˆR l, R l, ˆR l,m and ˆR l,c The easure is based on the probabilities of the relative closeness of copeting estiators to an unknown paraeter First, we recall two definitions If T 1 and T 2 are two estiators of a coon paraeter θ, PMC of T 1 relative to T 2 is defined by (58) PMC(T 1,T 2 θ) = P( T 1 θ < T 2 θ ), θ Ω, where Ω is the paraeter space The estiator T 1 is called Pitan closer (with respect to θ) than T 2 if and only if PMC(T 1,T 2 θ) 1/2, θ Ω, with strict inequality holding for at least one θ Based on a Type-II right censored saple (X 1:n,X 2:n,,X r:n ) fro Exp(0,σ) distribution, Balakrishnan et al [8] copared BLUE and BLIE of σ in ters of PMC criterion Also, Balakrishnan et al [9] copared BLUP and BLIP of future order statistics under Type-II censoring for Exp(0,σ) distribution in ters of PMC criterion Now, we copare each two reconstructors of R l based on PMC criterion (i) PMC of ˆR l relative to R l (BLUR and BLIR): Fro (36), (43) and (58), we have π ˆR l, R l (l,;n) = P( ˆR l R l < R l R l ) = P ( R l 1 (R n R +1 ) R l < R l ) (R n R +1 ) R l ( + 1 l = P ( 1) 2 (R 2 n R +1 ) 1 (R +1 R l ) < + 1 l () 2 (R n R +1 ) 2 ) (R +1 R l ) ( ( Rn R = P < R +1 R l ( + 1 l) ) 1 ) It can be shown that (R n R +1 )/(R +1 R l ) has the F-distribution with 2( 1) and 2( + 1 l) degrees of freedo So, we express π ˆR l, R (l,;n) as l ( ) 2()( 1) (59) π ˆR l, R (l,;n) = F l 2(n 1),2(+1 l), ( + 1 l)(2n 2 1) where F a,b (x) stands for the cdf of F-distribution with a and b degrees of freedo at x (ii) PMC of ˆR l relative to ˆR l,m (BLUR and MLR): Fro (36), (51) and (58), we have π ˆR l, ˆR l,m (l,;n) = P( ˆR l R l < ˆR l,m R l ) = P ( R l 1 (R n R +1 ) R l < R +1 l + 1 ) (R n R +1 ) R l

8 1024 B Khatib and J Ahadi (510) ( + 1 l = F 2(n 1),2(+1 l) (2 1 + l ) 1 ) + 1 (iii) PMC of ˆR l relative to ˆR l,c (BLUR and CMR): Fro (36), (53) and (58), we have (511) π ˆR l, ˆR l,c (l,;n) = P( ˆR l R l < ˆR l,c R l ) { = P ( R l 1 (R n R +1 ) R l < R +1 (1 M } l,) (R n R +1 ) R l ( + 1 l = F 2(n 1),2(+1 l) (2 1 + (1 M ) 1 ) l,) (iv) PMC of R l relative to ˆR l,m (BLIR and MLR): Siilarly, we have (512) π R l, ˆR l,m (l,;n) = P( R l R l < ˆR l,m R l ) = P ( R l (R n R +1 ) R l < R +1 l + 1 ) (R n R +1 ) R l = F 2(n 1),2(+1 l) (( + 1 l + l + 1 (v) PMC of R l relative to ˆR l,c (BLIR and CMR): ( + 1 l (513) π R l, ˆR (l,;n) = F l,c 2(n 1),2(+1 l) (2 (vi) PMC of ˆR l,m relative to ˆR l,c (MLR and CMR): ( l (514) π ˆR l,c, ˆR l,m (l,;n) = F 2(n 1),2(+1 l) (2 ) 1 ) + (1 M ) 1 ) l,) (1 M l,) ) 1 ) Fro the above results, one can exaine the perforance of any two reconstructors in ters of PMC criterion by coparing the values of edian of the F-distribution with 2(n 1) and 2(+1 l) degrees of freedo and corresponding values in (59) (514), but, nuerical coputations are needed We have coputed the PMC probabilities in (59) (514) for n = 5,10,15,20 and all selected values of l and The results for n = 10 are presented in Tables 2 7, respectively, which exaine how uch a reconstructor is better than the other The nuerical results for n = 5,15 and 20 are available with the authors, and will be sent on request to interested readers

9 Best Linear Unbiased and Invariant Reconstructors for the Past Records 1025 Table 2 Values of π ˆR l, R l (l,;n) for n = Table 3 Values of π ˆR l, ˆR l,m (l,;n) for n = Table 4 Values of π ˆR l, ˆR l,c (l,;n) for n = Table 5 Values of π R l, ˆR l,m (l,;n) for n =

10 1026 B Khatib and J Ahadi Table 6 Values of π R l, ˆR l,c (l,;n) for n = Table 7 Values of π ˆR l,c, ˆR l,m (l,;n) for n = Fro our nuerical results and by epirical evidence, we observe that: (1) The BLIR (CMR) is Pitan-closer than BLUR (BLUR) for the following situations: (i) when n = 10,20, = n/2 + a and l = 1,2,,2a + 1,(a = 0,1,,n/2 2); (ii) when n = 5,15, = (n + 1)/2 + a and l = 1,2,,2a + 2,(a = 0,1,,(n + 1)/2 3); otherwise, the BLUR (BLUR) is Pitan-closer than the BLIR (CMR) specially for l = [See for exaple Table 2 (Table 4) when n = 10] (2) The MLR (CMR) is Pitan closer than BLUR (BLIR) in the following situations: (i) when n = 10,20, = n/2 + a and l = 1,2,,2a,(a = 1,,n/2 2); (ii) when n = 5,15, = (n + 1)/2 + a and l = 1,2,,2a + 1,(a = 0,1,,(n + 1)/2 3); otherwise, the BLUR (BLIR) is Pitan-closer than the MLR (CMR) specially for l = [See for exaple Table 3 (Table 6) when n = 10] (3) The MLR (MLR) is Pitan-closer than BLIR (CMR) for the following situations: (i) when n = 10,20, = n/2 + a and l = 1,2,,2a 1,(a = 1,,n/2 2); (ii) when n = 5,15, = (n+1)/2+a and l = 1,2,,2a,(a = 1,,(n+1)/2 3); otherwise, the BLIR (CMR) is Pitan-closer than the MLR (MLR) specially for l = [See for exaple Table 5 (Table 7) when n = 10]

11 Best Linear Unbiased and Invariant Reconstructors for the Past Records A real exaple To illustrate the perforance of the proposed reconstructors, we use a real data set concerning the ties (in inutes) between 48 consecutive telephone calls to a copany s switchboard which is presented by Castillo et al [11] where the authors found that the exponential distribution gave an adequate fit Table 8 contains the corresponding data Table 8 Ties (in inutes) between 48 consecutive calls Fro the data of Table 8, six upper records extracted which are 134, 168, 186, 220, 320 and 325 Assuing that only R +1,,R 6 have been observed, we show that how one can reconstruct the issing record values and copare the results with the exact values With this in ind, we copute different point reconstructors presented in the paper for the lost record value R l (1 l ) Using (36), (43), (51) and (53) the values of the BLUR, BLIR, MLR and CMR are obtained, respectively The results are presented in Table 9 for = 3 and 4 and 1 l Table 9 The nuerical values of point reconstructors l Exact value ˆR l R l ˆR l,m ˆR l,c For = 3, the observed records are R 4 = 220,R 5 = 320 and R 6 = 325 Based on these observations we have reconstructed the first three records Fro Table 9, it is observed that for = 3 the point reconstructors are reasonably close to the exact values When = 4, it is assued that two records R 5 = 320 and R 6 = 325 are observed In this case, the point reconstructors are not close to the exact values but not too far It should be entiond that one exaple does not tell us uch ore Acknowledgeent The authors would like to thank the Editor in Chief and the referees for their careful reading and valuable suggestions that iproved the original version of the anuscript References [1] J Ahadi and N Balakrishnan, Pitan closeness of record values to population quantiles, Statist Probab Lett 79 (2009), no 19,

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