On Bayesian prediction of future median generalized. order statistics using doubly censored data. from type-i generalized logistic model

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1 Journal of Statistical and Econometric Methods, vol.2, no.1, 213, ISS: (print), (online) Scienpress Ltd, 213 On Bayesian prediction of future median generalized order statistics using doubly censored data from type-i generalized logistic model Tahani A. Abushal 1 Abstract This paper is concerned with the problem of deriving expressions for the Bayesian predictive survival functions for the median of future sample of generalized order statistics having odd and even sizes. Both of the informative and future samples are drawn from a population whose distribution is truncated type-i generalized logistic distribution TTIGL (β, α, τ). Doubly type II censored data and two sample technique have been used here. Bayesian prediction intervals using two independent samples, based on informative prior is obtained. Bayesian prediction intervals for: upper order statistics and upper records are considered as special cases. umerical computations based on simulation study are given to illustrate the performance of the procedures. Keywords: Bayesian prediction; Double censoring; Logistic distribution; Twosample prediction; Order statistics; Records; Generalized order statistics 1 Department of Mathematics, Umm AL-Qura University, Saudi Arabia taabushal@uqu.edu.sa Article Info: Received : December 2, 212. Revised : January 11, 213 Published online : March 1, 213

2 62 On Bayesian prediction of future median generalized... 1 Introduction In Kamps (1995) generalized order statistics (GOS) have been introduced as a unified approach to several models of ordered random variables. Such models are ordinary order statistics (OOS) [David and agaraja (23), Arnold, Balakrishnan and agaraja (1992)], records [Ahsanullah (1994) and Arnold, Balakrishnan and agaraja (1998)], sequential order statistics [Cramer and Kamps (1996)] and ordering via truncated distributions and censoring schemes [Kamps (1995)]. Kamps s book (1995) gave several applications such as a variety of disciplines, recurrence relation for moments of order statistics and characterizations. Several authors utilized the GOS in their work, such authors, among others, are Ahsanullah (2), Habibullah and Ahsanullah (2), Kamps and Gather (1997), Keseling (1999), Cramer and Kamps (2), Pawlas and Szynal (21), Ahmad and Fawzy (23), AL-Hussaini and Ahmad (23a,b), AL-Hussaini (24),Ahmad (27, 21) and Jaheen (22, 25). Several authors have predicted future order statistics and records from homogeneous and heterogeneous populations that can be represented by single or finite mixtures of distributions. For a good survey see, AL-Hussaini and Jaheen (1995, 1996, 1999), AL-Hussaini and Ahmad (23b), Ali Mousa (23) and AL-Hussaini (1999, 21). The logistic distribution is one of the oldest growth models. The truncated logistic distribution plays a role in a variety of applications, also, the type I generalized logistic distribution has applications in the theoretical and practical fields. For more details on the logistic and half-logistic distributions, see Balakrishnan (1985, 1992), Balakrishnan and Wang (1991) and Balakrishnan and Chan (1992). AL-Angary (1997) introduced the truncated type I generalized logistic distribution which was denoted by T T IGL(β, γ, α) and described some of its properties. Some studies have discussed the truncated type I generalized logistic distributions such as, AL-Hussaini and Ateya (23, 25) and AL-Hussaini et al.(26). A random variable X is said to have a truncated type I generalized logistic distribution with vector of parameters Θ = (β, α, τ) if it s probability density function (PDF) is given by f(x; θ) = α τ(1 2 α ) exp[ (x β)/τ][1 + exp[ (x β)/τ]] (α+1), x β, (β, τ >, α > ). (1)

3 Tahani A. Abushal 63 The reliability and hazard rate functions are given by R(x) = 1 ω α, (2) 1 2 α r θ (x) = α(ω 1)ω α 1, (3) τ(1 ω α ) where ω = 1 + e (x β)/τ, x β. In our study we will take β =, then the vector of parameters will be θ = (α, τ). For a value x i of the random variable X, let ω i = 1 + exp x i r, ε i (α, τ) = 1 ω i α, η i (α, τ) = (ω i 1)ω i α 1 ε i (α, τ) So, (1), (2) and (3) can be written in the following forms (with β = ) as f(x i ; α, τ) = α τ (1 2 α ) 1 ε i (α, τ)η i (α, τ), x i >, (τ >, α > ). (5) The reliability and hazard rate functions are given, respectively, by (6) and (7). (4) R(x i ) = (1 2 α ) 1 ε i (α, τ), (6) and r θ (x i ) = α τ η i(α, τ). (7) We will write ε i, η i instead of ε i (α, τ), η i (α, τ). Suppose that X 1, X 2,..., X n is a random sample (rs) of size n drawn from a population whose cdf is F (x) and pdf is f(x). Let X 1;n,m,k,..., X n;n,m,k be the corresponding GOS, where m 1, k 1, see Kamps (1995). It was shown by Ahmad and Abu-Shal (28) that the joint P DF of the GOS X r1 ;n,m,k,..., X rl ;n,m,k, can be written, for < r 1 < < r l < 1, r =, r l+1 = n + 1, x =, x rl +1 =, as f r1,...,r l (x r1,..., x rl ) = C(i; r l )[h m (F θ (x r1 )) h m ()] r 1 1 [F θ (x rl )] γr l 1 [F θ (x ri )] m f θ (x ri )][h m (F θ (x ri+1 )) h m (F θ (x ri ))] r i+1 r i 1 ]f θ (x rl ), (8) l 1 [ i=1

4 64 On Bayesian prediction of future median generalized... for F 1 θ (+) < x r1 x rl < F 1 (1), where l 1 C(i; r l ) = C rl 1/ (r i+1 r i 1)! i=1 θ i= r l C rl 1 = γ i, γ i = k + (n i)(m + 1). (9) h m (z) = (1 z) m+1 /(m + 1), ln(1 z), m = 1 m 1 The joint P DF of the first r GOS X 1;n,m,k,..., X r;n,m,k, for 1 r n can be obtained if we choose r 1 = 1, r 2 = 2,..., r l = r in (8), see Kamps (1995). If we choose, in (8) and (9), r 1 = s, r 2 = s + 1,..., r l = s + l 1 r, we can easily show that f s,...,r (x s,..., x r ) = = ( 1) s 1 k r (s 1)! C r 1 (s 1)! [h r 1 m(f θ (x s )) h m ()] s 1 [ [F θ (x i )] m f θ (x i )][F θ (x r )] γr 1 f θ (x r ), ( 1) s 1 r 1 C r 1 (m + 1) s 1 (s 1)! [ [F θ (x i )] m f θ (x i )]f θ (x r )[F θ (x r )] γr 1 l= i=s i=s s 1 ω (s) l [F θ (x s )] (s l 1)(m+1), m 1, r 1 [ln(f θ (x s ))] s 1 [F θ (x r )] k 1 f θ (x r ) i=s r θ (x i ), m = 1, (1) where ω (s) l = ( 1) l( ) s 1 l and γr = k + (m + 1)(n r). Suppose that n items are simultaneously put on a life test and that for some reasons, the first s 1 failure times were not observed. The observed failure times are start only from the sth to the rth failure time, 1 < s < r < n. These ordered observations are referred to as a doubly type II censored data. Type II censoring is obtained when s = 1 and the complete sample is obtained when s = 1 and r = n. For more about doubly censored sample, reader is referred to Khen et al. (211, 21), among others. Suppose that X s < X s+1 < < X r,where 1 < s < r < n, where X i X i;n,m,k, i = 1, 2,..., n be a doubly type-ii censored random sample (informative). Let Y 1 < Y 2 < < Y,where Y i Y i;,m,k, i = 1, 2,...,, M >, K > be a second independent generalized

5 Tahani A. Abushal 65 ordered random sample (of size ) of future observations drawn from the same distribution. Based on such a doubly type II censored observations, we want to predict the future median (unobserved) Y Y. Then for ξ being a positive integer 1 we have: Y = { Y ξ;,m,k, = 2ξ 1 Y ξ;,m,k +Y ξ+1;,m,k, 2 = 2ξ. (11) 2 Density functions of future median 2.1 The case of odd let Y In the case of the odd future sample size where = 2ξ 1, ξ = 1, 2, 3,...,, donete the median of future generalized order statistics. For a given θ, the P DF of Y f Y (y θ) = is given, see Kamps (1995), by C ξ 1 (ξ 1)!(M + 1)! [F θ(y)] γ ξ 1 f θ (y)g ξ 1 M [F θ(y)], (12) where ξ Cξ 1 = γj, γj = K + ( j)(m + 1) j=1 g M (y) = h M (y) h M (), y (, 1) h M (y) = (1 y) M+1 /(M + 1), M 1 ln(1 y), M = 1. (13) By substituting Eqn.(13) in Eq.(12), yields f Y (y θ) = C ξ 1 (ξ 1)!(M + 1)! [F θ(y)] γ ξ 1 f θ (y)[h M (F θ (y)) h M (F θ ())] ξ 1. (14) By expanding [h M (F θ (y)) h M (F θ ())] ξ 1 binomially, f Y (y θ) can be written as f Y (y θ) [F θ (y)] γ ξ 1 f θ (y)σ ξ 1 j= w(j) ξ [h M (F θ (y))] j, (15)

6 66 On Bayesian prediction of future median generalized... By making use of Eq.(13) and (15), we obtain [F θ (y)] γ ξ 1 f θ (y)σ ξ 1 j= f Y (y θ) ω(j) ξ [F θ (y)] j(m+1), M 1 [F θ (y)] K 1 f θ (y)[ln(f θ (y))] ξ 1, M = 1. (16) By substituting Eqn.(5),(6) and (7) in Eq.(16), we obtain. α η τ y[(1 2 α ) 1 ε y ] γ ξ ξ 1 j= ω(j) ξ [(1 2 α ) 1 ε y ] j(m+1), M 1 f Y (y θ) α η τ y[(1 2 α ) 1 ε y ] K [ln[(1 2 α ) 1 ε y ]] ξ 1, M = 1. where γ ξ = K + ( ξ)(m + 1) and ω(j) ξ = ( 1) j( ) ξ 1 j. (17) 2.2 The case of even It can be shown when is even, that the P DF of the median Y given θ, is given by ξ j= f Y (y θ) ω j(ξ)ψ j (y θ), M 1, ϖ(y θ), M = 1, for a (18) where ψ j (y θ) = y [F θ (z)] j(m+1)+m [1 F θ (2y z)] γ ξ+1 1 f θ (2y z)f θ (Z)dz, M 1, ϖ(y θ) = y H θ (z)h θ (2y z)[s(z)] ξ 1 [1 F θ (2y z)] K dz, M = 1. By substituting Eqn.(4),(5) and (6) in Eq.(19), we obtain y. (19) α 2 ψ j (y θ) = τ [(1 2 2 α ) 1 ε z ] (M+1)(j+1) [(1 2 α ) 1 ε 2y z ] γ ξ+1 ηz η 2y z dz, M 1, y. (2) α 2 ϖ(y θ) = τ [(1 2 2 α ) 1 ε 2y z ] K [ln[(1 2 α ) 1 ε z (α, τ)]] ξ 1 η z η 2y z dz, M = 1.

7 Tahani A. Abushal 67 Based on the GOS X s;n,m,k, X s+1;n,m,k,..., X r;n,m,k, for s r n, the likelihood function can be written, see Ahmad and Abu-Shal (28), as r 1 L(θ x ) [h m (F θ (x s )) h m ()] s 1 [ [F θ (x i )] m f θ (x i )][F θ (x r )] γr 1 f θ (x r ) i=s [ r 1 i=s [F θ(x i )] m f θ (x i )][F θ (x r )] γr 1 f θ (x r ) s 1 l= = ω(s) l [F θ (x s )] (s l 1)(m+1), m 1, [ln(f θ (x s ))] s 1 [F θ (x r )] k 1 f θ (x r ) r 1 i=s r θ(x i ), m = 1, (21) where Θ = (α, τ) and x = (x s;n,m,k, x s+1;n,m,k,..., x r;n,m,k ) = (x s,..., x r ). Using Eqs.(4),(5),(6) and (7) in Eq.(21), we get ( α τ )r s+1 [ε (1 2 α ) γ 1 r] γ r+1 [ r i=s εm+1 i η i ] s 1 l= ω(s) l L(α, τ x ) (1 2 α ) l(m+1) ε (m+1)(s l 1) s, m 1, ( α τ )r s+1 [ln εs ] s 1 ε k (1 2 α ) k 1 2 r[ r α i=s η i], m = 1. (22) Suppose that the conjugate prior density,which is measured by a function π(α, τ) given by π(α, τ) = π 1 (τ α)π 2 (α). (23) Suppose that π 1 (τ α) is Gamma (c 1, α),π 2 (α) is Gamma (c 2, c 3 ) with respective densities π 1 (τ α) α c 1 τ c 1 1 exp( τα), α, τ >, (c 1 > ), (24) π 2 (α) α c 2 1 exp( c 3 α), α >, (c 2, c 3 > ). (25) It then follows, by substituting (24) and (25) in (23), that the prior P DF of α and τ is given by π(α, τ) α c 1+c 2 1 τ c 1 1 exp[ α(τ + c 3 )], α, τ >, (c 1, c 2, c 3 > ), (26) where c 1, c 2 and c 3 are the prior parameters (or hyper-parameters). Using the likelihood function (22) and the prior (26) the posterior probability density function of α and τ for given informative data, say, π (α, τ x ) is given by π (α, τ x ) L(α, τ; x )π(α, τ). (27)

8 68 On Bayesian prediction of future median generalized... Using the likelihood function (22) and the prior (26) the posterior pdf of α and τ can be written using (27) as α I τ [ε r ] γ r+1 (1 2 α ) γ 1 exp[ α(τ + c 3 )][ r i=s εm+1 i η i ] s 1 π l= (α, τ x ) ω(s) l (1 2 α ) l(m+1) ε (m+1)(s l 1) s, m 1 α I τ (1 2 α ) k exp[ α(τ + c 3 )][ln εs ] s α ε k r[ r i=s η i], m = 1, (28) where I = c 1 + c 2 + r s and = c 1 r + s 2. 3 Bayesian prediction 3.1 The case of odd future sample size By making use of Eqn.(28) and (17), yields the Bayes predictive density function of the future median Y, = 2ξ 1, ξ = 1, 2, 3,...,, given the (r s + 1) gos s, denoted by h Y (y θ) as h Y (y x ) = f Y (y θ)π (θ x )dθ, y > x r. (29) Θ By making use of Eqs.(28) and (17)in (29), we obtain α I+1 τ 1 ε γ r+1 r η y exp[ α(τ + c 3 )][Π r i=sε m+1 η i ][(1 2 α ) 1 ε y ] γ ξ ξ 1 j= f Y (y θ)π (θ x ) where s 1 l= ζ(ξ,s) l,j ε (m+1)(s l 1) s, m 1, M 1, (1 2 α ) γ l+1 [(1 2 α ) 1 ε y ] j(m+1) α I+1 τ 1 η y (1 2 α ) k exp[ α(τ + c 3 )][(1 2 α ) 1 ε y ] K [ln εs (1 2 α ) ]s 1 ε k r[π r i=sη i ][ln[(1 2 α ) 1 ε y ]] ξ 1, m = 1, M = 1, (3) ζ (ξ,s) l,j = ( 1) l+j ( s 1 l )( ξ 1 j i ). (31) To obtain (1 τ)1 Bayesian prediction interval BPI for a future generalized order statistic Y, say (L, U), we solve simultaneously the following two nonlinear equations, numerically, P [Y > L x ] = L h Y (y θ)dy = 1 τ 2, (32)

9 Tahani A. Abushal 69 P [Y > U x ] = U h Y (y x )dy = τ 2. (33) Equation (32) and (33), can be solved by using ewton-raphson iteration form as follows: L L j+1 = L j j h Y (y x )dy (1 τ ) 2, (34) h Y (L j x ) U j+1 = U j U j h Y (y x )dy τ 2, (35) h Y (U j x ) where the initial values L o,u o can be taken equal to x r. The integrals in (34) and (35) can be obtained using the routine QDAGI in IMSL Order statistics case The Bayes prediction density function of the future median Y, = 2ξ 1, ξ = 1, 2, 3,...,, given the informative sample x s,..., x r, can be written from (29) and (3), when m =, k = 1,M = and K = 1 for as Y as h Y (y θ) = A 1 α I+1 τ 1 ε n r r η y exp α(τ+c 3) [Π r i=sε i η i ] ξ 1 s 1 [(1 2 α ) 1 ε y ] ξ+1 ζ (ξ,s) j,l (1 2 α ) (n l) j= l= [(1 2 α ) 1 ε y ] j ε s l 1 s dαdτ, (36) where A 1 is a normalizing constant. Using the iteration forms (34) and (35), the routine QDAGI in IMSL to compute the double and triple integrals we obtain the prediction bounds of Y Record values case Making use of Eq.(3), yields the Bayes predictive density function of the future median Y, = 2ξ 1, ξ = 1, 2, 3,..., when m = 1, k = 1,M = 1 and K = 1 for Y as h Y (y x ) = A 1 [Π r i=sη i ][ln[(1 2 α ) 1 ε y ]] ξ 1 [ln α I+1 τ 1 η y ε y ε r (1 2 α ) 2 exp[ α(τ + c 3 )] ε s (1 2 α ) ]s 1 dαdτ, (37)

10 7 On Bayesian prediction of future median generalized... Using the iteration forms (34) and (35), the routine QDAGI in IMSL to compute the double and triple integrals we obtain the prediction bounds of Y. 3.2 The case of even future sample size Substituting Eqs.(18) and (28) in the integrand of (29), with = 2ξ, ξ = 1, 2,...,, we obtain Where h Y (y x ) = y Θ f Y (y x )π (θ x )dθ, y >, (38) αi+2 τ 2 η z η 2y z [ε r ] γ r+1 (1 2 α ) γ 1 exp α(τ+c 3) [Π r i=sε m+1 i η i ][(1 2 α ) 1 ε 2y z ] γ ξ s 1 l= [(1 2 α ) 1 ε z ] (M+1)(j+1) (1 2 α ) l(m+1) f Y (y x )π (θ x ) ε s (m+1)(s l 1) dz, m 1, M 1 y αi+2 τ 2 η z η 2y z ε k r(1 2 α ) k exp α(τ+c 3) [(1 2 α ) 1 ε 2y z ] K [ln[(1 2 α ) 1 ε z ]] ξ 1 [Π r i=sη i ][ln εs ] s 1 dz, m = 1, M = α ξ j= ζ(ξ,s) j,l (39) Order statistics case Making use of Eq.(39), the Bayes predictive density function of the future median Y, = 2ξ, ξ = 1, 2, 3,..., when m =, k = 1,M =, K = 1, for Y is h Y (y x ) = A 1 s 1 l= ξ j= α I+2 τ 2 η z η 2y z (1 2 α ) (n 1) exp α(τ+c 3) ε n r+2 r [Π r i=sε i η i ][(1 2 α ) 1 ε 2y z ] ξ+1 ω j (ξ)ω l (s) [(1 2 α ) 1 ε z ] j+1 (1 2 α ) l ε s l 1 s dαdτ. (4)

11 Tahani A. Abushal 71 s = 1 The Baysian prediction bounds of future order statistics in caseis case of h Y (y x ) = A 1 [Π r i=sε i η i ][(1 2 α ) 1 ε 2y z ] ξ+1 [α I+2 τ 2 η z η 2y z ε n r+2 r (1 2 α ) (n 1) exp α(τ+c 3) ξ ω j (ξ)[(1 2 α ) 1 ε Z ] j+1 ]dαdτ, (41) j= Using the iteration forms (34) and (35), the routine QDAGI in IMSL to compute the double and triple integrals we obtain the prediction bounds of Y Record values case Making use of Eq.(39), yields the Bayes predictive density function of the future median Y, = 2ξ 1, ξ = 1, 2, 3,..., when m = 1, k = 1,M = 1, K = 1 is h Y (y x ) = A 2 where A 1 2 = α I+2 τ 2 η z η 2y z ε r (1 2 α ) 1 exp α(τ+c 3) [Π r i=sη i ] [(1 2 α ) 1 ε 2y z ][ln[(1 2 α ) 1 ε z ]] ξ 1 [ln 1 2 α ]s 1 dαdτ, (42) y α I+2 τ 2 η z η 2y z ε r (1 2 α ) 1 exp α(τ+c 3) [Π r i=sη i ] [(1 2 α ) 1 ε 2y z ][ln[(1 2 α ) 1 ε z ]] ξ 1 ε s [ln 1 2 α ]s 1 dαdτdz. (43) Using the iteration forms (34) and (35), the routine QDAGI in IMSL Library to compute the double and triple integrals we obtain the prediction bounds of Y. ε s Remark 3.1. (1) It may be noted that if s = 1, we obtain a type II censored sample technique and if we choose s = 1 and r = n, our results reduce to the complete sample case. (2) In some situtions one can not observe some initial values of any sample, so he must be chosen the doubly type II censoring scheme.

12 72 On Bayesian prediction of future median generalized... 4 umerical computations In this section, we obtain the interval predictors of future median OOS and ordinary upper record when the underlying population has T T IGL distribution according to the following steps. 4.1 OOS (m =, k = 1) Based on a given doubly Type-II censored sample, We will consider here the case which represents the OOS. The 95% Bayesian prediction bounds for the future median, and their actual (simulated) prediction levels, are obtained according to the following steps: (1) A random sample of sizes (1, 3, 5) are generated from TTIGL distribution. The sample is ordered and the upper 1%, 2%, 25%,3%,4% and 5% are censored where the future sample size is fixed and set to 5, 8. The computation are carried out for the hyper-parameters cases (c1, c2, c3) are set to (.7, 1.3,.85), the data generated from TTIGL (,.977, 1.25) and (,.8, 2.239). (2) Based on these doubly Type-II censored data, the 95% Bayesian prediction bounds for the future (unobserved) are then calculated by solving Eqs.(34) and (35) for the lower and upper bounds with τ = 95% for different values of, when = 2ξ 1 is odd and = 2ξ is even. (3) For 1, generated future ordered samples each of size, from TTIGL density, the simulated prediction levels of Y are then calculated. The prediction are conducted on the basis of a doubly type II censored samples and type II censored samples. The results are presented in Tables 1-2.

13 Tahani A. Abushal 73 Table 1: The lower limit (LL), the upper limit (UL), the length of the BPI and the percentage coverage of the 95% BPI for the future median Y when = 2ξ 1 is odd or = 2ξ from TTIGL(β =, α =.977, τ = 1.25). Case Y s = 1 s = 2 s = 3 (LL, UL) % (LL, UL) % (LL, UL) % Length Length Length n = 1 Y3 (.115,1.564) 91.7 (.59,1.799) 88.5 (.342,1.818) 85.7 r = Y4 (.123,1.4843) 85.7 (.95,1.8716) 86.5 (.123,1.994) n = 1 Y3 (.245,1.491) 91.8 (.115,1.5916) 92.8 (.368,1.785) 9.7 r = Y4 (.14,1.315) 89.5 (.1281,1.6311) 92.3 (.125,1.8145) n = 3 Y5 (.942,1.97) 88.7 (.7163,1.883) 87.4 (.7542,1.965) 89.3 r = Y6 (.855,.88645) 84.4 (.963,.94552) 85.5 (.851,.9885) n = 3 Y5 (.421,.9848) 92.7 (.5,1.96) 92.5 (1.3,2.93) 84.5 r = Y6 (.94,.8765) 89.5 (.8,.8614) 88.2 (.94,.9165) n = 3 Y5 (.82,.7944) 94.7 (.65,.857) 93.5 (1.42,1.9199) 86.5 r = Y6 (.814,.76142) 95.5 (.744,.832) 97.2 (.865,.894) n = 5 Y5 (.641,1.93) 92.7 (.441,1.2135) 92.5 (.613,1.2187) 91.5 r = Y6 (.81,.86142) 84.5 (.7114,.913) 88.2 (.85,.9885) n = 5 Y5 (.416,.9289) 92.7 (1.41,2.135) 86.5 (1.41,2.48) 84.5 r= Y6 (.6499,.7723) 93.5 (.963,.94552) 96.2 (.7114,.913) n = 5 Y5 (.91,.8485) 95.7 (.631,1.237) 96.5 (1.43,2.64) 87.5 r= Y6 (.9625,.7455) 92.5 (.8,.7614) 92.2 (.672,.7863)

14 74 On Bayesian prediction of future median generalized... Table 2: The lower limit (LL), the upper limit (UL), the length of the BPI and the percentage coverage of the 95% BPI for the future median Y when = 2ξ 1 is odd or = 2ξ from TTIGL (β =, α =.8, τ = 2.239). Case Y s = 1 s = 2 s = 3 (LL, UL) % (LL, UL) % (LL, UL) % Length Length Length n = 1 Y3 (1.7371,5.12) 9.7 (1.6711,5.8874) 91.5 (2.587,6.5371) r = Y4 (1.4135,5.5819) 91.7 (1.5289,5.1217) 91.5 (2.9,6.7622) n = 1 Y3 (2.6371,5.6468) 88.3 (1.97,5.9) 85.8 (3.7371,6.489) 96.7 r = Y4 (2.1621,5.4787) 9.5 (2.5581,6.617) 86.3 (3.618,6.535) n = 3 Y5 (1.7986,5.397) 86.7 (1.98,6.62) 89.4 (3.7986,8.671) 87.7 r = Y6 (1.174,5.9458) 88.3 (.175,4.1468) 9.5 (3.39,7.49) n = 3 Y5 (2.95,6.7) 87.2 (2.7987,6.396) 92.5 (2.731,7.17) 81.5 r = Y6 (1.692,4.6411) 93.5 (1.12,4.137) 92.2 (2.131,6.756) n = 3 Y5 (1.896,4.76) 92.7 (1.78,3.59) 92.5 (2.718,5.2182) 92.5 r = Y6 (1.183,3.45) 94.5 (1.179,3.646) 93.2 (2.16,5.6) n = 5 Y5 (1.76,5.96) 96.7 (2.7987,6.47) 96.5 (4.31,1.398) 96.5 r = Y6 (2.143,5.1247) 94.5 (2.275,5.168) 95.2 (5.117,9.358) n = 5 Y5 (1.93,3.993) 96.7 (2.887,6.76) 96.5 (2.97,6.2969) 96.5 r = Y6 (1.1178,3.2128) 94.5 (2.174,5.1458) 96.2 (2.9572,6.9641) n = 5 Y5 (1.7987,3.396) 96.7 (1.18,4.367) 95.5 (2.887,5.5666) 96.5 r = Y6 (1.174,3.1458) 96.5 (1.1142,3.1899) 97.2 (2.175,4.1458)

15 Tahani A. Abushal OURV (m = 1, k = 1) Our interest is in the median future recored, because generating of record sample takes a relatively longer time than the OOS, we use n = 1, 15 from TTIGL (α =.8, τ = 2.239) distribution and apply 2% and 5% censoring. The future sample size is fixed. The vector of hyperparameters (c 1, c 2, c 3 ) are same as in OOS example (.7, 1.3,.85). Based on these doubly Type-II censored data, the 95% Bayesian prediction bounds for the future (unobserved) are then calculated by solving Eqs.(3.6) and (3.7) for the lower and upper bounds. For 1, generated future recored samples each of size = 4, 5, from TTIGL density, the percentage coverage for the simulated prediction levels of Y are then calculated. The results are presented in Table 3. Table 3: The lower limit (LL), the upper limit (UL), the length of BPI and the percentage coverage of the 95% BPI for the future median Y of ordinary upper record values when = 2ξ 1 is odd or = 2ξ from TTIGL (β =, α =.8, τ = 2.239). Case Y s = 1 s = 2 (LL, UL) % (LL, UL) % Length Length n = 1 Y3 (3.635,7.5723) 88.7 (3.1134,7.1229) 89.1 r = Y2 ( ,7.9231) 92.5 (2.157,6.4831) n = 1 Y3 (2.9617,6.7875) 93.6 (2.4493,6.1373) 89.7 r= Y2 (3.9838,7.6229) 93.1 (3.144,6.6493) n = 15 Y3 (1.2967,4.2712) 84.7 (2.2593,5.448) 92.5 r = Y2 (1.3983,4.5923) 85.6 (2.8332,5.1774) n = 15 Y3 (2.1489,4.8712) 92.6 (2.658,5.6841) 92.7 r= Y2 (3.24,5.3824) 94.1 (2.958,5.7516)

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