Prediction for Progressively Type-II Censored Competing Risks Data from the Half-Logistic Distribution

Essam K. AL-Hussaini; Alaa H. Abdel-Hamid; Atef F. Hashem

doi:10.2991/jsta.d.200224.004

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Volume 19, Issue 1, March 2020, Pages 36 - 48

Prediction for Progressively Type-II Censored Competing Risks Data from the Half-Logistic Distribution

Authors

Essam K. AL-Hussaini¹, Alaa H. Abdel-Hamid²^{, *}, Atef F. Hashem²

¹Department of Mathematics, Faculty of Science, Alexandria University, Egypt

²Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Egypt

^*Corresponding author. Email: hamid_alh@science.bsu.edu.eg

Corresponding Author

Alaa H. Abdel-Hamid

Received 13 April 2018, Accepted 21 December 2019, Available Online 5 March 2020.

DOI: 10.2991/jsta.d.200224.004 How to use a DOI?
Keywords: Maximum likelihood predictor; Bayesian prediction; Competing risks model; Progressive type-II censoring; Half-logistic distribution; Two-sample prediction; Simulation
Abstract: Point and interval predictions of the s-th order statistic in a future sample are discussed. The informative sample is assumed to be drawn from a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz and half-logistic distributions. The informative and future samples are progressively type-II censored, under competing risks model, and assumed to be obtained from the same population. A special attention is paid to the half-logistic distribution. Using six different progressive censoring schemes, numerical computations are carried out to illustrate the performance of the procedure. An illustrative example based on real data is also considered. The biases, mean squared prediction errors of the maximum likelihood predictors, coverage probabilities and average interval lengths of the Bayesian prediction intervals are computed via a simulation study.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Prediction is an important problem in the statistical inference. It is the problem of inferring the values of unknown observables (future observations), or functions of such observables, from current available (informative) observations. One-sample and two-sample schemes are two commonly used schemes of prediction. A predictor could be a point or an interval predictor. Prediction has many applications in the field of quality control, reliability, medical sciences, business, engineering and has been studied by many authors, including [1–9]. AL-Hussaini et al. [10] obtained Bayesian one-sample prediction of future order statistics from the half-logistic distributions, based on progressively type-II censored sample under competing risks model.

In reliability and survival analysis, engineering, demographic, actuarial literature, econometric, biological or medical studies, the units might fail owing to one of various causes. These causes compete in order to fail the units. This is called in the statistical literature “competing risks.” The data for the competing risks model consist of the lifetime of the failed unit and an indicator variable denoting the cause of failure. In this article, we study the competing risks model under the assumption of only two independent causes of failure and the lifetimes of n units to be tested are independent and identically distributed (iid). We assume that, for i=1,…,n, Ti1 and Ti2 are two iid random variables (RVs) and Ti=min{Ti1,Ti2} where Tij denotes the lifetime of unit i when it fails because of cause j, j=1,2.

Let, for j=1,2, the cumulative distribution function (CDF), Fj(t), of a RV Tij be of the form

Fj(t)≡Fj(t;βj)=1−exp−ujt,0≤b1≤t≤b2≤∞,0,otherwise,(1)

where βj is a positive parameter and uj(t)≡uj(t;βj)=−ln[1−Fj(t)] is a non-negative, continuous, monotone increasing and differentiable function of t such that uj(t)→0,ast→b1+ and uj(t)→∞,ast→b2−, j=1,2. In particular, any CDF with positive domain (suitable for life testing) is a special case of (1). For example, the Weibull, exponential, Rayleigh, compound Weibull (Burr type XII), compound exponential (Lomax), compound Rayleigh, Pareto, beta, Gompertz and compound Gompertz distributions are all special cases of (1). See [1,11,12]. It may be remarked that distributions, defined over the whole real line may be represented by (1) with the appropriate choice of uj(t). The probability density function (PDF), fj(t), corresponding to CDF (1), is given by

fj(t)≡fj(t;βj)=uj′(t)exp−uj(t),(2)

where uj′(t) is the hazard rate function (HRF) and exp [−uj(t)] is the survival function. The prime denotes derivative with respect to t, throughout.

The CDF of Ti is given by

F(t)≡F(t;β)=1−∏j=12[1−Fj(t)]=1−exp−ut,(3)

where

u(t)≡u(t;β)=∑j=12uj(t),(4)

and β=(β1,β2) is a vector of positive parameters.

The PDF f (t), corresponding to CDF (3), is given by

f (t)≡f (t;β)=u′texp−ut.(5)

In medical or industrial applications, censoring usually applies when the experimenter is unable to get total information on lifetimes for each unit or reducing the total test time and the associated cost. Type-I and type-II are two commonly used censoring schemes (CSs), see, for example, [13, 14, 15]. These types of censoring cannot allow the experimenter to remove units from a life test at various stages during the experiment. The experimenter can overcome this problem by using progressive type-II censoring which is considered to be a generalization of type-II censoring. It allows the experimenter to remove units from a life test at various stages during the experiment. For more details on progressive censoring, see [16,17].

In this paper, based on two-sample prediction technique, we discuss point and interval predictions of the s-th order statistic in a future sample based on progressively type-II censored sample generated from a general class of distributions under competing risks model. The results are then applied to the half-logistic population.

The rest of the article is organized as follows: In Section 2, progressive type-II censoring is described. In Section 3, the maximum likelihood predictors (MLPs) are provided. Section 4 presents the Bayesian two-sample prediction method. The half-logistic distribution is considered in Section 5. In Section 6, numerical computations and simulations are given. Concluding remarks are finally presented in Section 7.

2. PROGRESSIVE TYPE-II CENSORING

The progressive type-II censoring under competing risks model can be applied as follows:

Suppose that n units are put on the experiment.
Suppose m(<n) and CS R1,…,Rm are fixed before the experiment and n=m+∑i=1mRi, where Ri represents the number of removed units at the i-th failure, i=1,…,m.
The experiment terminates when the m-th failure occurs at which Rm=n−m−∑i=1m−1Ri surviving units are all removed from the experiment.
The data from progressively type-II censored sample under competing risks model are as follows: (t1:m:n;a1;R1)<⋯<(tm:m:n;am;Rm) where a1,…,am denote the m causes of failure.

Note that the complete sample case is achieved when Ri=0, i=1,…,m and n=m. But, if Ri=0, i=1,…,m−1 and Rm=n−m, then progressive type-II CS reduces to traditional type-II CS.

Based on Equations (1) and (2), and the progressively type-II censored sample, the likelihood function is then given by, see [18],

L(β;t)∝∏i=1m∏j=12hj(ti)I(ai=j)∏l=12(1−Fl(ti))Ri+1 =∏j=12∏r=1mjuj′(trj)exp−∑i=1m(Ri+1)u(ti),(6)

where hj(ti)=fj(ti)∕(1−Fj(ti)) is the HRF, ti=min{ti1,ti2}, i=1,…,m, t = (t1,…,tm), mj=∑i=1mI(ai=j) is the number of failures due to cause j, j=1,2, m=m1+m2 and

I(Ai=j)=1,Ai=j,0,otherwise,

and Ai is a discrete RV with realization ai.

3. MAXIMUM LIKELIHOOD PREDICTOR

Suppose that (T1:m:n;a1;R1),…,(Tm:m:n;am;Rm) is an informative progressively type-II censored sample of size m obtained from a sample of size n with progressive CS (R1,…,Rm) under competing risks model. Suppose also that (Y1:m⋆:n⋆;a1⋆;R1⋆),…,(Ym⋆:m⋆:n⋆;am⋆⋆;Rm⋆⋆) is a future (unobserved) independent progressively type-II censored sample of size m⋆ obtained from a sample of size n⋆ with progressive CS (R1⋆,…,Rm⋆⋆) under competing risks model, where a1⋆,…,am⋆⋆ denote the m⋆ causes of failure. Our object, in this section, is to obtain MLP of the s-th order statistic, Ys, in the future sample, s=1,…,m⋆, based on an informative progressively type-II censored sample.

Following Balakrishnan et al. [19] and Kamps and Cramer [20], the conditional PDF of Ys, s=1,…,m⋆ is given by

g (ys|β)=Bs−1⋆ f ( ys)∑ℓ=1scℓ1−F ( ys)γℓ−1,(7)

where

Bs−1⋆=∏ℓ=1sγℓ,γℓ=n⋆−∑q=1ℓ−1(Rq⋆+1),(ifℓ=1,γ1=n⋆),cℓ=∏d=1(d≠ℓ)s1γd−γℓ,(ifs=1,c1=1).

The conditional PDF (7) of Ys, using Equations (3) and (5), becomes

g(ys|β)=Bs−1⋆u′(ys)∑ℓ=1scℓexp−γℓu(ys).(8)

The predictive likelihood function (PLF) of Ys and β is given by, see, for example, [5],

g⋆(ys,β;t)=g(ys|β)L(β;t) =Bs−1⋆u′(ys)∑ℓ=1scℓexp−γℓu(ys)×∏j=12∏r=1mjuj′(trj)exp−∑i=1m(Ri+1)u(ti). (9)

The logarithm of PLF (9) is then given by

lng⋆=lnBs−1⋆+lnu′(ys)+ln∑ℓ=1scℓexp−γℓu(ys)+∑j=12∑r=1mjlnuj′(trj)−∑i=1m(Ri+1)u(ti). (10)

The predictive maximum likelihood estimates (PMLEs) (β1^β2^) of (β1, β2) and MLP ys^ of ys can be obtained by setting to zero the first partial derivatives of (10) with respect to (β1, β2) and ys and then solve the resultant equations using any iteration method, the predictors yield.

A (1−α⋆)100% prediction intervals (PIs) for future order statistic Ys can be constructed, using the MLP ys^, as follows

ys^−zα⋆∕2var(ys^),ys^+zα⋆∕2var(ys^),

where zα⋆∕2 is the percentile of standard normal distribution with right-tale probability of α⋆∕2, var(ys^) can be calculated using the inverse of observed Fisher information matrix, see, for example, [21,22].

4. BAYESIAN TWO-SAMPLE PREDICTION

In the following section, we discuss the Bayesian predictive intervals for future order statistics under progressively type-II censored competing risks data. Suppose that both of the two parameters β1 and β2 are unknown. Suppose also that the prior belief of the experimenter is that they are independent and each of them has a gamma distribution. Therefore, the joint prior density for the parameters is of the form

π(β1,β2)=π1(β1)π2(β2),(11)

where

π1(β1)=α2α1Γ(α1)β1α1−1exp−α2β1,β1>0,(α1,α2>0),(12)

π2(β2)=α4α3Γ(α3)β2α3−1exp−α4β2,β2>0,(α3,α4>0).(13)

Waller and Waterman [23] showed that the family of gamma distribution may be used as priors in Bayesian reliability analysis. Thus, gamma prior density may be rich enough to cover the prior belief of the experimenter.

From (12) and (13), Equation (11) becomes

π(β1,β2)=Dβ1α1−1β2α3−1exp−(α2β1+α4β2),β1,β2>0,(α1,α2,α3,α4)>0,(14)

where D=α2α1α4α3∕(Γ(α1)Γ(α3)).

From (6) and (14), the joint posterior PDF of β1 and β2 is then given by

π∗(β1,β2|t)=K−1ηexp−δ,(β1,β2)>0,(15)

where

K=∫0∞∫0∞ηexp−δdβ1dβ2,η=β1α1−1β2α3−1∏j=12∏r=1mjuj′(trj),δ=α2β1+α4β2+∑i=1m(Ri+1)u(ti).

Using Equations (8) and (15), the Bayesian predictive PDF of Ys, s=1,…,m⋆, is then given by

f⋆(ys|t)=∫0∞∫0∞g(ys|β1,β2)π∗(β1,β2|t)dβ1dβ2 =K−1Bs−1⋆∑ℓ=1scℓ∫0∞∫0∞ηu′(ys)exp−(δ2+γℓu(ys))dβ1dβ2. (16)

The Bayesian prediction intervals (BPIs) for Ys, s=1,…,m⋆, are obtained by evaluating P(Ys>v|t) for some given values of v. It follows from (16) that

P(Ys>υ|t)=K−1Bs−1★S(υ),υ>0,(17)

where

S(υ)=∑ℓ=1scℓγℓ∫0∞∫0∞ηexp⁡[−(δ+γℓu(υ))]dβ1dβ2.(18)

Since 1=P(Ys>0|t)=K−1Bs−1⋆S(0), then K−1Bs−1⋆=1S(0). So that

P(Ys>υ|t)=S(υ)S(0).(19)

Notice that

S(0)=∑ℓ=1scℓγℓ∫0∞∫0∞ηexp−δdβ1dβ2.

A100τ% BPIs for Ys, s=1,…,m⋆, can be obtained by solving the following two nonlinear equations for lower and upper bounds LB and UB, respectively,

P(Ys>LB|t)=1+τ2,P(Ys>UB|t)=1−τ2,

which are equivalent to the two equations

S(LB)−1+τ2S(0)=0,S(UB)−1−τ2S(0)=0.(20)

5. APPLICATION TO A HALF-LOGISTIC DISTRIBUTION

In this section, we discuss MLP and BPI for the s-th order statistics Ys under a half-logistic distribution. The half-logistic (or folded-logistic) distribution is one of the family of logistic distributions. It has been applied as a life testing model by Balakrishnan [24]. Many authors, including Balakrishnan and Puthenpura [25], Balakrishnan and Chan [26], Rosaiah et al. [27] and Kim and Han [28] have studied the half-logistic distribution.

Let X be a logistic RV with CDF given by

F X(x)=1+exp−θx−1,−∞<x<∞,(θ>0),

where θ is a scale parameter.

If T=|X|, and βj≡θ then T has a half-logistic distribution, the CDF for which is

F Tj(t)=1−exp−βjt1+exp−βjt,t>0,βj>0,(21)

and its PDF is given by

fTj(t)=2βjexp⁡[−βjt](1+exp⁡[−βjt])2,(22)

which is monotone decreasing on [0,∞). Therefore, the PDF under competing risks model that corresponds to (5) takes the form

f (t)= ∑j=12βj1+exp−βjt∏j=1221+expβjt.(23)

The PDFs given by Equations (22) and (23) and the HRFs are plotted in Figure 1. It can be noticed from this figure that the PDFs are all decreasing functions with mode at 0, while the HRFs are increasing constant functions.

It may be noticed that Equation (21) can be obtained from (1) by putting

ujt=ln1+expβjt2.(24)

Using Equation (24), the MLP and BPI for Ys under a half-logistic distribution are given by solving (10) and (20), respectively.

Remark 5.1.

The half-logistic distribution includes one parameter which facilitates the calculations required in the estimation and prediction methods. It has also an advantage rather than the exponential distribution since the first has increasing hazard rate while the latter has constant hazard rate.

6. SIMULATION STUDY AND ILLUSTRATIVE EXAMPLES

6.1. Simulation Study

The following steps are followed to generate progressively type-II censored samples from CDF (21).

For given values of the prior parameters (α1, α2, α3, α4), which are chosen to satisfy the following conditions of unbiased estimates,
EB1=α1α2=β1,EB2=α3α4=β2,
where E denotes the expectation and B1 and B2 are two RVs with realizations β1 and β2, respectively, generate values for the parameters (β1, β2), using Equations (12) and (13), see [29,30].
For given values of n and m (1 <m<n), generate two independent random samples each of size n from Uniform (0, 1) distribution, (U1j, …, Unj), j=1,2.
Generate two random samples (t1j,…,tnj), j=1,2, from CDF (21) as follows
tij=−1βjln1−Uij1+Uij,i=1,…,n,j=1,2.
Calculate ti such that, ti= min{ti1,ti2}, i=1,…,n.
For given values of the CS R1,…,Rm apply the progressive type-II censoring as described in Section 2, and hence obtain the ordered sample (t1:m:n,…,tm:m:n), which represents progressively type-II censored sample under competing risks model.

6.2. Simulation Procedure

In this subsection, a Monte Carlo simulation study is carried out in order to determine MLPs and BPIs for Ys, s=1,…,m⋆. The following six CSs are applied in the generation of the informative and future sample:

CS1:
Ri=1,i=1,…,n−m,Ri=0,otherwise,Ri⋆=1,i=1,…,n⋆−m⋆,Ri⋆=0,otherwise,
which means that we remove one unit after each observed failure of the first n−mn⋆−m⋆ failures in the informative (future) sample.
CS2:
Ri=n−m,i=1,Ri=0,otherwise,Ri⋆=n⋆−m⋆,i=1,Ri⋆=0,otherwise,
which means that we remove n−mn⋆−m⋆ units after the first observed failure in the informative (future) sample.
CS3:
Ri=n−m,i=m2,Ri=0,otherwise,Ri⋆=n⋆−m⋆,i=m⋆2,Ri⋆=0,otherwise,
which means that we remove n−mn⋆−m⋆ units after the middle observed failure in the informative (future) sample.
CS4:
Ri=n−m,i=m,Ri=0,otherwise,Ri⋆=n⋆−m⋆,i=m⋆,Ri⋆=0,otherwise,
which means that we remove n−mn⋆−m⋆ units after the last observed failure in the informative (future) sample.
CS5:
Ri=n−m2,i=1,m,Ri=0,otherwise,Ri⋆=n⋆−m⋆2,i=1,m⋆,Ri⋆=0,otherwise,
which means that we remove n−m2(n★−m★2) units after the first and last observed failures in the informative (future) sample.
CS6:
Ri=n−m3,i=1,m2,m,Ri=0,otherwise,Ri⋆=n⋆−m⋆3,i=1,m⋆2,m⋆,Ri⋆=0,otherwise,
which means that we remove n−m3(n★−m★3) units after the first, middle and last observed failures in the informative (future) sample.

Through the simulation procedure, the values m=18 (with n=30), in the informative sample, and m⋆=10 (with n⋆=10 and 16), in the future sample, have been chosen.

Based on 1000 informative samples, MLPs for Ys, s=2,4,6,8,10, biases and mean squared prediction errors (MSPEs) are presented in Table 1. A 95% PIs for Ys, s=2,4,6,8,10, in addition to the average interval lengths (AILs) of the PIs, are also included in Table 1.

	n* = m* = 10						n* = 16 and m* = 10
CS	Ys	MLP	Bias	MSPE	PI	AIL	MLP	Bias	MSPE	PI	AIL
1	Y2	0.2074	−0.1699	0.0952	(0.0938, 0.3210)	0.2272	0.1364	−0.1200	0.0458	(0.0618, 0.2109)	0.1490
	Y4	0.6553	−0.1538	0.1655	(0.4202, 0.8903)	0.4701	0.4359	−0.1159	0.0840	(0.2771, 0.5947)	0.3176
	Y6	1.1984	−0.1580	0.2800	(0.8206, 1.5761)	0.7555	0.8713	−0.1391	0.1761	(0.5904, 1.1523)	0.5619
	Y8	1.9042	−0.2027	0.5185	(1.3437, 2.4646)	1.1209	1.5818	−0.2235	0.4636	(1.1059, 2.0576)	0.9518
	Y10	3.2687	−0.5587	2.1717	(2.3248, 4.2126)	1.8878	2.9423	−0.5790	2.1348	(2.0808, 3.8038)	1.7230
2	Y2	0.2044	−0.1676	0.0928	(0.0938, 0.3150)	0.2212	0.1613	−0.1524	0.0732	(0.0725, 0.2501)	0.1776
	Y4	0.6535	−0.1534	0.1651	(0.4223, 0.8847)	0.4624	0.5979	−0.1582	0.1563	(0.3837, 0.8122)	0.4285
	Y6	1.1902	−0.1570	0.2759	(0.8219, 1.5586)	0.7367	1.1237	−0.1610	0.2653	(0.7720, 1.4755)	0.7035
	Y8	1.8977	−0.2020	0.5142	(1.3472, 2.4483)	1.1010	1.8550	−0.2074	0.5154	(1.3183, 2.3917)	1.0734
	Y10	3.2552	−0.5577	2.1585	(2.3404, 4.1700)	1.8296	3.2073	−0.5602	2.1627	(2.3067, 4.1080)	1.8013
3	Y2	0.2059	−0.1688	0.0941	(0.0924, 0.3195)	0.2270	0.1319	−0.1160	0.0427	(0.0590, 0.2047)	0.1457
	Y4	0.6555	−0.1536	0.1656	(0.4204, 0.8905)	0.4701	0.3850	−0.1008	0.0642	(0.2445, 0.5255)	0.2811
	Y6	1.1930	−0.1574	0.2769	(0.8148, 1.5711)	0.7563	0.7796	−0.1549	0.1753	(0.5242, 1.0351)	0.5109
	Y8	1.9166	−0.2041	0.5254	(1.3502, 2.4831)	1.1329	1.5016	−0.2393	0.4711	(1.0386, 1.9647)	0.9261
	Y10	3.2943	−0.5639	2.2038	(2.3432, 4.2455)	1.9023	2.9263	−0.5896	2.2054	(2.0529, 3.7997)	1.7468
4	Y2	0.2071	−0.1697	0.0956	(0.0909, 0.3233)	0.2324	0.1321	−0.1161	0.0428	(0.0583, 0.2059)	0.1476
	Y4	0.6611	−0.1550	0.1682	(0.4184, 0.9038)	0.4854	0.3856	−0.1010	0.0647	(0.2424, 0.5288)	0.2864
	Y6	1.1850	−0.1563	0.2741	(0.8050, 1.5650)	0.7600	0.6730	−0.0965	0.0956	(0.4522, 0.8938)	0.4417
	Y8	1.9130	−0.2038	0.5244	(1.3457, 2.4804)	1.1347	1.0037	−0.0970	0.1375	(0.6959, 1.3114)	0.6156
	Y10	3.2265	−0.5529	2.1219	(2.2794, 4.1737)	1.8942	1.3496	−0.1011	0.1855	(0.9511, 1.7481)	0.7970
5	Y2	0.2070	−0.1696	0.0948	(0.0926, 0.3213)	0.2287	0.1436	−0.1278	0.0521	(0.0643, 0.2230)	0.1587
	Y4	0.6579	−0.1542	0.1664	(0.4218, 0.8940)	0.4722	0.4681	−0.1219	0.0942	(0.2974, 0.6387)	0.3412
	Y6	1.1942	−0.1574	0.2775	(0.8182, 1.5703)	0.7520	0.8334	−0.1180	0.1447	(0.5684, 1.0984)	0.5300
	Y8	1.9129	−0.2037	0.5227	(1.3540, 2.4718)	1.1178	1.2810	−0.1255	0.2246	(0.9010, 1.6610)	0.7600
	Y10	3.2281	−0.5528	2.1292	(2.2968, 4.1594)	1.8626	1.8263	−0.1527	0.3604	(1.2994, 2.3532)	1.0539
6	Y2	0.2079	−0.1704	0.0957	(0.0927, 0.3266)	0.2302	0.1386	−0.1224	0.0478	(0.0618, 0.2153)	0.1535
	Y4	0.6567	−0.1539	0.1663	(0.4208, 0.8931)	0.4728	0.4335	−0.1130	0.0811	(0.2748, 0.5922)	0.3174
	Y6	1.1750	−0.1549	0.2692	(0.8049, 1.5451)	0.7402	0.8020	−0.1210	0.1418	(0.5403, 1.0636)	0.5232
	Y8	1.9221	−0.2047	0.5271	(1.3592, 2.4850)	1.1258	1.3202	−0.1461	0.2625	(0.9218, 1.7185)	0.7967
	Y10	3.2717	−0.5605	2.1771	(2.3342, 4.2091)	1.8749	2.0127	−0.2023	0.5063	(1.4302, 2.5952)	1.1651

Table 1

MLPs for Ys with their Biases and MSPEs, in addition to 95% PIs based on 1000 informative samples.

The following steps are followed to determine the AILs and coverage probabilities (COVPs) of BPIs for Ys, s=2,4,6,8,10.

Generate a progressively type-II censored random sample under competing risks model as shown in Subsection 6.1. This sample will be called “informative sample.” Based on this sample, calculate BPIs for Ys, s=2,4,6,8,10.
Generate another sample (will be called “future sample”) and assign the values of Ys, s=2,4,6,8,10, and then determine whether the values of Ys, s=2,4,6,8,10 lie in their corresponding BPIs or not.
Repeat the above two steps 1000 times to induce 1000 BPIs for Ys, s=2,4,6,8,10.
Calculate the average of the BPIs and hence calculate the AILs of the BPIs.
Calculate the COVPs of the BPIs as follows
COVP=Number of BPIs that includeYs1000.

The prior parameter values α1=0.30,α2=0.75,α3=0.57 and α4=0.95 have been taken so as to generate β1=0.4 and β2=0.6, using (12) and (13). The 95% BPIs for Ys, s=2,4,6,8,10, AILs and COVPs of the BPIs are presented in Table 2.

		n* = m* = 10			n* = 16 and n* = 10
CS	Ys	BPI	AIL	COVP	BPI	AIL	COVP
1	Y2	(0.0503, 1.1472)	1.0969	95.4	(0.3021, 0.7611)	0.7290	96.1
	Y4	(0.2460, 1.9967)	1.7507	95.4	(0.1660, 1.4125)	1.2465	94.8
	Y6	(0.5349, 2.9466)	2.4118	94.2	(0.3939, 2.3073)	1.9134	95.2
	Y8	(0.9552, 4.3893)	3.4341	95.7	(0.7864, 3.9453)	3.1589	95.4
	Y10	(1.7136, 8.2085)	6.4948	95.1	(1.5304, 7.9319)	6.4014	96.1
2	Y2	(0.0503, 1.1438)	1.0936	95.6	(0.0403, 0.9910)	0.9507	95.9
	Y4	(0.2444, 1.9728)	1.7283	95.7	(0.2193, 1.8554)	1.6360	94.9
	Y6	(0.5383, 2.9472)	2.4089	94.0	(0.5097, 2.8744)	2.3647	95.4
	Y8	(0.9607, 4.3864)	3.4257	94.8	(0.9236, 4.3003)	3.3767	96.6
	Y10	(1.7299, 8.2343)	6.5044	95.9	(1.6826, 8.1171)	6.4344	95.0
3	Y2	(0.0506, 1.1567)	1.1061	96.0	(0.0313, 0.7442)	0.7128	96.4
	Y4	(0.2454, 1.9937)	1.7483	94.8	(0.1501, 1.2732)	1.1231	95.0
	Y6	(0.5434, 2.9973)	2.4540	94.8	(0.3460, 2.1649)	1.8189	94.9
	Y8	(0.9490, 4.3704)	3.4214	96.3	(0.7366, 3.8634)	3.1268	93.4
	Y10	(1.7089, 8.1976)	6.4887	94.7	(1.4874, 7.9135)	6.4262	95.6
4	Y2	(0.0506, 1.1604)	1.1098	95.9	(0.0307, 0.7315)	0.7008	94.1
	Y4	(0.2430, 1.9829)	1.7399	95.5	(0.1472, 1.2538)	1.1066	95.8
	Y6	(0.5422, 3.0050)	2.4628	94.8	(0.3051, 1.7518)	1.4467	95.4
	Y8	(0.9418, 4.3597)	3.4179	95.0	(0.5059, 2.3433)	1.8374	95.6
	Y10	(1.7081, 8.2391)	6.5310	96.2	(0.7673, 3.1182)	2.3509	96.2
5	Y2	(0.0510, 1.1638)	1.1128	95.6	(0.0349, 0.8324)	0.7975	94.9
	Y4	(0.2445, 1.9835)	1.7390	95.0	(0.1774, 1.4987)	1.3213	96.7
	Y6	(0.5405, 2.9770)	2.4365	93.9	(0.3800, 2.1550)	1.7750	95.9
	Y8	(0.9524, 4.3757)	3.4233	95.6	(0.6366, 2.9208)	2.2842	95.7
	Y10	(1.7248, 8.2572)	6.5323	95.2	(1.0010, 4.0860)	3.0850	95.7
6	Y2	(0.0511, 1.1665)	1.1154	94.0	(0.0335, 0.7978)	0.7642	95.4
	Y4	(0.2428, 1.9702)	1.7273	94.5	(0.1629, 1.3767)	1.2139	94.7
	Y6	(0.5403, 2.9783)	2.4380	95.2	(0.3668, 2.1147)	1.7479	95.9
	Y8	(0.9589, 4.4068)	3.4479	94.6	(0.6592, 3.1088)	2.4496	95.7
	Y10	(1.7203, 8.2415)	6.5212	95.3	(1.0867, 4.6203)	3.5335	96.4

Table 2

The AILs and COVPs of 95% BPIs for Ys. Prior parameter values: α1=0.30, α2=0.75, α3=0.57 and α4=0.95. Population parameter values: β1=0.4 and β2=0.6.

6.3. Illustrative Examples

6.3.1. Example 1

Based on the values of β1 and β2, generated above, generate two independent random samples each of which of size n=30 from CDF (21) and then apply step 4 in Subsection 6.1 to them in order to obtain a random sample of size n=30 under competing risks model. Apply CS1 to this sample to obtain a progressively type-II censored sample of size m=18. This sample is called the informative sample.

Assume that (i) (n⋆,m⋆)=(10,10) (no censoring) and (ii) (n⋆,m⋆)=(16,10) (six values are censored). Based on CS1 we generate another progressively type-II censored sample under competing risks model, using CDF (21), as before. This sample is called the future sample. Table 3 displays, in the second column, the original random sample of size 30 generated from CDF (21) due to the first cause of failure (β1=0.4), while that due to the second cause of failure (β2=0.6) is presented in the third column. The values of these two samples are then compared and the minimum values are recorded in the fourth column. The values of progressively type-II censored samples of sizes 18 are presented in the fifth columns. The sample in the last column is called informative sample. Table 4 shows some selected values of the future order statistics which will be proposed to determine their MLPs and BPIs, while Table 5 shows MLPs and 95% BPIs for the selected values in Table 4. The PMLEs of β1 and β2 are β1^=0.3048 and β2^=0.6950 when n⋆=10 and β1^=0.3049 and β2^=0.6949 when n⋆=16.

i	Ti1	Ti2	Ti	Ti;18;30
1	5.59979	5.76755	5.59979	0.01603
2	1.73549	1.63035	1.63035	0.15559
3	1.38032	2.71452	1.38032	0.18830
4	1.48707	2.01749	1.48707	0.31341
5	8.95048	3.23940	3.23940	0.34363
6	0.48371	9.14179	0.48371	0.41885
7	0.79873	1.35164	0.79873	0.46795
8	5.42455	0.57271	0.57271	0.57271
9	4.07025	2.72627	2.72627	0.70169
10	4.22138	3.51768	3.51768	0.72990
11	2.89722	0.15559	0.15559	0.85833
12	8.68340	0.31341	0.31341	1.00249
13	3.77322	3.73408	3.73408	1.82999
14	1.53726	3.16222	1.53726	2.21175
15	3.17939	1.43613	1.43613	2.22394
16	0.85833	1.15286	0.85833	2.36294
17	5.37478	0.41885	0.41885	3.01668
18	7.13561	2.22394	2.22394	6.14015
19	4.79655	1.00249	1.00249	———
20	4.20190	0.70169	0.70169	———
21	4.63921	2.21175	2.21175	———
22	0.46795	1.70203	0.46795	———
23	2.93812	2.36294	2.36294	———
24	0.34363	1.15989	0.34363	———
25	1.54094	0.01603	0.01603	———
26	6.14015	6.22725	6.14015	———
27	1.82999	5.68222	1.82999	———
28	1.42938	0.72990	0.72990	———
29	3.74519	3.01668	3.01668	———
30	5.85944	0.18830	0.18830	———

Table 3

Informative progressively type-II censored samples. Prior parameter values: α1=0.30,α2=0.75,α3=0.57 and α4=0.95. Population parameter values: β1=0.4 and β2=0.6.

	Y_s
n*	m*	Y2	Y4	Y6	Y8	Y10
10	10	0.0520	0.8016	0.9920	1.5103	3.7583
16	10	0.0520	0.6533	0.8259	1.1179	5.8339

Table 4

Selected values for Ys.

	n* = m* = 10		n* = 16 and m* = 10
Ys	MLP	BPI	MLP	BPI
Y2	0.2160	(0.0525, 1.1930)	0.1358	(0.0334, 0.7895)
Y4	0.6818	(0.2527, 2.0421)	0.4572	(0.1687, 1.4314)
Y6	1.2314	(0.5576, 3.0651)	0.9021	(0.4029, 2.3538)
Y8	1.9692	(0.9986, 4.5919)	1.6373	(0.8119, 4.0709)
Y10	3.3635	(1.8121, 8.7346)	3.0640	(1.6033, 8.3531)

Table 5

MLPs and 95% BPIs for the values of Ys that given in Table 4.

6.3.2. Example 2

Our object, in this example, is to predict with the s-th order statistic, Ys, in the future sample, s=1,…,m⋆, based on a data set originally analyzed by Hoel [31]. The data were obtained from a laboratory experiment in which 77 male mice exposed to a radiation dose of 300 roentgens at 5 to 6 weeks of age. The cause of death for each mouse was determined by autopsy to be thymine lymphoma, reticulum cell sarcoma or other causes. For the purpose of analysis, we restrict the causes of death to two causes only considering reticulum cell sarcoma as cause 1 and combining the other causes of death as cause 2. Kundu et al. [32] generated a progressively type-II censored sample from the original data with m = 25. The CS is given by Ri=2,i=1,…,24 and R25=4. The progressively type-II censored sample used by Kundu et al. [32], Pareek et al. [33] and Cramer and Schmiedt [18] is given by

(40, 2, 2), (42, 2, 2), (62, 2, 2), (163, 2, 2), (179, 2, 2), (206, 2, 2), (222, 2, 2), (228, 2, 2),

(252, 2, 2), (259, 2, 2), (318, 1, 2), (385, 2, 2), (407, 2, 2), (420, 2, 2), (462, 2, 2), (517, 2, 2), (517, 2, 2), (524, 2, 2), (525, 1, 2), (536, 1, 2), (558, 1, 2), (605, 1, 2), (612, 1, 2), (620, 2, 2), (621, 1, 4),

where the first component represents the lifetime, the second component indicates the cause of death and the third component denotes the censoring value.

The lifetimes of the mice are assumed to be independent and subject to the exponential distribution in Kundu et al. [32], Weibull distribution in Pareek et al. [33] and Lomax distribution in Cramer and Schmiedt [18].

It can be easily shown that the above progressively type-II censored data could come from CDF (3) with uj(t), given by (24), using Kolmogorov–Smirnov test. The empirical CDF used in this test may be written as, see [14],

F~(ti:m:n)=1−∏j=1i(1−p~j),i=1,…,m,

where

p~j=1n−∑k=2jRk−1−j+1,j=1,…,m.

From the above data and using CDF (3) with uj(t), given by (24), it can be shown that the maximum likelihood estimates (MLEs) of β1 and β2 are given, respectively, by β1~=0.000481 and β2~=0.001116. The Kolmogorov–Smirnov test statistic =0.22687 and the P-value =0.99982. Therefore, there is a good possibility for the given data to be come from CDF (3) with uj(t), given by (24). Under the above progressively type-II censored data, the logarithm of likelihood Equation (6) with uj(t), given by (24) is plotted in Figure 2. It can be noticed from this figure that the plane attains its maximum at β1~=0.000481 and β2~=0.001116.

The prior parameter values have been taken to be α1=0.0030,α2=2.9000,α3=0.0071 and α4=5.1000 to generate population parameter values β1~=0.000481 and β2~=0.001116 using (12) and (13). Suppose that m⋆=10 (with n⋆=10 and n⋆=16 later on) and applying CS4. Table 6 shows MLPs for Ys, in the future sample, s=2,4,6,8,10. The PMLEs of β1 and β2 are β1^=0.0.000497 and β2^=0.001152 when n⋆=10 and 16. Table 7 shows 95% BPIs for Ys, in the future sample, s=2,4,6,8,10. For 100,000 generated future ordered samples each of which of size m⋆=10 under competing risks model using CDF (3) with uj(t), given by (24), considering the parameters β1~=0.000481 and β2~=0.001116. The COVPs of the BPIs for Ys, s=2,4,6,8,10 have been also calculated and presented in Table 7.

		Y_s
n*	m*	Y2	Y4	Y6	Y8	Y10
10	10	130.9690	413.3470	746.4020	1193.3700	2037.9600
16	10	79.5712	245.7490	425.3320	625.3520	857.8680

Table 6

MLPs for Ys. Prior parameter values: α1=0.0030,α2=2.9000,α3=0.0071 and α4=5.1000. Population parameter values: β~1=0.000481 and β~2=0.001116.

	n* = m* = 10		n* = 16 and m* = 10
Ys	BPI	COVP	BPI	COVP
Y2	(31.6553, 703.4732)	96.0	(19.4418, 449.9195)	95.9
Y4	(153.0676, 1197.9496)	96.7	(90.4819, 740.9591)	96.7
Y6	(338.9109, 1792.2874)	97.1	(190.2302, 1042.1497)	97.1
Y8	(608.4114, 2681.1188)	97.5	(313.3242, 1378.8600)	97.5
Y10	(1105.0914, 5116.6316)	97.2	(463.0192, 1781.5689)	97.9

Table 7

The COVPs in % and 95% BPIs for Ys.

7. CONCLUDING REMARKS

In this paper, under competing risks model and based on a progressively type-II censored informative sample generated from the half-logistic distribution, we have discussed MLPs and BPIs for future progressively type-II censored competing risks data from the same population. An illustrative example based on simulated data and another one based on real data have been considered to investigate the performance of prediction.

From Table 1 we observe the following:

For all considered CSs, by increasing the index s the MSPEs and AILs increase.
In the fourth column in which we have considered the progressive censoring in future sample, the MSPEs and AILs give less values with respect to the last three CSs than those for the above CSs (CS1, CS2, CS3).

From Table 2 we observe the following:

The COVPs of the BPIs are quite close to the nominal confidence levels 95%.
For all considered CSs, by increasing the index s the AILs increase.
In the fourth column in which we have considered the progressive censoring in future sample, better results for the BPIs, through smaller values for the AILs and closer values for the COVPs (near to 95%) have been obtained. This is more clear in the last three CSs.
CS4, in which we remove all the surviving units after the last failure, gives better results with respect to the AILs, than the other CSs.
Numerical computations show that the above results do not change by changing the values of the prior parameters.
If the hyperparameters are unknown, “simple empirical Bayes estimators” may be obtained by using “past samples,” see [34]. Alternatively, one could use hierarchical Bayes approach in which a suitable prior for the vector of hyperparameters is used, see [35].

CONFLICT OF INTEREST

Essam. K. AL-Hussaini contributes with the review of the paper. Alaa H. Abdel-Hamid contributes with the idea and preparation of the paper. Atef. F. Hashem contributes with derivation (calculation) of the analytical (numerical) results in addition to writing the paper.

REFERENCES

1.E.K. AL-Hussaini, J. Stat. Plan. Inf., Vol. 79, 1999, pp. 79-91.

2.E.K. AL-Hussaini, J. Stat. Plan. Inf., Vol. 113, 2003, pp. 15-24.

3.M.A.M. Ali Mousa, J. Stat. Comput. Simul., Vol. 71, 2001, pp. 163-181.

4.Z.F. Jaheen, Commun. Stat. Simul. Comput., Vol. 32, 2003, pp. 663-676.

5.I. Basak, P. Basak, and N. Balakrishnan, Comput. Stat. Data Anal., Vol. 50, 2006, pp. 1313-1337.

6.I. Basak and N. Balakrishnan, Sankhyā., Vol. 71, 2009, pp. 222-247.

7.M.Z. Raqab, A. Asgharzadeh, and R. Valiollahi, Comput. Stat. Data Anal., Vol. 54, 2010, pp. 1732-1743.

8.M.M. Mohie El-Din and A. Shafay, Stat. Pap., Vol. 54, 2013, pp. 287-307.

9.A.H. Abdel-Hamid and E.K. AL-Hussaini, J. Stat. Comput. Simul., Vol. 84, 2014, pp. 1297-1312.

10.E.K. AL-Hussaini, A.H. Abdel-Hamid, and A.F. Hashem, J. Egypt. Math. Soc., Vol. 23, 2015, pp. 190-196.

11.E.K. AL-Hussaini and M.I. Osman, J. Stat. Comput. Simul., Vol. 58, 1997, pp. 121-144.

12.A.H. Abdel-Hamid and E.K. AL-Hussaini, Metrika., Vol. 66, 2007, pp. 213-231.

13.N.R. Mann, R.E. Schafer, and N.D. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York, NY, USA, 1974.

14.W.Q. Meeker and L.A. Escobar, Statistical Methods for Reliability Data, Wiley, New York, NY, USA, 1998.

15.J.F. Lawless, Statistical Models and Methods for Lifetime Data, second ed., Wiley, New York, NY, USA, 2003.

16.N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods and Applications, Birkhauser Publishers, Boston, MA, USA, 2000.

17.N. Balakrishnan and E. Cramer, The Art of Progressive Censoring: Applications to Reliability and Quality, Birkhäuser, New York, NY, USA, 2014.

18.E. Cramer and A.B. Schmiedt, Comput. Stat. Data Anal., Vol. 55, 2011, pp. 1285-1303.

19.N. Balakrishnan, E. Cramer, and U. Kamps, Stat. Probab. Lett., Vol. 54, 2001, pp. 301-315.

20.U. Kamps and E. Cramer, Statistics., Vol. 35, 2001, pp. 269-280.

21.W. Nelson, Accelerated Testing: Statistical Models, Test Plans and Data Analysis, Wiley, New York, NY, USA, 1990.

22.A.H. Abdel-Hamid and E.K. AL-Hussaini, J. Comput. Appl. Math., Vol. 235, 2011, pp. 5259-5271.

23.R.A. Waller and M.S. Waterman, SIAM Rev., Vol. 20, 1978, pp. 820-856.

24.N. Balakrishnan, J. Stat. Comput. Simul., Vol. 20, 1985, pp. 287-309.

25.N. Balakrishnan and S. Puthenpura, J. Stat. Comput. Simul., Vol. 25, 1986, pp. 193-204.

26.N. Balakrishnan and P.S. Chan, Comput. Stat. Data Anal., Vol. 13, 1992, pp. 123-141.

27.K. Rosaiah, R.R.L. Kantam, and B. Srinivasa Rao, Calcutta Stat. Assoc. Bull., Vol. 61, 2009, pp. 183-196.

28.C. Kim and K. Han, Stat. Pap., Vol. 51, 2010, pp. 375-387.

29.E.K. AL-Hussaini and A.H. Abdel-Hamid, Commun. Stat. Simul. Comput., Vol. 33, 2004, pp. 963-982.

30.A.H. Abdel-Hamid, J. Egypt. Math. Soc., Vol. 16, 2008, pp. 75-98.

31.D.G. Hoel, Biometrics., Vol. 28, 1972, pp. 475-488.

32.D. Kundu, N. Kannan, and N. Balakrishnan, N. Balakrishnan and C.R. Rao (editors), Handbook of Statistics, Elsevier, Amsterdam, Netherlands, Vol. 23, 2004, pp. 331-348.

33.B. Pareek, D. Kundu, and S. Kumar, Comput. Stat. Data Anal., Vol. 53, 2009, pp. 4083-4094.

34.J. Maritz and T. Lwin, Empirical Bayes Methods, second, Chapman and Hall, London, England, 1989.

35.J.M. Bernardo and A.F.M. Smith, Bayesian Theory, Wiley, New York, NY, USA, 1994.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 1
Pages: 36 - 48
Publication Date: 2020/03/05
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
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TY  - JOUR
AU  - Essam K. AL-Hussaini
AU  - Alaa H. Abdel-Hamid
AU  - Atef F. Hashem
PY  - 2020
DA  - 2020/03/05
TI  - Prediction for Progressively Type-II Censored Competing Risks Data from the Half-Logistic Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 36
EP  - 48
VL  - 19
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200224.004
DO  - 10.2991/jsta.d.200224.004
ID  - AL-Hussaini2020
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