American Journal of Theoretical and Applied Statistics
Volume 4, Issue 5, September 2015, Pages: 329-338

Prediction Intervals for Progressive Type-II Right-Censored Order Statistics from Two Independent Sequences

M. M. Mohie El-Din1, M. S. Kotb1, W. S. Emam2

1Department of Mathematics, Faculty of Science, Al-Azhar, University, Nasr City, Cairo, Egypt

2Department of Basic Science, Faculty of Engineering, British, University in Egypt, Al-Shorouq City, Cairo, Egypt

Email address:

(M. M. M. El-Din)
(M. S. Kotb)
(W. S. Emam)

To cite this article:

M. M. Mohie El-Din, M. S. Kotb, W. S. Emam. Prediction Intervals for Progressive Type-II Right-Censored Order Statistics from Two Independent Sequences. American Journal of Theoretical and Applied Statistics. Vol. 4, No. 5, 2014, pp. 329-338. doi: 10.11648/j.ajtas.20150405.13


Abstract: This article discusses the problem of predicting future progressive Type-II right censored order statistics based on progressive Type-II right-censored, ordered statistics, record values and current records that observed from the past X-sequence. Such coverage probabilities of the prediction intervals are exact and don’t depend on the sampling distribution F. Finally, a real life time data were given to breakdown the insulating fluid between electrodes which is used to illustrate the derived results.

Keywords: Prediction Intervals, Progressive Censoring, Order Statistics, Records, Coverage Probability, Prediction Coefficient


1. Introduction

Statistical prediction is the derivation issue of the unknown values of future sample based on available observations. Prediction problems can be generally categorized as One-sample prediction, two-sample prediction and multi-sample prediction. Type II-sample prediction is governed here. In this type, predicting of future variables based on another independent observed sample. Also, the predictor can be either point or interval. The prediction (confidence) intervals can be one-sided or two-sided. Moreover, the predictor can be parametric (if it depends on the distribution parameters) or nonparametric (distribution-free prediction type). Distribution-free two-sided prediction intervals (DFPIs) are interest of natural in this article. Many contexts have taken place in the DFPIs direction in recent years by using several assumption; see, for example [1-11]. Recently, DFPIs for future progressively Type-II right-censored order statistics sample (PCOs) (which it’s cdf purposed by Kamps and Cramer [12]) based on k-records [13]. Here, the DFPIs derivation for a future PCOs (that developed by Balakrishnan et al. [16]) based on different observed samples (PCOs; order statistics; upper records and largest current records).

This paper is organized as follows, section 2 contains some preliminaries. In section 3, the DFPIs discussion of a future PCO based on PCOs, order statistics, upper records and current records. In section 4, a real life time data are considered and numerical computations are given to illustrate most the results which are derived. Finally, the conclusion of this study is given in Section 5.

2. Preliminaries

Suppose  denote the observed lifetimes of identical units of size  that placed on a reliability experiment of life-test. Assume that, these units are independent and identically distributed (iid) from a continuous population with cumulative distribution function (cdf)  and probability density function (pdf) . Further, independently of X-sample, let  another unobserved sample of size  that withdrawn from the same population, consider we intend to study  units-out-of-m and remove from the life-test at different time points (progressively) the remaining  units from these unobserved iid random variables (r.v’s). Specifically, immediately following the first observed failure , remove from right at random  surviving units from the test; next, immediately observe following the second failure , remove at random  surviving units from the test, and so on; finally, after the  observed failure , remove all the remaining surviving units (that did not failed yet) . Such  the number of failures (that were already observed) and the progressive censoring scheme  are pre-fixed, the scheme  is said to be progressive Type-II right-censoring, and the ordered values  were obtained from these observations are referred to PCOs. For more details concerning a PCOs, see, for example, Balakrishnan and Aggarwala [14], Balakrishnan [15]. The marginal pdf of the  PCO,  that developed by Balakrishnan et al. [16], given by

(1)

where , , , , and . The survival function of  can be easily obtained as

(2)

It remains to refer that, not only the prediction coefficients  and  which are stated here depend on the subscripts but also depend on observed sample size , future PCOs size  and the progressive censoring scheme , i.e.

3. DFPIs for Single PCO

In this section, we intend to construct  DFPIs for the  PCO,  from the future PCOs of size  out of  from Y-sequence of the form , such that, the lower  and the upper  bounded are observed from the following four schemes: PCOs, order statistics, upper records and current records, each case has separately discussion in special subsection, such that the corresponding coverage probabilities are free of the parent distribution .

3.1. Based on PCOs

Table 1. values of  for  and  for some choices of  and .

      j       j      
      4 6 8 10 4 6 8 10

3 1 0.659 0.885 0.974 0.995 0.781 0.875 0.884 0.885

  2 0.444 0.670 0.733 0.741 0.424 0.517 0.527 0.527

  3 0.202 0.429 0.492 0.500 0.152 0.245 0.255 0.255

5 1 0.312 0.676 0.918 0.993 0.688 0.926 0.973 0.976

  2 0.265 0.629 0.872 0.946 0.494 0.731 0.778 0.782

  3 0.159 0.523 0.766 0.840 0.232 0.469 0.516 0.520
20:15 9 1 0.116 0.435 0.811 0.985 0.284 0.698 0.945 0.996

  2 0.108 0.427 0.804 0.978 0.255 0.669 0.916 0.968

  3 0.077 0.396 0.772 0.946 0.164 0.577 0.825 0.877

11 1 0.031 0.198 0.581 0.940 0.139 0.501 0.864 0.993

  2 0.030 0.197 0.580 0.939 0.131 0.492 0.855 0.984

  3 0.023 0.191 0.574 0.932 0.092 0.454 0.817 0.946
15:8 4 1 0.345 0.691 0.916 0.989 0.655 0.906 0.968 0.974

  2 0.284 0.629 0.824 0.928 0.469 0.720 0.781 0.788

  3 0.163 0.509 0.734 0.807 0.223 0.474 0.536 0.542

6 1 0.070 0.277 0.624 0.934 0.398 0.780 0.962 0.996

  2 0.065 0.272 0.619 0.929 0.336 0.718 0.900 0.934

  3 0.045 0.252 0.600 0.910 0.197 0.579 0.760 0.794
25:15 4 1 0.690 0.913 0.962 0.966 0.800 0.905 0.915 0.915

  2 0.477 0.700 0.748 0.753 0.459 0.563 0.573 0.574

  3 0.218 0.441 0.489 0.494 0.169 0.274 0.284 0.284

6 1 0.432 0.804 0.967 0.995 0.710 0.937 0.976 0.978

  2 0.359 0.732 0.895 0.923 0.510 0.737 0.776 0.778

  3 0.206 0.578 0.741 0.769 0.237 0.464 0.503 0.505

Let  be an observed sequence of PCOs of size  from the X-sequence with the progressive censoring scheme . Suppose we are interested in obtaining  DFPIs for  of the form , such that the coverage probability  being free of the parent distribution .

Lemma 1. Let  and  be two independent PCOs from continuous cdf . then , , is DF one-sided PI for the future , with the corresponding prediction coefficient , that does not depend on the sampling distribution , and is given by

(3)

Proof: Upon (1) and (2), we can write

(4)

Table 2. values of  for  and  for some choices of  and .

      j       j      
      r+3 r+4 r+5 r+6 r+3 r+4 r+5 r+6

15:10 4 r-3 0.472 0.584 0.684 0.768 0.504 0.616 0.714 0.794

  r-2 0.449 0.561 0.660 0.744 0.477 0.589 0.687 0.768

  r-1 0.405 0.417 0.617 0.701 0.428 0.541 0.638 0.719

6 r-3 0.199 0.273 0.356 0.444 0.372 0.477 0.579 0.674

  r-2 0.192 0.267 0.350 0.438 0.355 0.459 0.562 0.657

  r-1 0.177 0.252 0.335 0.423 0.319 0.423 0.526 0.621

8 r-3 0.057 0.088 0.127 0.179 0.267 0.358 0.457 0.557

  r-2 0.056 0.086 0.126 0.177 0.254 0.345 0.444 0.544

  r-1 0.052 0.082 0.123 0.174 0.231 0.322 0.421 0.521
25:20 10 r-3 0.525 0.627 0.716 0.789 0.636 0.713 0.770 0.808

  r-2 0.475 0.577 0.666 0.739 0.549 0.625 0.682 0.721

  r-1 0.406 0.508 0.597 0.670 0.442 0.519 0.576 0.615

14 r-3 0.430 0.545 0.656 0.755 0.619 0.697 0.754 0.792

  r-2 0.398 0.513 0.624 0.723 0.535 0.613 0.670 0.708

  r-1 0.352 0.466 0.577 0.677 0.434 0.512 0.569 0.607

18 r-3 0.305 0.442 0.603 0.769 0.659 0.740 0.792 0.820

  r-2 0.293 0.430 0.591 0.757 0.576 0.656 0.708 0.736

  r-1 0.272 0.409 0.570 0.736 0.472 0.552 0.604 0.632

Theorem 1. Under the assumption of lemma 1, then , , is a DFPIs for , whose coverage probability is free of the parent distribution .

Proof: Based on lemma 1, and by assuming that the , are continuous r.v’s, we obtain

(5)

Under the assumptions of theorem 1, we can choose  and  so that  exceeds the desired confidence level , tables (1 and 2) presents values of the prediction coefficient  for two different censoring schemes of the past PCOs ; , respectively, for some choices of and.

3.2. Based on Order Statistics

In the following, the prediction discussion of future PCOs based on observed order statistics. Let  denote sequences of the lifetimes of reliability experiment units, we shall assume that these variables are iid from an absolutely continuous population with cdf. Suppose  are usual order statistics that obtained from these iidr.v’s, for more details about order statistics, see, Arnold et al. [17]. The marginal pdf and the survival function of  are easily expressed in terms of and , respectively as.

(6)

(7)

Lemma 2.

Based on X-sample observations, suppose we are interested in obtaining  DFPIs for  from a future Y-sample of the form , such that, the coverage probability . We refer to the interval  as  PI for . For a given , if  and  are continuous r.v’s, we get

(8)

Upon the conditioning argument, in the non-parametric prediction procedure by assuming that  and  are continuous r.v’s also, we then have

(9)

Such that,  represent the prediction coefficient which does not depend on the parameters of the parent distribution , wich it’s depends only on the r.v’s positions (the indices  and ). Here,  and  are the lower and upper bounds of the prediction interval for  respectively.

Theorem 2. Let  be  order statistic from an observed random sample of size  with continuous cdf. If ,  is the th PCOs from a future unobserved random sample of size -out-of  with the same cdf, then , , is a DFPI for , whose coverage probability is free of  and the corresponding prediction coefficient  is given by

(10)

Proof: Such  is continuous, it is known from (7) and (8) that

(11)

By conditioning on , and from (6) and (9), it follows that

Equation (10) can be obtained directly by simplifying the previous beta constants, we have presented the values of  that given by (10) for  and for some selected values of the integers  and  in table 3. Thus,  is a  DFPI for the future  PCOs from a future unobserved random sample of size , with  given by (10) which does not depend on .

3.3. Based on Upper Records

Here, The prediction discussion of future PCOs based on observed upper records. Let  be a sequence of iid continuous r.v’s with cdf . An observation  is defined to be an upper record if  for every , one may refer to the books by Arnold et al.[17], Nevzorov[18], Gulati and Padgett[19] and Ahsanullah[20]. Let us denote the  upper record value by . Then the survival function of of the  is given by

(12)

Table 3. The values of  for and for some selected choices of and.

      j       j

4 6 8 10

4 6 8 10
     

     

12 3 1 0.3547 0.6120 0.7927 0.8922 12 3 1 0.6749 0.8952 0.9495 0.9555
    2 0.2629 0.5203 0.7009 0.8004     2 0.4548 0.6751 0.7294 0.7354
    3 0.1384 0.3957 0.5764 0.6758     3 0.2064 0.4266 0.4810 0.4870
  6 1 0.0423 0.1445 0.3189 0.5345   6 1 0.2486 0.6218 0.9035 0.9934
    2 0.0376 0.1398 0.3142 0.5298     2 0.2195 0.5927 0.8744 0.9643
    3 0.0250 0.1272 0.3017 0.5173     3 0.1392 0.5124 0.7941 0.8840
  9 1 0.0012 0.0079 0.0329 0.0981   9 1 0.0325 0.1915 0.5537 0.9254
    2 0.0011 0.0079 0.0329 0.0980     2 0.0311 0.1902 0.5523 0.9241
    3 0.0008 0.0076 0.0326 0.0977     3 0.0237 0.1828 0.5449 0.9166
30 3 1 0.6285 0.7569 0.7870 0.7922 30 3 1 0.7547 0.7910 0.7928 0.7929
    2 0.3641 0.4925 0.5226 0.5277     2 0.3236 0.3599 0.3617 0.3618
    3 0.1456 0.2740 0.3041 0.3093     3 0.0916 0.1279 0.1297 0.1298
  6 1 0.4396 0.7421 0.8968 0.9487   9 1 0.5836 0.9105 0.9882 0.9942
    2 0.3377 0.6402 0.7948 0.8467     2 0.4713 0.7982 0.8758 0.8819
    3 0.1801 0.4826 0.6372 0.6891     3 0.2517 0.5786 0.6563 0.6623
  7 1 0.3446 0.6643 0.8682 0.9517   18 1 0.0712 0.3740 0.8006 0.9881
    2 0.2782 0.5979 0.8018 0.8854     2 0.0687 0.3715 0.7981 0.9855
    3 0.1582 0.4779 0.6817 0.7653     3 0.0527 0.3555 0.7822 0.9696
  8 1 0.2581 0.5711 0.8169 0.9387   19 1 0.0493 0.3056 0.7455 0.9817
    2 0.2165 0.5296 0.7753 0.8972     2 0.0478 0.3042 0.7441 0.9803
    3 0.1299 0.4430 0.6887 0.8106     3 0.0376 0.2939 0.7339 0.9700
  20 1 0.0003 0.0043 0.0276 0.1098   20 1 0.0329 0.2423 0.6822 0.9725
    2 0.0003 0.0043 0.0276 0.1098     2 0.0321 0.2415 0.6814 0.9717
    3 0.0003 0.0042 0.0276 0.1097     3 0.0258 0.2352 0.6751 0.9654
     

     

12 3 1 0.3547 0.6120 0.7927 0.9082 12 3 1 0.6298 0.8841 0.9483 0.9555
    2 0.2629 0.5203 0.7009 0.8164     2 0.4956 0.7499 0.8141 0.8213
    3 0.1384 0.3957 0.5764 0.6919     3 0.2307 0.4850 0.5492 0.5564
  6 1 0.0423 0.1445 0.3189 0.6066   6 1 0.2053 0.5868 0.8921 0.9927
    2 0.0376 0.1398 0.3142 0.6019     2 0.1953 0.5769 0.8821 0.9827
    3 0.0250 0.1272 0.3017 0.5894     3 0.1299 0.5114 0.8167 0.9172
  9 1 0.0012 0.0079 0.0329 0.1441   9 1 0.0233 0.1686 0.5301 0.9197
    2 0.0011 0.0079 0.0329 0.1441     2 0.0231 0.1684 0.5298 0.9195
    3 0.0008 0.0076 0.0326 0.1438     3 0.0185 0.1638 0.5253 0.9149
30 3 1 0.6285 0.7569 0.7870 0.7924 30 3 1 0.7427 0.7904 0.7928 0.7929
    2 0.3641 0.4925 0.5226 0.5280     2 0.4113 0.4590 0.4614 0.4615
    3 0.1456 0.2740 0.3041 0.3095     3 0.1184 0.1662 0.1686 0.1686
  6 1 0.4396 0.7421 0.8968 0.9532   9 1 0.5209 0.8934 0.9867 0.9942
    2 0.3377 0.6402 0.7948 0.8513     2 0.4727 0.8451 0.9385 0.9460
    3 0.1801 0.4826 0.6372 0.6937     3 0.2661 0.6386 0.7319 0.7394
  7 1 0.3446 0.6643 0.8682 0.9604   18 1 0.0507 0.3348 0.7793 0.9863
    2 0.2782 0.5979 0.8018 0.8940     2 0.0505 0.3346 0.7790 0.9861
    3 0.1582 0.4779 0.6817 0.7740     3 0.0411 0.3252 0.7697 0.9767
  8 1 0.2581 0.5711 0.8169 0.9537   20 1 0.0222 0.2112 0.6561 0.9692
    2 0.2165 0.5296 0.7753 0.9122     2 0.0222 0.2111 0.6561 0.9691
    3 0.1299 0.4430 0.6887 0.8256     3 0.0188 0.2078 0.6527 0.9658

Table 4. The values of  for and for some selected choices of and.

      j       j

4 6 8 10

4 6 8 10
     

     

12 2 1 0.6042 0.7845 0.8475 0.8638 12 3 1 0.3547 0.6548 0.9206 0.9548
    2 0.3708 0.5511 0.6142 0.6303     2 0.2629 0.5630 0.8289 0.8631
    3 0.1606 0.3409 0.4040 0.4202     3 0.1384 0.4385 0.7043 0.7385
  3 1 0.4826 0.7542 0.8927 0.9421   4 1 0.2043 0.4922 0.9018 0.9834
    2 0.3484 0.6200 0.7585 0.8079     2 0.1650 0.4529 0.8625 0.9440
    3 0.1753 0.4470 0.5855 0.6349     3 0.0958 0.3837 0.7933 0.8748
  4 1 0.3207 0.6223 0.8397 0.9456   5 1 0.1002 0.3210 0.8310 0.9879
    2 0.2562 0.5578 0.7752 0.8811     2 0.0856 0.3064 0.8164 0.9734
    3 0.1444 0.4460 0.6634 0.7692     3 0.0537 0.2745 0.7845 0.9414
  5 1 0.1848 0.4540 0.7227 0.8995   6 1 0.0423 0.1827 0.7218 0.9793
    2 0.1578 0.4270 0.6957 0.8725     2 0.0376 0.1780 0.7171 0.9746
    3 0.0972 0.3664 0.6351 0.8119     3 0.0250 0.1655 0.7046 0.9621
30 7 1 0.5024 0.8221 0.9485 0.9761 30 10 1 0.1281 0.4423 0.9279 0.9959
    2 0.3933 0.7130 0.8393 0.8670     2 0.1134 0.4277 0.9133 0.9812
    3 0.2096 0.5292 0.6556 0.6832     3 0.0743 0.3885 0.8741 0.9420
  8 1 0.4118 0.7629 0.9357 0.9820   11 1 0.0852 0.3530 0.8990 0.9964
    2 0.3380 0.6891 0.8619 0.9082     2 0.0769 0.3447 0.8907 0.9881
    3 0.1920 0.5430 0.7159 0.7622     3 0.0522 0.3200 0.8660 0.9634
  9 1 0.3260 0.6891 0.9093 0.9808   12 1 0.0546 0.2726 0.8621 0.9956
    2 0.2778 0.6408 0.8610 0.9326     2 0.0501 0.2681 0.8576 0.9911
    3 0.1667 0.5297 0.7499 0.8215     3 0.0351 0.2531 0.8425 0.9761
     

     

12 3 1 0.3698 0.6748 0.8707 0.9437 12 3 1 0.4593 0.8300 0.9422 0.9554
    2 0.2780 0.5830 0.7790 0.8519     2 0.3536 0.7243 0.8365 0.8497
    3 0.1458 0.4508 0.6467 0.7197     3 0.1799 0.5506 0.6628 0.6759
  6 1 0.0469 0.2010 0.5012 0.8312   6 1 0.0824 0.4469 0.8425 0.9891
    2 0.0422 0.1963 0.4965 0.8265     2 0.0762 0.4408 0.8364 0.9830
    3 0.0283 0.1824 0.4825 0.8126     3 0.0515 0.4160 0.8116 0.9582
  9 1 0.0014 0.0158 0.0980 0.3857   9 1 0.0039 0.0932 0.4416 0.8964
    2 0.0013 0.0157 0.0980 0.3856     2 0.0038 0.0931 0.4415 0.8963
    3 0.0010 0.0154 0.0976 0.3853     3 0.0030 0.0922 0.4407 0.8955
30 3 1 0.6379 0.7685 0.7910 0.7928 30 3 1 0.6845 0.7866 0.7927 0.7929
    2 0.3735 0.5040 0.5265 0.5284     2 0.3962 0.4983 0.5045 0.5046
    3 0.1469 0.2775 0.3000 0.3018     3 0.1419 0.2440 0.2502 0.2503
  6 1 0.4583 0.7979 0.9365 0.9612   10 1 0.2255 0.7496 0.9722 0.9967
    2 0.3564 0.6960 0.8346 0.8593     2 0.2063 0.7304 0.9531 0.9775
    3 0.1888 0.5284 0.6670 0.6917     3 0.1331 0.6572 0.8798 0.9043
  7 1 0.3633 0.7364 0.9324 0.9764   11 1 0.1649 0.6820 0.9601 0.9978
    2 0.2969 0.6700 0.8660 0.9100     2 0.1537 0.6709 0.9489 0.9867
    3 0.1680 0.5412 0.7372 0.7812     3 0.1036 0.6207 0.8988 0.9365
  8 1 0.2754 0.6565 0.9110 0.9826   12 1 0.1168 0.6090 0.9422 0.9980
    2 0.2338 0.6149 0.8695 0.9411     2 0.1105 0.6027 0.9359 0.9917
    3 0.1400 0.5211 0.7756 0.8473     3 0.0774 0.5696 0.9028 0.9586
  9 1 0.2004 0.5663 0.8743 0.9822   13 1 0.0801 0.5333 0.9182 0.9974
    2 0.1753 0.5413 0.8492 0.9571     2 0.0767 0.5299 0.9148 0.9940
    3 0.1102 0.4761 0.7841 0.8920     3 0.0556 0.5088 0.8937 0.9729

Theorem 3. Let  be  upper record from an observed random sample of size  with continuous cdf. If ,  is the th PCOs from a future unobserved sample of size  with the same cdf, then , , is DFPI for the future  PCOs, , whose coverage probability is free from , and is given by

(13)

Proof: For a given y, by assuming that the records  are continuous r.v’s, and from (12) and (8) it is now easy to write (see,[6])

(14)

The same method as a proof of Theorem 1; and from (14) and (9), we readily obtain

(15)

The values of  are presented in table 5 for  and for some selected choices of and. is a  PIs for the future  PCOs, such  given by (13).

3.4. Based on Current Records

Suppose  is a sequence of iid continuous r.v’s with cdf. Let us denote the  upper records by  (with). Now, let  be the largest observation, at the time when the  upper record occurs in the -sequence. Then, the marginal density and the survival functions of  (see Arnold et al. [21], p. 276), are given respectively by

(16)

(17)

DFPI of future PCOs based on observed upper records are discussed as the same method of Theorem 1 In the following, let  be a future  PCOs from the Y-sequence that we are interested in obtaining  PIs for it of the form  such that  is the  largest record and .

Lemma 3.

Let  be  largest current record from an observed random sample with continues cdf. If ,  is the th PCOs from a future unobserved random sample of size  with the same cdf, then , , is DF one-sided PI for the future , with the corresponding prediction coefficient , that does not depend on , and is given by

(18)

Proof: From (17) and (1), we get

(19)

Table 5. The values of  for and for some selected choices of and.

      j       j

15 25 35

15 25 35
     

     

:25 10 1 0.3700 0.3700 0.3700 50:45 30 1 0.6514 0.6514 0.6514
    2 0.0858 0.0858 0.0858     2 0.2910 0.2910 0.2910
    3 0.0150 0.0150 0.0150     3 0.0980 0.0980 0.0980
  15 1 0.5669 0.5669 0.5669   40 1 0.8693 0.8693 0.8693
    2 0.2152 0.2152 0.2152     2 0.6125 0.6125 0.6125
    3 0.0622 0.0622 0.0622     3 0.3516 0.3516 0.3516
  20 1 0.7938 0.7938 0.7938   43 1 0.9346 0.9346 0.9346
    2 0.4373 0.4373 0.4373     2 0.7659 0.7659 0.7659
    3 0.1982 0.1982 0.1982     3 0.5396 0.5396 0.5396
  25 1 0.9599 0.9599 0.9599   45 1 0.9769 0.9769 0.9769
    2 0.8484 0.8491 0.8484     2 0.9025 0.9038 0.9025
    3 0.7606 0.7606 0.7606     3 0.7687 0.7687 0.7700

Theorem 4. Under the assumption of lemma 3, then , , is DFPI for the future , with the corresponding prediction coefficient  does not depend on .

Proof: Based on lemma 3, and by assuming that the current records  are continuous r.v’s, we get

(20)

The values of  are presented in table 6, for  and for some selected choices of and. Such  is given by (20) which does not depend on the parent distribution .

Table 6.The values of  for  and for some selected choices of and.

      j j
      15 25 35 45 15 25 35 45

30:25 10 1 0.145 0.145 0.145 0.145 0.154 0.154 0.154 0.154

  2 0.046 0.046 0.046 0.046 0.050 0.810 0.870 0.932

  3 0.012 0.012 0.012 0.012 0.014 0.014 0.014 0.014

15 1 0.331 0.331 0.331 0.331 0.339 0.339 0.339 0.339

  2 0.160 0.160 0.160 0.160 0.166 0.166 0.166 0.166

  3 0.066 0.066 0.066 0.066 0.070 0.070 0.070 0.070

20 1 0.590 0.590 0.590 0.590 0.596 0.596 0.596 0.596

  2 0.403 0.403 0.403 0.403 0.409 0.409 0.409 0.409

  3 0.244 0.244 0.244 0.244 0.250 0.250 0.250 0.250

25 1 0.899 0.924 0.924 0.924 0.900 0.925 0.925 0.925

  2 0.840 0.865 0.866 0.866 0.842 0.867 0.868 0.868

  3 0.759 0.783 0.784 0.784 0.761 0.786 0.787 0.787
50:45 30 1 0.427 0.427 0.427 0.427 0.429 0.429 0.429 0.429

  2 0.234 0.234 0.234 0.234 0.236 0.236 0.236 0.236

  3 0.110 0.110 0.110 0.110 0.111 0.111 0.111 0.111

35 1 0.580 0.580 0.580 0.580 0.582 0.582 0.582 0.582

  2 0.387 0.387 0.387 0.387 0.389 0.389 0.389 0.389

  3 0.226 0.226 0.226 0.226 0.227 0.227 0.227 0.227

40 1 0.757 0.757 0.757 0.757 0.758 0.758 0.758 0.758

  2 0.606 0.606 0.606 0.606 0.608 0.608 0.608 0.608

  3 0.442 0.442 0.442 0.442 0.443 0.443 0.443 0.443

45 1 0.914 0.956 0.957 0.957 0.914 0.956 0.957 0.957

  2 0.877 0.920 0.921 0.921 0.877 0.920 0.921 0.921

  3 0.821 0.864 0.865 0.865 0.821 0.864 0.865 0.865

4. Real Life Data Example

The real life-observations that given in Nelson [22] are considered in this section to illustrate the derived results. These data which was also used in Lawless ([23], p. 185), The observation is data time to breakdown of an insulating fluid between electrodes at a voltage of 34 kV (minutes). The 19 times to breakdown are contained in the following table

0.96 4.15 0.19 0.78 8.01 31.75 7.35 6.50 8.27 33.91
32.52 3.16 4.85 2.78 4.67 1.31 12.06 36.71 72.89  

The observed PCOs from the data  as  with the progressive censoring scheme  can be:

0.19 0.78 0.96 2.78 3.16 4.15 4.85
6.50 7.35 8.01 8.27 31.75 32.52 33.91

Also, the usual order statistics values that obtained from the sample  will be:

0.19 0.78 0.96 1.31 2.78 3.16 4.15 4.67 4.85 6.50
7.35 8.01 8.27 12.06 31.75 32.52 33.91 36.71 72.89

Moreover, we can obtain from the sample  the following eight upper records:

0.19 0.96 4.15 8.01 31.75 33.91 36.71 72.89

Based on these fourteen PCOs with , the results of DFPI of future th PCOs ,  from unobserved PCOs of size  with  from a future Y-sample of size 15 are displayed in table 7. Also, based on the above order statistics obtained, DFPI of future th PCOs ,  from a future unobserved random sample of size  with  from a future Y-sample of size 15, with corresponding prediction coefficient of at least 0.80 are displayed in table 8. In table 9, we have presented the values of for , for some selected choices of  based on the real observed upper records. Such  is given by (13).

Table 7. PIs for future  PCOs  from a future unobserved random sample of size  from Y-sample, based on the real observed PCOs.

 

0.5170 6

0.7318
 

0.6251 7

0.6423
 

0.6987 8

0.4364
 

0.7203 9

0.2186
 

0.8012 10

0.0614

Table 8. DFPI for future PCOs , based on the real observed ordered statistics sample .

1

  6

0.8042
2

0.8045 7

0.8246
3

0.8181 8

0.8778
4

0.8186 9

0.8891
5

0.8295 10  

Table 9. DFPI for some future  PCOs  from a future unobserved random sample of size  from Y-sample, based on above upper records.

10

60

20     0.1980 70     0.6848
30    

80     0.7674
40     0.3958 90     0.8085
50    

       

5. Conclusion

In this article, based on progressive Type-II right-censored, ordered statistics, record values and current records, distribution-free PIs for future progressively Type-II right censored order statistics are derived. The obtained results in all tables show that the DFPI decreases with decreasing PI length. Also, the corresponding coefficient increases with decreasing the past sample size. Moreover it may be noted from the results, the prediction coefficients for a future PCOs  increase as  increase, even if the censored observations size  increase. Based on results of tables (5,6 and 8), DFPI for PCOs based on upper (current) records may be advised only for higher  PCOs.


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