Metodoloski zvezki, Vol. 13, No. 1, 2016, 1-15
Interval Prediction of Order Statistics Based on Records by Employing Inter-Record Times: A Study Under Two Parameter Exponential
Distribution
Morteza Amini1 and S.M.T.K. MirMostafaee2
Abstract
In this note, we propose a parametric inferential procedure for predicting future order statistics, based on record values, which takes inter-record times into account. We utilize the additional information contained in inter-record times for predicting future order statistics on the basis of observed record values from an independent sample. The two parameter exponential distribution is assumed to be the underlying distribution.
1 Introduction
Suppose Y1,... ,Ym are independent and identically distributed (iid) observations from an absolutely continuous cumulative distribution function (cdf) F, possessing probability density function (pdf) f. The order statistics of the sample Y1,..., Ym, represented by Yi:m < ■ ■ ■ < Ym:m, are obtained by arranging the sample in an increasing order. Order statistics have been used in a wide range of applications, including robust statistical estimation, detection of outliers, characterization of probability distributions, goodness-of-fit tests, entropy estimation, analysis of censored samples, reliability analysis, quality control and strength of materials. A useful survey of available results until 2003 is given in the book of David and Nagaraja (2003).
Let X1,X2,... be a sequence of iid random variables, independent of and identically distributed to Y1. An observation Xj is called an upper (lower) record value if its value exceeds (resp. falls below) those of all the previous observations, that is the nth upper (resp. lower) record value, Un (resp. Ln), is defined as XTn, where T1 = 1, with probability 1, and Tn = min{j : j > Tn-1, Xj > XTn-1} (resp. Tn = min{j : j > Tn-1, Xj < XTn-1}), for n > 1. Throughout this paper we
department of Statistics, School of Mathematics, Statistics and computer Science, College of Science, University of Tehran, P.O. Box 14155-6455, Tehran, Iran; morteza.amini@ut.ac.ir
2 Department of Statistics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran; m.mirmostafaee@umz.ac.ir
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Amini and MirMostafaee
deal with upper record values for a predictive inference. Similar results can be obtained for the case of lower record values. The inter-record time statistic, defined as
As = Ts+1 - Ts, s > 1,
is the number of observations between sth and (s + 1)th record values. For more details we refer the reader to Arnold et al. (1998). Record data arise in a wide variety of practical situations including industrial stress testing, finance, meteorological analysis, hydrology, seismology, sporting and athletic events, and mining surveys.
The problem of predicting future observations has been extensively studied in the literature and several parametric and non-parametric procedures are developed for prediction. In many practical data-analytic situations, one is interested in constructing a prediction interval on the basis of available observations. There are situations in which the available observations and the predictable future observation are of the same type. The prediction of future records on the basis of observed records from the same distribution and prediction of order statistics based on order statistics are studied, among others, by Dunsmore (1983), Nagaraja (1984), Chou (1988), Awad and Raqab (2000), Raqab and Balakrishnan (2008) and the references therein.
Recently, Ahmadi and Balakrishnan (2010), Ahmadi and MirMostafaee (2009), Ah-madi et al. (2010) and MirMostafaee and Ahmadi (2011), discussed the prediction of future records from a Y-sequence based on the order statistics observed from an independent X-sequence, and vice versa.
In predicting future order statistics on the basis of observed record statistics, sometimes the available observations also include inter-record times which can be utilized as additional information to improve the predictive inference. In other words, when both record values and the inter-record times are available, it would be nice to employ the information included in both records and record times. Feuerverger and Hall (1998) emphasized that "However, the record times and record values jointly contain considerably more information about F than the record values alone." Actually, applying the additional information about record times is not a new subject and several authors focused on inference based on both record values and record times, see for example Samaniego and Whitaker (1986), Lin et al. (2003), Doostparast (2009), Doostparast and Balakrishnan (2013), Kizilaslan and Nadar (2014) and MirMostafaee et al. (2016).
In this paper, a two parameter exponential distribution, Exp(p, a), with pdf
f (x; a) = 1 e-{x-^)/a, x > ^ G R, a > 0,	(1.1)
a
is considered as the underlying distribution. We write Z ~ Exp(p, a) if the pdf of Z can be expressed as (1.1). Note that ^ and a are the location and scale parameters, respectively. Throughout this paper we assume that both parameters, ^ and a, are unknown.
Now, suppose that Y1, ■ ■ ■ ,Ym constitute a future random sample from a two parameter exponential distribution, i.e. Yi, ■ ■ ■ , Ym ~ Exp(^, a) and Yi:m < ■ ■ ■ < Yj:m are the corresponding order statistics of this sample. In addition, Ym = m-1 Y:m denotes the mean of this future sample. If Y1, ■ ■ ■ ,Ym denote the times to failure of m independent units in a lifetime test, then Ym can be interpreted as the mean time on test of these failed units. We assume that the available data include the observed upper record
Interval Prediction of Order Statistics...
3
values, U1, ■ ■ ■ , Un, given the inter-record times, (Ai,..., An-1). We emphasize that these record values are assumed to be extracted from a sequence of iid random variables {Xj,j = 1,2, ■ ■ ■ } where Xj ~ Exp(p, a) for j = 1,2, ■ ■ ■. Moreover, the sequence {Xj, j = 1, 2, ■ ■ ■ } and the sample {Yi, i = 1, ■ ■ ■ , m} are statistically independent. Note that n is the number of the observed record values and depends on the experiment, however, m is the sample size of the future observations and it can be considered arbitrary. In addition, n and m are unrelated. The problem of interest is to obtain conditional prediction intervals for jth future order statistic, Yj:m, as well as for the mean, Ym, in a future sample on the basis of the available data. We compare our conditional prediction intervals with the unconditional ones proposed by Ahmadi and MirMostafaee (2009) and observe an improvement over the predictive inference without inter-record times. Therefore, we consider two cases: (a) The informative data contain only the upper record values, (b) The informative data contain the upper record values and the inter-record times, and then we observe that case (b) has some predictive inferential improvement in comparison with case (a).
The rest of the paper is organized as follows. Some general preliminaries are presented in Section 2. Conditional prediction intervals for the future jth order statistic, Yj:m, and the mean of the future sample, Ym, based on record values of given inter-record times for the two parameter exponential distribution are studied in Sections 3 and 4. An illustrative example and some concluding remarks are involved in Sections 5 and 6. The R codes for computing some results of the paper are given in the appendix.
2 Preliminaries
In this section, we present some general preliminary results used in future sections. Given upper record values u1,..., un-1, which are observed and extracted from the sequence {Xj; j > 1}, inter-record times A1,..., An-1 are independent geometrically distributed random variables with success probabilities F(ui), i = 1,..., n — 1. Furthermore, the record values U1,... ,Un form a Markov Chain with adjacent transition pdf equal to the left truncated pdf of the underlying distribution, see Arnold et al. (1998). Thus, the joint distribution of Un = (U1,..., Un) and An = (A1,..., An-1) is
fUnAn (Un, Sn) = J] f (Ui)[F (Ui)]'1-1/K),	(2.1)
i=1
where un = (u1,..., un) G Xn, in which X is the support of X and Sn = (51,..., 5n-1) G Nn-1, see Samaniego and Whitaker (1986) and Arnold et al. (1998) page 169. We emphasize that An contains n — 1 positive integer-valued discrete random variables and 8n is the observed vector of An. By integrating (2.1) with respect to (w.r.t.) u1,...,un, we can easily prove the following result.
Lemma 1 The joint probability mass function of A1,..., An-1 is
n-1
Pa„ (¿n) = Pr(An = Sn) = ^ cj (n' 5n)[(a1 (n, j, Sn) + 1)(a1(n, j, Sn) + an(n,j, Sn) + 2)]--
j=1
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Amini and MirMostafaee
where
"j-2 / n-ji-1 \ n-j-2 / n-j N
cj(n, ¿„) = (-1)n-j-1 u E * H E ^
jl=0 \t=n-j + 1 / j2=0 Vi=j2 + 2 ./ n-j	n-1
ai(n,j, 6n) = E ^ - 1' an(n,j, ¿n) = E
t=1	t=n-j+1
in which we assume for a > t=a = 0 and J}b=a ¿t = 1.
In this paper, we need the conditional distribution of U1 and Un given by An = ¿n as follows.
Lemma 2 The conditional pdfof U1 and Un given An = ¿n is
n-1
1
fuuun\ A„ (ui,un| ¿n) = [Pa„ (¿n)]-1 ^ Cj (n, ¿„)[F(ui)]ai	)/ (u„),
j=1
where Cj(n, ¿n), a1(n,j, 8n), an(n,j, ¿n) and Pa„ (¿n) are as in Lemma 1.
The proof of Lemma 2 is straightforward by integrating (2.1) w.r.t. u2,... ,un-1 and dividing the obtained equation by PA„ (¿n).
3 Conditional prediction intervals for order statistics
In this section, the goal is to find a conditional prediction interval for Yj:m when the observed U1,... ,Un are available given An = ¿n for the two parameter exponential distribution.
To this end, we consider the pivotal quantity
w =	(31)
Note that the pivotal quantity Wj is the same as the one considered by Ahmadi and MirMostafaee (2009). This quantity is location and scale invariant namely it is free of both unknown parameters i.e. the location parameter / and the scale parameter a. It is also a simple function of both observed and future statistics, so that the future statistic can be derived from it easily. Ahmadi and MirMostafaee (2009) found the unconditional distribution of Wj while we present the conditional distribution of Wj given An = ¿n, (i.e. the inter-record times are assumed to be known and fixed) in the following theorem.
Theorem 1 The conditional cdfof Wj in (3.1) given An = ¿n is for w > 0
m n-1 l ai(n,ji, Sn) a„(n,ji, Sn)	Sn)^ ^a„(n,ji,	l ^
Fw3\An(w^n) = EEE E E 1 lyj,^) ~
l=j ji = 1 j2=0 j3=0	J4 = 0	n'
xcji (n, ¿n)[(j2 + m - l + j3 + j4 + 2)((j2 + m - l)w + j + 1)]-1,
Interval Prediction of Order Statistics...
5
and for w < 0
FWj |A„ M^ = ^^^ I]	I] l (— 1) j +:3+:4 p/% ) j2
l=jjl = 1 j2=0 33 = 0	34 = 0	AnV nJ
xcji (n, ¿„)[(j2 + m - l + j3 + j4 + 2)(j4 + 1 - w(j3 + j4 + 2))] —1,
where a1(n,j1, 6n) and an(n,j1, 6n) are defined in Lemma 1 and PAn (5n) is the joint mass function of A1,..., An—1 which is also given in Lemma 1.
Proof: Letting J*^ = (Un - U1)/a, U* = (U1 - ß)/a and Yj*m = (Yj:m - ß)/a, we may write
i'^O i'^O
FWj | An (wl^n) = J j0 %m (vw + U)fu*,J*A | An (u, v|^n) du dv	(3-2)
00
For t > 0, we have
i=j
Also, from Lemma 2, we obtain
m
(t) = £ (7) (1 - e—')l
"')::- e—<)le—<m—l>'.	(3.3)
n-1
/u.,j:i|A„(u,v| ¿n) = [Pa„(¿n)]-^ Cj(n, ¿n)[1 —e-u]ai (n'3',")[1 —e-("+v)]a"(n35n)e-(2u+v).
3=1
(3.4)
Hence, by substituting (3.4) and (3.3) in (3.2) and using the binomial expansions, we have for w > 0,
m n-1 l ai(n,ji, Sn) a„(n,ji, Sn) /m\/ai(n,ji, ¿„)\/a„(n,ji, S„)\f l\ (n s )
F	(w| A )	V	V	{l){	33	A	J4	3Cji (n 0n)
| An	^	¿^ -(-1)32+33+34 Pa (S )-
l=jji=132=0 33=0	34=0	V '	AnV n7
x	/ e-(3'2+m-l+33+34 + 2)«e-((32+m-l)w+34+1)u d^ dV
00
and therefore we naturally arrive at the desired expression. Similarly, we may attain the expression for FWj| An(w|Sn) when w < 0 after substituting (3.4) and (3.3) in (3.2) by noting that the integral w.r.t. u must be taken from — vw to to.	□
Let wY(n, m, j; Sn) be the 7th conditional quantile of W3 given An = Sn, i.e.
Pr(Wj < w7(n,m,j; Sn)| An = Sn) = Y-
To find 100(1 — a)% two-sided conditional prediction intervals for l}:m based on record values given An = Sn, we have to find the conditional quantiles wai (n,m,j; Sn) and w1-a2 (n, m, j; Sn), for a1 + a2 = a, 0 < a < 1, i = 1, 2, numerically.
Now, a 100(1 — a)% conditional prediction interval for Y}:m based on record values given An = Sn, is given by
(U1 + wai(n,m,j; Sn)(Un — U1), U1 + w1-Q2(n,m,j; Sn)(Un — U1)). (3.5)
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Amini and MirMostafaee
Table 1: The values of wo.025(3, m, j), wo.975(3, m,j), wo.975(3, m, j) - wo.o25(3,m, j),
wo.025(3, m, j; Sn), wo.975(3, m, j; Sn), wo.975(3, m, j; Sn) - wo.o25(3, m, j; Sn), for m = 10,20, j = 5,7,10 (for m = 10), j = 12,17,20 (for m = 20) and different values of
Sn-
	m		10			20	
	j	5	7	10	12	17	20
Unconditional	W0.025	-3.671	-2.814	-0.907	-3.140	-1.760	-0.380
	W0.975	1.278	2.635	9.761	1.827	4.767	12.186
	w0.975 — w0.025	4.949	5.449	10.668	4.967	6.527	12.566
Sn = (1, 2)	W0.025	-1.097	-0.500	0.249	-0.651	0.055	0.464
Pa„ (Sn) = 0.0833	W0.975	1.766	3.502	11.868	2.459	6.108	14.586
	w0.975 — w0.025	2.863	4.002	11.619	3.110	6.053	14.122
Sn = (1, 3)	W0.025	-1.288	-0.652	0.201	-0.827	-0.025	0.420
Pa„ (Sn) =0.05	W0.975	1.290	2.627	9.481	1.786	4.675	11.690
	W0.975 — W0.025	2.578	3.279	9.280	2.613	4.700	11.270
Sn = (1, 4)	W0.025	-1.427	-0.774	0.160	-0.965	-0.098	0.386
Pa„ (Sn) = 0.0333	W0.975	1.022	2.106	7.984	1.398	3.793	9.872
	W0.975 — W0.025	2.449	2.880	7.824	2.363	3.891	9.486
Sn = (2, 3)	W0.025	-2.181	-1.267	0.045	-1.538	-0.320	0.324
Pa„ (Sn) =0.0167	W0.975	1.027	2.413	10.212	1.502	4.669	12.787
	W0.975 — W0.025	3.208	3.680	10.167	3.040	4.989	12.463
Sn = (2,4)	W0.025	-2.330	-1.409	-0.008	-1.697	-0.415	0.289
Pa„ (Sn) =0.0119	W0.975	0.823	1.976	8.880	1.193	3.896	11.163
	W0.975 — W0.025	3.153	3.385	8.888	2.890	4.311	10.874
Conditionally on Sn, we get more information about the unknown parameters ^ and a, or generally more information about F, which leads to better prediction intervals for Yj:m. It is noted that conditioning on inter-record times does not decrease the length of the prediction interval necessarily and increase or decrease in the location and scale of the interval depend on the values of Sn. For the purpose of illustration, consider the conditional quantiles of Wj, which are computed and tabulated in Table 1, for a = 0.05, n = 3, m = 10, 20, j = 5,7,10 (m = 10), j = 12,17,20 (m = 20) and some values of Sn. The values of unconditional quantiles of Wj in Table 1 are taken from Ahmadi and MirMostafaee (2009), Tables 3 and 4. By comparing the entries of Table 1, one can observe that for a few cases, the conditional prediction intervals have bigger lengths, especially when we predict the biggest future order statistic, i.e. Ym:m. But note that in the most cases the conditional intervals are shorter than the unconditional ones for different values of Sn, so we may conclude that generally the conditional prediction approach leads to shorter (and hence better) prediction intervals in average for different values of Sn and this can be considered as an improvement.
Interval Prediction of Order Statistics...
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4 Conditional Prediction Intervals for the mean of future sample
The problem of constructing a conditional prediction interval for Ym on the basis of observed U1,..., Un, given An = Sn, using the pivotal quantity
V™ = I-! •	(41)
is considered for the two parameter exponential distribution in this section. Note that the pivotal quantity Vm has been also considered by Ahmadi and MirMostafaee (2009) and its unconditional distribution has been obtained by them. Moreover, Vm is also location and scale invariant and therefore is free of the unknown location and scale parameters. The following theorem presents the conditional distribution function of Vm given An = 8n.
Theorem 2 The conditional distribution function of Vm in (4.1) given An = 6n is
m-1 n-1 l ai(n,ji, Sn) a„(n,ji, Sn) (ai(n,ji, ¿„)\(a„(n,ji,	l
FV_|A.(xiin) = 1 -E jTjr	1 (jj.)«
l=0 ji = 1 j2=0 js=0	j4=0	V '	AnV n>
^ Cji (n , ¿n)xj2mlr(l - j + 1)r(j2 + 1)
X (m + j3 + j4 + 2)l-j2+1(mx + j4 + 1)j2+1'
for x > 0, and
n-1 al(n,jl, Sn) a„(n,ji, S„) ( 1)js+j4 (ai(n,ji, S„)\(a„(n,ji, S„)\ (n r )
(*) = EE E (-1) ( jS K j4 K (n' 'n)
^ j=0 j4=0 Pa„ (¿n)(2+ j3 + j4)[j4 + 1 - (2+ j3 + j4)x]
m-1 n-1 l ai(n,ji, Sn) a„(n,ji, Sn) l—j2 (ai(n,ji, Sn)\ (a„(n,ji, 6„)W l
- EEE E E EVjS j4
(-1)53+54+55 PA (¿n)/l
1=0 ji=1 j2=0 j3=0	j4=0 j5=0 V 7	AnV n'
x Cj (n, ¿n)xj2+j5m'r(l - j2 + 1)r(j2 + j5 + 1)
X j5l(m + j3 + j4 + 2)1-j2-j5 + 1 [j4 + 1 - (j3 + j4 + 2)x]j2+j5 + 1 ' for x < 0, where a1(n, j1, 6n) and an(n, j1, 6n) are given in Lemma 1
Proof: Let = (Un - Ux)/o, UJ" = (U1 - v)/a and F' = (F™ - Note that
Fym\ A„ (x|^n)^ / (vx + u)/ur,j;il A„ CM^ du dV (4.2)
00
where fuj 11 An (u, v| ¿n) is given in (3.4). Since mF' ~ r(m, 1), that is for t > 0
FYm (t) = 1 - 'I: ^	(4.3)
1=0
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Amini and MirMostafaee
so by substituting (3.4) and (4.3) in (4.2) and using the binomial expansions, we get for
x < 0
n-1 ai(n,ji, Sn) a„(n,ji, Sn) fa1(n,j1, S„)\fa„(n,ji, S„)\ c (n s )
rp	, ,J5 n _ V^	V^	V^	^	j3	A	j4	/tj1 °n>
FVm | A„ (x|Sn)~	^
j1 = 1 j3 = 0	j4=0	n
/»TO /»to
x / / e-(j3+j4+2)"e-(j4+1)v du dv
./0 J-vx
m-1 n-1 l ai(n,ji, Sn) a„(n,ji, 5„) /ai(n,ji, 6„)\/a„(n,ji, 5„)w l \
- ^ ^ ^ X/ X/ j3 j4
(-1)j3+j4 PA (¿Jll 1=0 ji = 1 j2=0 j3 = 0	j4 = 0	7	'
/»TO /»TO
j (n, ¿„)xj2m1 / / e-(m+j3+j4+2)"e-(mx+j4+1)vu1-j2 vj2 du dv
./0 J-vx
n-1 ai(n,j'i, 5„) an(n,ji, Sn) ( —1)j3+j4^ ai (n,ji, |"a„(n,ji, Cj (n £ £ £ j3 j4 ji ' n
ji = 1 j3=0 j4=0 PAn (¿n)(2+ 33 + j4)[j4 + 1 - (2+ 33 + 34)*]
m-1 n-1 I ai(n,ji, Sn) a„(n,ji, 5„) Ij /ai(n,j'i, ¿„)\^a„(n,ji, 6n)\/ 1 \
__ ^ ^ ^	^	^	V^ ^	j3	A	j4	AjV
(
1=0 ji = 1 j2=0 j3=0	j4=0 j5=0
(-1)j3+j4+j5 PAn (¿n)/l
x Cji (n, ¿n)xj2+j5mly(l - 32 + 1) fTO e-(j4+1-(j3+j4+2)x)vvj2+j5 dv
j5l(m + 33 + 34 +2)1-j2-j5 + 1 J0
and therefore we naturally attain the desired result. Similarly, we may deduce the desired expression for FVm | A„ (x|Sn) when x > 0.	□
To find conditional prediction interval for Ym based on records given An = Sn, we have to find the conditional quantiles of Vm given An = Sn, vai (n, m; Sn) and
v1-a2 (n, m; Sn), for a1 + a2 = a, 0 < a < 1, i = 1, 2, numerically, where
Pr(Vm < v7(n, m; Sn) | An = Sn) = YA 100(1 — a)% conditional prediction interval for Ym based on record values given An = Sn then is
(U + Vai (n, m; Sn)(Un — U1),U1 + V1-a2(n, m; Sn)(Un — U1)). (4.4) An illustrative example has been presented in Section 5.
5 An illustrative example
x
In this section, we illustrate the proposed procedures by considering a real data set. A rock crushing machine has to be reset if, at any operation, the size of rock being crushed
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Table 2: 95% CPIs and UPIs for Yi2:20, Y>0:20 and Y20 for Example 1.
CPI
UPI
Yi2:20 (0,24.17836)
(0, 54.061745)
Y20:20 (13.290315,183.67385) (0,307.85602)
Y20
(0,26.233175) (0,61.32183)
is larger than any that has been crushed before. The following data given by Dunsmore (1983) are the sizes dealt with up to the third time that the machine has been reset:
9.3, 0.6, 24.4, 18.1, 6.6, 9.0, 14.3, 6.6, 13.0, 2.4, 5.6, 33.8. The record values were the sizes at the operation when resetting was necessary. Dunsmore (1983) assumed that these data follow an Exp(0, a) distribution. Clearly, we have
Consider a future sample of size m = 20. We want to find equi-tailed 95% conditional prediction intervals (CPIs) for Yi2:2o, Y20:20 and Y20 using (3.5) and (4.4) and compare these intervals with unconditional ones (UPIs). The results are given in Table 2. Note that some lower bounds have got negative values, which were replaced by zero. We can see that the conditional prediction intervals are shorter than the corresponding unconditional ones.
6 Concluding remarks
In this paper, we found prediction intervals for the future order statistics based on record values, given record time statistics, when the underlying distribution is two parameter exponential. These intervals have the advantage of utilizing more information embedded in the observed sequence in comparison with their corresponding unconditional ones obtained by Ahmadi and MirMostafaee (2009). These ideas can be extended to the non-parametric and the Bayesian context. The conditional point predictors are also of interest. Work on these problems is currently under process and we hope to report these findings in future papers.
Ui = 9.3, U2 = 24.4, U3 = 33.8,
T1 = 1, T2 = 3, T3 = 12,
Ai = 2, and A2 = 9.
Acknowledgement
We are very grateful to the respected editor and the respected referees for their insightful comments and suggestions which have led to this improved version.
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[15]	MirMostafaee, S.M.T.K., Amini, M. and Balakrishnan, N. (2016): Exact nonpara-metric conditional inference based on k-records, given inter k-record times. Journal of the Korean Statistical Society, Accepted.
[16]	Nagaraja, H.N. (1984): Asymptotic linear prediction of extreme order statistics. Annals of the Institute of Statistical Mathematics, 36, 289-299.
[17]	Raqab, M.Z. and Balakrishnan, N. (2008): Prediction intervals for future records. Statistics and Probability Letters, 78, 1955-1963.
[18]	Samaniego, F.J. and Whitaker, L.R. (1986): On estimating population characteristics from record-breaking observations. I. parametric results. Naval Research Logistics Quarterly, 33, 531-543.
Appendix
Here, we present the R codes for computing the conditional cumulative distribution functions of Wj, (see Theorem 1) and Vm (see Theorem 2). R functions for computing the unconditional cumulative distribution functions of Wj and Vm (see Ahmadi and MirMostafaee, 2009) are also given.
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooo cjn function oooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
cjn=function(n,j,delta){
z=(-ir(n-j-1)
z1=n-j+1
z2=j-2
z4=n-j-2
z5=n-j
s=1
if(z2>=0 & z1>=0){ for(j1 in 0:z2){ z3=n-j1-1
ss=ifelse(z3>=z1,sum(delta[z1:z3]),0)
s=s*ss }}
t=1
if(z4>=0){ for(j2 in 0:z4){ z6=j2+2
tt=ifelse(z5>=z6,sum(delta[z6:z5]),0)
t=t*tt }}
return(z/t/s)
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Amini and MirMostafaee
}
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M:n c- c- -r\ v-/--\"K -n"K-i~l-i-l-T7 r\-F n^ "I -I- -n 9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-ooooooooo Mass piUbablliiy U! Delta ooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
pdelta=functiun(n,delta){
n1=n-1
pdel=0
foi(jj in 1:n1){ nj=n-jj
nj1=n-jj+1 A=cjn(n,jj,delta)
a1=ifelse(nj>=1,sum(delta[1:nj]),0)-1 an=ifelse(n1>=nj1,sum(delta[nj1:n1]),0) C=(a1 + 1)* (a1+an+2)
pdel=pdel+A/C }
ietuin(pdel) }
9-9-9-9-9-9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'^ ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
9-9-9-9-9-9-9-9-9-	r*. -F TaT	9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-
ooooooooo CUnditiUnal cdf Uf W	ooooooooooooooooooooooo
9-9-9-9-9-9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9'9-ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
Fw=function(n,j,m,w,delta){
n1=n-1
pw=0
foi(l in j:m){ foi(j1 in 1:n1){ for(j2 in 0:l){ nj1=n-j1+1 nj=n-j1
a1=ifelse(nj>=1,sum(delta[1:nj]),0)-1 an=ifelse(n1>=nj1,sum(delta[nj1:n1]),0) foi(j3 in 0:a1){ for(j4 in 0:an){
A=chouse(m,l)*chouse(a1,j3)*chouse(an,j4)*chouse(l,j2)
*((-1)"(j2+j3+j4))*cjn(n,j1,delta)/pdelta(n,delta)
B=j2+m-l+j3+j4+2
if(w<0) C=B*(j4+1-w*(j3+j4+2))
if(w>=0) C=B*(w*(j2+m-l)+j4+1)
pw=pw+A/C }}}}}
Interval Prediction of Order Statistics...
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return(pw) }
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
2-2-2-2-2-2-2-2-2-	\T	2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-
ooooooooo LUnUitlUndl CU! of V	ooooooooooooooooooooooo
2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2^
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Fv=functiun(n,m,v,Ueltd){
pv=0
n1=n-1
m1=m-1
if(v>=0){
fur(l in 0:m1){
fur(j1 in 1:n1){
for(j2 in 0:l){
nj1=n-j1+1
nj=n-j1
d1=ifelse(nj>=1,sum(Ueltd[1:nj]),0)-1 dn=ifelse(n1>=nj1,sum(Ueltd[nj1:n1]),0) fur(j3 in 0:a1){ for(j4 in 0:an){
A=chuuse(d1,j3)*chuuse(an,j4)*choose(l,j2)/factorial(l) /pUelta(n,Ueltd)*((-1)~(j3 + j4))
B=cjn(n,j1,Uelta)*(v"j2)*(m"l)*gammd(l-j2+1)*gammd(j2+1) /((m+j3+j4+2)~(l-j2+1))/((m*v+j4+1)~(j2+1))
pv=pv+A*B }}}}}}
if(v>=0) pv=1-pv
pv1=0
pv2 = 0
if(v<0){
fur(j1 in 1:n1){
nj1=n-j1+1
nj=n-j1
d1=ifelse(nj>=1,sum(Ueltd[1:nj]),0)-1 an=ifelse(n1>=nj1,sum(Ueltd[nj1:n1]),0) fur(j3 in 0:a1){ fur(j4 in 0:an){
A=((-1)"(j3+j4))*choose(a1,j3)*chuuse(an,j4)*cjn(n,j1,Ueltd) /pUelta(n,Ueltd)/(2+j3+j4)/(j4+1-v*(2+j3+j4))
pv1=pv1+A }}}
fur(l in 0:m1){ fur(j1 in 1:n1){ fur(j2 in 0:l){
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Amini and MirMostafaee
nj1=n-j1+1 nj=n-j1
a1=ifelse(nj>=1,sum(delta[1:nj]),0)-1 an=ifelse(n1>=nj1,sum(delta[nj1:n1]),0) for(j3 in 0:a1){ for(j4 in 0:an){ lj2=l-j2
for(j5 in 0:lj2){
A=choose(a1,j3)*choose(an,j4)*choose(l,j2)/factorial(l) /pdelta(n,delta)*((-1)~(j3+j4+j5)) B=cjn(n,j1,delta)*(v~(j2 + j5))*(m~l)*gamma(l-j2+1) *gamma(j2+j5+1)/factorial(j5)/((m+j3+j4+2)"(l-j2-j5+1)) /((j4+1-v*(j3+j4+2))"(j2+j5+1))
pv2=pv2+A*B }}}}}}
pv=pv1-pv2 }
return(pv) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% unconditional cdf of W %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
FwU=function(n,j,m,w){ pw=0
if(w<0) pw=(m-j + 1) * ((1-w)"(1-n))/(m+1) if(w>=0){ ss=0 j1=j-1
for(i in 0:j1){
ss=ss+choose(j1,i)*((-1)"i)*((1+w*(m-j+i+1))"(1-n))
/(m-j+i+1)/(m-j+i+2) }
pw=1-j*choose(m,j)*ss }
return(pw) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% unconditional cdf of V %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
FvU=function(n,m,v){ pv=0
if(v<0) pv=((1-v)~(1-n))/((1 + 1/m)~m)
if(v>=0){
Interval Prediction of Order Statistics...
15
m1=m-1
s1=0
s2=0
for(i in 0:m1){ nn=n+i-2
s1=s1+choose(nn,i)*((1-1/(m*v+1))~i)*((1/(m*v+1))~(n-1)) * ((m/(m+1))"(m-i))
s2=s2+choose(nn,i)*((1-1/(m*v+1))~i)*((1/(m*v+1))~(n-1)) }
pv=s1+1-s2 }
return(pv) }