Asymptotic Efficiencies of the MLE Based on Bivariate Record Values from Bivariate Normal Distribution

Transcription

1 JIRSS 013 Vol. 1, No., pp 35-5 Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 Asymptotc ffcences of the ML Based on Bvarate Record Values from Bvarate Normal Dstrbuton Morteza Amn 1, Jafar Ahmad 1 Department of Statstcs, School of Mathematcs, Statstcs and Computer Scence, College of Scence, Unversty of Tehran, Iran. Department of Statstcs, Ferdows Unversty of Mashhad, Iran. Abstract. Maxmum lkelhood ML estmaton based on bvarate record data s consdered as the general nference problem. Assume that the process of observng k records s repeated m tmes, ndependently. The asymptotc propertes ncludng consstency and asymptotc normalty of the Maxmum Lkelhood ML estmates of parameters of the underlyng dstrbuton s then establshed, when m s large enough. The bvarate normal dstrbuton s consdered as an hghly applcable example n order to estmate the parameter θ = µ 1,, µ, σ by ML method of estmaton based on mk bvarate record data. Asymptotc varances of the ML estmators are calculated by dervng the Fsher nformaton matrx about θ contaned n the vector of the frst k bvarate record data. As another applcaton, we concerned the problem of breakng boards of Glck 1978, Amer. Math. Monthly, 85, -6 by consderng three dfferent samplng schemes of breakng boards and we computed the relatve asymptotc effcences of ML estmators based on these three types of data. Keywords. Addtvty, bvarate dstrbuton, Fsher nformaton matrx, nverse samplng, Jensen s nequalty. MSC: 6F1, 6N0, 6F10. Morteza Amn morteza.amn@ut.ac.r, Jafar Ahmadahmad-j@um.ac.r Receved: Aprl 01; Accepted: September 01

2 36 Amn and Ahmad 1 Introducton Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 In a sequence of par-wse random varables, the values of one varable that exceed or fall below the prevous values of that varable are called record values and the correspondng value of other varable to each exceedng value s the concomtant of that record value. The bvarate sequence s then termed the sequence of bvarate record values. The reader s referred to Arnold et al for more detals on unvarate and bvarate record data. stmaton of parameters of bvarate normal dstrbuton based on bvarate ordered random varables s concerned by a few authors. Harrell and Sen 1979, Qnyng He and Nagaraja 009 and the references theren, consder the problem of estmaton based on bvarate order statstcs under the bvarate normal dstrbuton. Chacko and Thomas 008 obtaned best lnear unbased estmators of the parameters of the bvarate normal dstrbuton n terms of only concomtants of record values. The problem of dervng Fsher nformaton contaned n bvarate record values was studed by Amn and Ahmad 007, 008. Amn and Ahmad 009 studed the addtonal nformaton by consderng nter-record tmes. In ths paper we consder ML estmaton based on bvarate record data. Snce for a sngle record-breakng sample, the number of record data are rare, t s napplcable to nvestgatng the asymptotc propertes of the estmators when the number of records n a sequence gets large. In nonparametrc settng, Samanego and Whtaker 1988 extend the sngle sample results to the mult-sample case and studed the consstency of the nonparametrc maxmum-lkelhood estmator of the samplng dstrbuton. In ths samplng scheme, one repeat an nverse samplng scheme for achevng record data m tmes. The asymptotc propertes, then, can be concerned by lettng m to get large. A balanced scheme ncludes m ndependent sequences each wth the frst k record data. We show that n such a scheme, the ML estmators are consstent and asymptotcally normal as m tends to nfnty. One of the most mportant bvarate dstrbutons n statstcal nference s the bvarate normal dstrbuton. As an applcaton of the proposed estmaton we estmate the parameter θ = µ 1,, µ, σ by ML estmaton method based on mk bvarate record data. Asymptotc varances of the ML estmators are calculated by dervng the Fsher nformaton matrx about θ contaned n the vector of the frst k bvarate record data.

3 Asymptotc ffcences of the ML Based on Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 As another applcaton of the aforementoned nferental problem consder lfe testng problems n whch testng of an tem s destructve and costly. If the tems are expensve, one can set up the experment so that only the low lfe unts are destroyed. As an example, one may consder the example of testng the breakng strength of wooden beams cted n Gulat and Padgett 1994 a, b and Glck 1978, n whch beams are replaced when they do not break untl the pressure reaches the mnmum prevously observed breakng tme. In other words, only the lower record values are observed. Now, suppose that we want to ft a model for the relatonshp between the breakng strength as the response varable and some other characterstcs of wooden beams as predctors such as elastcty. Here our only observatons are the lower records of breakng strength and ther assocated concomtants. In such stuatons the problem of estmaton based on record values and ther concomtants s n a hgh degree of mportance. Suppose we want to observe the strength X and elastcty Y of mk beams by a destructve lfe test. One can consder, the followng three dfferent samplng schemes for testng the wooden beams: A: Test the beams by replacng them when they do not break untl the pressure reaches the mnmum prevously observed breakng tme. Contnue the experment untl breakng k beams. Repeat the whole experment m tmes. B: Perform test A and also save the number of beams remanng unbroken between the th and the + 1 th breakng beams n each replcaton. C: Smply test mk beams untl breakng. Test A and B are smlar tests, except that we observe an addtonal varable whch are record tmes. Indeed, n test B, we consder the addtonal nformaton that the strength of the beams remanng unbroken between the th and the + 1 th breakng beams s greater than the strength of the th breakng beam. In test C the expermenter observe a bvarate random sample of sze mk. Although, n tests A and B, more beams are tested, the number of destroyed beams n the three tests s equal. However, the tests A and B are more economcal tests, snce the more low lfe tems are destroyed. The queston s that whch samplng scheme leads to more precse estmators of populaton parameters?. Here, the relatve asymptotc effcences of the ML estmators, based on the above three types of

4 38 Amn and Ahmad Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 data, can be calculated usng the Fsher nformaton matrx about θ contaned n the vector of the frst k bvarate record data. The paper s organzed as follows. Secton contans some notatons. In Secton 3, the man results of ML estmaton and lmt theorems are proposed. Secton 4, concerns the ML estmaton of the parameters of the bvarate normal dstrbuton, usng bvarate record values. Fnally, we calculate the Fsher nformaton matrces and asymptotc relatve effcences. Notatons Let {X, Y, 1} be a sequence of..d. par-wse random varables wth the absolutely contnuous cumulatve dstrbuton functon cdf F X,Y x, y; θ and the correspondng pdf f X,Y x, y; θ. Also f X x; θ and F X x; θ denote the margnal pdf and cdf of X, respectvely and F X x; θ = 1 F X x; θ. The sequence of lower records and ther concomtants s defned as R, R ] = X T, Y T, 1, where T 1 = 1 wth probablty one and for, T = mn{j : j > T 1, X j < X T 1 }. Suppose = T +1 T 1, = 1,,..., k 1, k = 0, are the number of trals needed to obtan new records, whch are called nter-record tmes. Let us denote and Rk = R 1,..., R k, k = 1,..., k, Ck = R 1],..., R k], R m = {R 1j,..., R kj, j = 1,..., m}, C m = {R 1]j,..., R k]j, j = 1,..., m}, m = { 1j,..., kj, j = 1,..., m}, where R j, R ]j, j stands for the th record data n the jth sequence, = 1,..., k, j = 1,..., m. The jont pdf of R m and C m s see Arnold et al., 1998 f Rm,C m r, c; θ = m k j=1 =1 k 1 f X,Y r j, c j ; θ/ F X r j ; θ]. 1 =1

5 Asymptotc ffcences of the ML Based on Usng 1 the jont pdf of R m, C m and m s gven by m k f Rm,C m, m r, c, δ; θ = f X,Y r j, c j ; θ{ F X r j ; θ} δ j. j=1 =1 Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 So, the condtonal probablty mass functon of m gven R m, C m s gven by f m R m,c m δ r, c; θ = m k 1 F X r j ; θ] δ j F X r j ; θ. 3 j=1 =1 We consder lower bvarate records to derve the results of ths paper. Smlar results also hold for the case of upper bvarate record data. Hereafter, we wll call lower records, smply, records. 3 Man Results Suppose that the assumptons of Secton hold. Also, let θ = θ 1,..., θ l Θ R l and the data nvolves R m, C m, m. Usng 1 and, the lkelhood equatons based on bvarate records only are j=1 =1 θ s log f X,Y r j, c j ; θ k 1 j=1 =1 θ s log F r j ; θ = 0, s = 1,..., l. Also the lkelhood equatons based on bvarate records and nter-record tmes are gven by j=1 =1 log f X,Y r j, c j ; θ+ θ s j=1 =1 δ j log θ F r j ; θ = 0, s = 1,..., l. s The followng theorems establsh consstency and asymptotc normalty of the roots of the above equatons f exst, as m tends to nfnty. Theorem 3.1. Let ˆθ m be the ML of θ based on m ndependent sequences of the frst k bvarate record values, then ˆθ m s a consstent estmator for θ as m. Proof. Let θ 0 be the value of θ. Snce ˆθ m s the ML estmator of θ, we have log f Rm,C m ϱ m, ς m ; ˆθ m log f Rm,C m ϱ m, ς m ; θ, for all θ Θ. 4

6 40 Amn and Ahmad On the other hand by Jensen s nequalty we have θ0 log f R,Cr, s; θ fr,c r, s; θ log θ0 f R,C r, s; θ 0 f R,C r, s; θ 0 = 0. Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 So θ0 log f R,C r, s; θ θ0 log f R,C r, s; θ 0. Snce for all θ Θ 1 m log f R m,c m ϱ m, ς m ; θ tends almost everywhere to θ0 log f R,C r, s; θ as m, hence wth probablty one and for large enough m, we have 1 m log f R m,c m ϱ m, ς m ; θ 1 m log f R m,c m ϱ m, ς m ; θ 0, for all θ Θ. 5 Settng θ = θ 0 n 4 and θ = ˆθ m n 5, t s deduced that ˆθ m tends to θ 0 wth probablty one, as m. Theorem 3.. Let ˆθ m be as n Theorem 3.1, then the asymptotc dstrbuton of mˆθ m θ s N l 0, I 1 Rk,Ck θ as m, where θ s the nverse of Fsher nformaton matrx n Rk, Ck. I 1 Rk,Ck Proof. The Taylor s expanson of θ r log f Rm,Cm ϱ m, ς m ; θ for r = θ=ˆθ 1,..., l around an arbtrarly θ 1 s equal to log f Rm,C θ m ϱ m, ς m ; θ r θ=ˆθ = log f Rm,C θ m ϱ m, ς m ; θ r θ=θ1 l + ˆθ s θ 1 log f Rm,C θ r θ m ϱ m, ς m ; θ, s s=1 θ=θ where θ s a sequence n Θ whch tends n probablty to θ 1. The left hand sde of the above equalty s equal to zero and hence θ log f R m,c m ϱ m, ς m ; θ θ=θ1 l = ˆθ s θ 1 log f Rm,C θθ m ϱ m, ς m ; θ s s=1 = mˆθ θ 1 I Rk,Ck θ 1. θ=θ

7 Asymptotc ffcences of the ML Based on Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 On the other hand, θ log f R m,c m ϱ m, ς m ; θ θ=θ1 has a mean equal to zero and a varance equal to mi Rk,Ck θ 1. So, by strong low of large numbers, we have 1 m θ log f R m,c m ϱ m, ς m ; θ = m 1 m θ=θ1 θ log f R m,c m ϱ m, ς m ; θ 0 θ=θ1 has an asymptotc dstrbuton N l 0, I Rk,Ck θ 1 as m. Hence, mˆθm θ = I 1 Rk,Ck θ 1 m θ log f R m,c m ϱ m, ς m ; θ has an asymptotc dstrbuton N l 0, I 1 Rk,Ck θ as m. Remark 3.1. Smlar asymptotc results as presented n Theorems 3.1 and 3., are hold for ML estmators based on bvarate records and ther nter-record tmes by replacng f Rm,C m ϱ m, ς m ; θ wth f Rm,C m, m ϱ m, ς m, δ m ; θ and I R,C θ wth I R,C, θ n Theorems 3.1 and Bvarate Normal Parameter stmaton Suppose that m ndependent sequences each wth k bvarate records are observed and the samplng dstrbuton s bvarate normal dstrbuton wth parameter θ = µ 1,, µ, σ, ρ. Let R j, R ]j denote the th bvarate record n the jth sequence. Also, denote the standard hazard rate functon and standard reverse hazard rate functon of X, respectvely, by h 0 x = f X x; θ 0 / F X x, θ 0 and r 0 x = f X x; θ 0 /F X x, θ 0, n whch θ 0 = 0, 1,, µ, σ, ρ. Our am s to obtan the ML of θ based on bvarate records. The lkelhood equatons for bvarate record values and record tmes are as follows: j=1 =1 Rj µ 1 ρ m j=1 =1 1 ρ R]j µ j=1 =1 σ Rj µ 1 h 0 = 0, 6

8 Downloaded from jrss.rstat.r at 1: on Monday September 17th Amn and Ahmad R ]j mkµ ρσ R σ j mkµ 1 = 0, 1 j=1 =1 j=1 =1 m k j=1 =1 ρ mk R j µ 1 m k j=1 =1 R ]j µ σ 1 σ +C θρ + 1 mkρ 3 = 0, where C θ = m j=1 k =1 Rj µ 1 R]j µ /σ1 σ, and +1 ρ mk1 ρ j=1 =1 mk1 ρ j=1 =1 R j µ 1 h 0 j=1 =1 Rj µ 1 + ρc θ Rj µ 1 = 0, 7 R]j µ + ρc θ = 0. For the case of bvarate record values wthout record tmes, the above equatons reman true except that 6 and 7 are replaced, respectvely, wth Rj µ 1 ρ m R]j µ σ and j=1 =1 1 ρ mk1 ρ +1 ρ j=1 =1 j=1 =1 j=1 =1 σ j=1 =1 r 0 Rj µ 1 = 0, Rj µ 1 + ρc θ R j µ 1 r 0 Rj µ 1 = 0. The roots of the above lkelhood equatons have no closed form and these equatons have to be solved numercally. However, t s not dffcult to carry out an numercal computaton usng mathematcal packages, e.g. Maple or Mathematca.

9 Asymptotc ffcences of the ML Based on Fsher Informaton Matrx Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 In order to compute the asymptotc relatve effcences of the ML estmator of θ based on m ndependent sequences each wth k bvarate records and tmes denoted wth RCT, wth correspondng estmator based on smlar mk bvarate records wthout tmes denoted wth RC, and the estmator based on a bvarate random sample of sze mk denoted wth IID, we derve the Fsher nformaton matrces n these three data bases. We have ki X,Y θ = I j ;, j = 1..., 5, such that I j = I j ; j and I 11 = k σ 1 1 ρ, I 1 = 0, I 13 = I = k1 ρ / σ ρ, I 3 = 0, I 4 = and I 33 = kρ σ 1 ρ, I 14 = 0, I 15 = 0, kρ 4σ 1 σ 1 ρ, I 5 = k σ 1 ρ, I 34 = 0, I 35 = 0, I 44 = k1 ρ / σ 4 1 ρ, I 45 = I 55 = k1 + ρ 1 ρ. kρ σ 1 ρ, kρ σ 1 1 ρ, We denote some moments of bvarate records from standard bvarate normal dstrbuton as follows α ] = R 0,1 R 0,1 ], α ] = R 0,1 ] and α = R, where R 0,1, R 0,1 ] s the th bvarate record from standard normal dstrbuton. Suppose the dstrbuton of X, Y s standard bvarate normal wth correlaton ρ for =1,,..., then Y = ρx + ϵ, 8 where the X and the ϵ are mutually ndependent and ϵ are normal dstrbuted wth zero mean and varance equal to 1 ρ. So by consderng X-record sequence we have for 1 R 0,1 ] = ρr 0,1 + ϵ ], 9

10 44 Amn and Ahmad where ϵ ] denotes the partcular ϵ assocated wth R 0,1. Snce X and ϵ are ndependent so the sequence R 0,1 s ndependent of ϵ ], the later beng mutually ndependent, each wth the same dstrbuton as ϵ. So we can conclude from 9 that for any fnte functon g. Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 R 0,1 ] gr0,1 = ρr 0,1 gr 0,1 and R 0,1 ] gr 0,1 = ρ R 0,1 gr 0,1 + 1 ρ gr 0,1. specally we have R 0,1 ] R0,1 = α ] = ρα and R 0,1 ] ] = α ] = ρ α + 1 ρ. The log-lkelhood functon of the frst k bvarate records s equal to where lθ; Rk, Ck = L θ; R, R ] log Φ =1 =1 R µ 1 L θ; R, R ] = 1 logσ1 1 ρ 1 ρ R µ 1 R] µ + σ σ ] R µ 1 R] µ ρ., Hence, we obtan I Rk,Ck θ = I j ;, j = 1..., 5, such that I j = I j ; j and I 11 = =1 µ L θ; R, R ] 1 1 σ1 r 0R 0,1 = 1 k 1 k σ1 1 ρ + =1 R 0,1 =1 r 0 R 0,1 r0r 0,1 =1 ],

11 Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 Asymptotc ffcences of the ML Based on and I = =1 1 = 1 σ 4 1 4σ = 1 σ σ1 L θ; R, R ] 3 =1 k ρ =1 R 0,1 r 0 R 0,1 { + =1 α =1 R 0,1 r 0 R 0,1 1 4 k + 1 3/4ρ 1 ρ =1 =1 α R 0,1 r 0R 0,1 } 3 4 ρ α ] =1 R 0,1 r 0 R 0, =1 ] 1 R 0,1 r 4 0R 0,1, =1 I 33 = µ L θ; R, R ] = =1 I 44 = = 1 σ 4 = 1 σ 4 =1 σ L θ; R, R ] { k ρ =1 k + ρ 41 ρ =1 α α ] ]. =1 ] R 0,1 r 0R 0,1 R 0,1 3 r 0 R 0,1 k σ 1 ρ, }] 3 4 ρ α ] =1 ] Amn and Ahmad 007 showed that I 55 = 1 ρ k =1 α + kρ 1 ρ.

12 46 Amn and Ahmad Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 Furthermore, we have I 1 = =1 1 σ 3 1 µ 1 σ1 L θ; R, R ] k 1 r 0 R 0,1 + R 0,1 r 0R 0,1 =1 { = 1 1 σ1 3 1 ρ α ρ { k 1 =1 =1 } α ] =1 1 r 0 R 0,1 + =1 =1 { = 1 1 ρ / σ1 3 1 ρ α 1 I 13 = I 14 = I 15 = I 3 = I 4 = I 5 = I 34 = I 35 = =1 =1 R 0,1 r 0 R 0,1 + =1 =1 =1 =1 =1 =1 =1 =1 L = µ 1 µ µ 1 σ L = µ 1 ρ L = σ1 µ L = σ1 L = σ σ1 = ρl µ σ L = µ ρ L = ] R 0,1 r 0R 0,1 =1 =1 r 0 R 0,1 }] R 0,1 r 0R 0,1 kρ σ 1 ρ, ρ σ 1 ρ ρ 1 ρ α, =1 α, =1 ρ σ 1 σ 1 ρ ρ 4σ 1 σ 1 ρ ρ σ 1 1 ρ ρ σ 3 1 ρ 1 σ 1 ρ =1 α, =1 =1 α, α, =1 α, =1 α, }],

13 Asymptotc ffcences of the ML Based on and Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 I 45 = =1 σ = ρl ] ρ σ 1 α ρ k. =1 The Fsher nformaton matrx n the frst k bvarate records and record tmes s equal to I Rk,Ck, k θ = I j ;, j = 1..., 5, such that I j = I j ; j. Snce the condtonal pdf f k Rk,Ck depends only on µ 1 and and s free of other parameters, we have for {, j; 1 5, 1 j 5, j} {1, 1,,, 1, }, Furthermore I j = I j θ θj log f R,C = I j. I 11 = 1 k σ1 1 ρ =1 = 1 k σ1 1 ρ + =1 { I = 1 k σ ρ =1 3 R 0,1 1 ΦR 0,1 4 =1 ΦR 0,1 1 R 0,1 1 ΦR0,1 4 =1 k + 1 3/4ρ 1 ρ = 1 σ =1 1 ΦR 0,1 ΦR 0,1 R 0,1 r 0 R 0,1 ΦR 0,1 =1 R 0,1 3 r 0 R 0,1 α h 0 R 0,1 h 0R 0,1 + =1 ] } 3 ρ α ] h 0R 0,1 α =1 =1 =1 ] ] r 0 R 0,1 h 0 R 0,1, R 0,1 r 0 R 0,1 ] R 0,1 r 0 R 0,1 h 0 R 0,1,

14 48 Amn and Ahmad Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 and I 1 = I 1 µ 1 σ1 log f R,C δ r, s = I 1 R µ 1 log Φ =1 = 1 σ 3 1 =1 =1 1 ΦR 0,1 1 ρ / 1 ρ ΦR 0,1 µ 1 σ1 R µ 1 µ 1 σ1 log 1 Φ α 1 =1 { k 1 =1 =1 r 0 R 0,1 R 0,1 r 0 R 0,1 R 0,1 h 0 R 0,1 r 0 R 0,1 4. Asymptotc relatve effcences In ths secton, we try to fnd an answer for the queston whch samplng scheme of A, B or C, ntroduced n Secton 1 leads to more precse estmators of populaton parameters?, for the specal case of bvarate normal dstrbuton. We use the asymptotc varances of the ML estmators whch are the trace tems of the nverse of the Fsher nformaton matrces contaned n the three types of data. Snce µ 1 and µ are locaton parameters, the Fsher nformaton matrces are free of µ 1 and µ. It can be seen that I 1 1, 1 s multple of σ1, I 1, s multple of σ1 4, I 1 3, 3 s multple of σ, I 1 4, 4 s multple of σ 4 and I 1 5, 5 s free of and σ, for I be each of the three mentoned nformaton matrces, where I 1, s the element of the th row and the th column of the matrx I 1. Thus, the asymptotc relatve effcency denoted by AR gven below does not depend on µ and σ, = 1,. Hence, wthout loss of generalty, we take = σ = 1 n evaluatng I 1 Rk,Ck }] θ, I 1 Rk,Ck, k θ and ki X,Y 1 θ usng the above formulas. For smplcty, we denote the ML estmate of θ based on Rk, Ck by ˆθ A, the ML of θ based on Rk, Ck, k by ˆθ B and the ML of θ based on the bvarate random sample of sze mk by ˆθ C. Asymptotc effcency of the ML usng bvarate record values and tmes wth respect to w.r.t. that usng bvarate random sample of the same sze, and AR of the ML usng bvarate record values and tmes w.r.t. that usng bvarate record.

15 Asymptotc ffcences of the ML Based on values wthout tmes are as follows: ARˆθ,B ; ˆθ,C = ki X,Y 1,, ; I 1 Rk,Ck, k Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 ARˆθ,B ; ˆθ,A = I 1 Rk,Ck, I 1 for = 1,..., 5. Rk,Ck, k,, Tables 1 to 5 show the values of AR of µ 1,, µ, σ and ρ, respectvely. These are smlar for negatve and postve values of ρ. Snce the values varatons for Tables 1 and w.r.t. ρ are very neglgble, they are shown n 8 decmal places. The other tables values are shown n 4 decmal places. Table 1: The values of ARˆµ 1,B ; ˆµ 1,C ARˆµ 1,B ; ˆµ 1,A. ρ k Table : The values of ARˆ,B ; ˆ,C ARˆ,B ; ˆ,A. ρ k

16 50 Amn and Ahmad Table 3: The values of ARˆµ,B ; ˆµ,C ARˆµ,B ; ˆµ,A. Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 ρ k Table 4: The values of ARˆσ,B ; ˆσ,C ARˆσ,B ; ˆσ,A. ρ k Table 5: The values of ARˆρ B ; ˆρ C ARˆρ B ; ˆρ A. ρ k

17 Asymptotc ffcences of the ML Based on Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 As we can see from the Tables 1 to 5, consderng nter-record tmes along by bvarate records, provdes a dstnct mprove n estmaton of θ. Also, plans A and B lead to more precse estmators of scale and correlaton parameters of bvarate normal dstrbuton. On the other hand, locaton parameters estmates are more precse based on plan C. Furthermore, smaller values of k, for locaton parameters, and greater values of k, for scale and correlaton parameters, lead to mprovement of the estmators based on plans A and B relatve to the estmators based on plan C. Also, ARˆσ,B ; ˆσ,C s relatve to ρ and ARˆµ,B ; ˆµ,C and ARˆρ B ; ˆρ C have reverse relaton wth ρ. References Abo-leneen, Z. A. and Nagaraja, H. N. 00, Fsher nformaton n an order statstc and ts concomtant. Ann. Inst. Statst. Math., 543, Abo-leneen, Z. A. 007, Fsher nformaton n generalzed order statstcs and ther concomtants. J. Stat. Theory Appl., 6, Abo-leneen, Z. A. and Nagaraja, H. N. 008, Fsher nformaton n order statstcs and ther concomtants n bvarate censored samples. Metrka, 67, Al-Saleh, M. F. and Al-Ananbeh, A. M. 007, stmaton of the means of the bvarate normal usng movng extreme ranked set samplng wth concomtant varable. Stat. Papers, 48, Amn, M. and Ahmad, J. 007, Fsher nformaton n record values and ther concomtants about the dependence and correlaton parameters. Statst. Probab. Lett., 77, Amn, M. and Ahmad, J. 008, Comparng Fsher nformaton n record values and ther concomtants wth random observatons. Statstcs, 4, Arnold, B. C., Balakrshnan, N., and Nagaraja, H. N., 1998, Records. New York: John Wley & Sons.

18 5 Amn and Ahmad Chacko, M., and Thomas P. Y. 008, stmaton of parameters of bvarate normal dstrbuton usng concomtants of record values. Stat. Papers, 49, Downloaded from jrss.rstat.r at 1: on Monday September 17th 018 Glck, N. 1978, Breakng record and breakng boards. Amer. Math. Monthly, 85, -6. Gulat, S. and Padgett, W. J. 1994a, Smooth nonparametrc estmaton of the dstrbuton and densty functons from record-breakng data. Commun. Stat. Theory. Meth., 3, Gulat, S. and Padgett, W. J. 1994b, Nonparametrc quantle estmaton from record-breakng data. Austral. Statst., 36, Harrell F.. and Sen P. K. 1979, Statstcal nference for censored bvarate normal dstrbutons based on nduced order statstcs. Bometrka, 66, He, Q. and Nagaraja, H. N. 009, Correlaton stmaton Usng Concomtants of Order Statstcs from Bvarate Normal Samples. Commun. Statst. Theory Meth., 381, Samanego, F. J. and Whtaker, L. R. 1988, On estmatng populaton characterstcs from record-breakng observatons, II: Nonparametrc results. Nav. Res. Logst. Q., 35, 1-36.