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Hybrid censoring samples in assessment the lifetime performance index of Chen distributed products

  • Majdah Mohammed Badr , Ahmed Ibrahim Shawky EMAIL logo and Gamal Amen Abd-Elmougod
Published/Copyright: November 21, 2019
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Open Physics
From the journal Open Physics Volume 17 Issue 1

Abstract

The performance and potential of a process of industrial products is assessed under lower specification limit L by lifetime performance index (CL). In this paper, under consideration the independent lifetimes Chen products with known one shape parameters the CL of the performance of a process is evaluated. For the hybrid censoring scheme the maximum likelihood (ML) estimate of CL is constructed as well as confidence interval for CL is developing. Also, Bayesian approach is adopted to estimates CL and credibly interval is constructed. Some theoretical results of hypothesis tests of CL is adopted. Finally, our obtaining results will be assessed and compared through Monte Carlo simulation study and numerical example.

Keywords: Chen distribution; Process capability indices; Hybrid progressive censoring; Bayesian estimation; Hypothesis tests
PACS: 02.50.Ga; 02.50.Ng

1 Introduction

The problem of determining the quality of the industrial products before they are put on the market requires some tests and calculation of some statistical measures. One of statistical measures of process capability analysis is called process capability index CL which can be used to describe the potential capability and process performance. The CL in industry service is utilized to assess whether the product quality meets the required level. For the applications of CL on electronic components see [1]. The minimum variance unbiased estimator of CL and confidence interval of exponential lifetime products are derived by [2]. Also, in normal lifetime case or exponential, one can see [3]. The applications of CL of Burr XII, Rayleigh and Weibull models discussed respectively by [4, 5] and [6].

In lifetime experiments, censoring is defined when the exact lifetimes of experimental units are known for some units but not all units. The common ones in lifetime testing are called type-I and typ-II censoring schemes. Firstly, in type-I censoring scheme the life testing experiment stop at prescribed time τ and the failure numbers occur randomly. In type-II censoring scheme, life testing experiment stop at prescribed number m of failures and the experiment time is random. The last two types of censoring schemes do not allow the experimental units to be removed at time other than the final point of the experiment. The general scheme which allows experimental units to be removed through the experiment is called progressive type-I or type-II censoring see [7]. The hybrid censoring scheme is defined as a joint of type-I and typ-II censoring schemes, for more details of hybrid censoring scheme see [8] and [9]. In type-II progressive hybrid censoring scheme, prior to the experiment, the censoring scheme R = (R1, R2, ..., Rm) and the total life testing experiment time τ are given. Type-II progressive hybrid censoring scheme is described as follows.

Suppose T1, T2, ..., Tn are n continuous identically distributed failure times of n independent units put in a life testing experiment. Let m, τ and R = {R1, R2, .... , Rm} be previously fixed. At the first failure T1;m,n. is recorded, the surviving units with size R1 are randomly removed from the experiment. When the second failure T2;m,n. is recorded, then randomly R2 removed from the experiment. The experiment continues until one of the two time (Tm;m,n. , τ) is observed and the reaming units are randomly removed from the experiment. The conventional of type-II progressive hybrid censoring scheme is more similar to type-I censoring. The ordered observed failure times are denoted by (T1;m,n., T2;m,n., .... , Tr;m,n.) and called type-II progressive hybrid censored order statistics with size r where r = m at Tm;m,n. < τ and r < m at Tm;m,n. > τ. Figures 1 and 2 show the schematic description of type-II progressive hybrid censoring. It is clearly that at τ → ∞ this scheme reduces to the ordinary type-II progressive censoring scheme

Figure 1 Schematic description of data under type-II progressive hybrid censoring scheme (r < m)
Figure 1

Schematic description of data under type-II progressive hybrid censoring scheme (r < m)

Figure 2 Schematic description of data under type-II progressive hybrid censoring scheme (r = m)
Figure 2

Schematic description of data under type-II progressive hybrid censoring scheme (r = m)

For given type-II progressive hybrid censoring data T = (T1;m,n., T2;m,n., .... , Tr;m,n.) the likelihood function of PDF f (t) and CDF F(t) is defined by

(1) L ( θ | t _ ) = n ! ( S ( τ ) ) δ j ∑ i = r + 1 m R i + 1 ( n − r ) ! ∏ i = 1 r f ( t i ) S ( t i ) R i ,

where θ is the vector of parameters, r is the number of failures occurred before τ,

(2) δ j = 0 , if  r = m 1 , if  r < m , ,

and

t 1 ; m , n < t 2 ; m , n < … < t r ; m , n .

Chen Distribution has a bathtub shaped or increasing failure rate function which is more useful in analysis of reliability. This type of distributions is discussed in different papers, for example, [10, 11, 12, 13] and [14]. Recently, estimating the parameters of Chen distribution under the assumption that data are type-I progressive hybrid censored [15]. In this paper, we discuss the Chen distribution with two parameters as the bathtub shape or increasing failure rate function presented by [16].

The random variable T is said to be Chen random variable with two shape and scale parameters a and b, respectively if it has probability density function (PDF ) given by

(3) f ( t ) = a b t a − 1 exp t a + b 1 − exp t a , t > 0 ,   a , b > 0 ,

and cumulative distribution function (CDF)

(4) F ( t ) = 1 − exp b 1 − exp t a .

Also, the reliability and hazard rate functions respectively are given by

(5) S ( t ) = exp b 1 − exp t a ,
(6) h ( t ) = a b x a − 1 exp x a .

The Chen distribution with different values of shape parameter a has the following properties

  1. For a < 1, Chen distribution has bathtub-shaped failure rate function.

  2. For a ≥ 1, Chen distribution has increasing failure rate function.

  3. For a = 1, Chen distribution becomes exponential power distribution.

The optimal parameters estimation of this distributions are discussed by [17] under doubly type-II censored scheme.

Statistical inference of CL for the Chen distribution with known shape parameter has been considered under type-II progressive hybrid censoring scheme. The concept of max p-value method discussed in [6] is adopted to select the optimum value of the shape parameter. Then, we constructed the ML estimator for CL and developed a testing procedure for the lifetime performance index of the products with Chen distribution. Also, Bayesian approach is developed of estimation procedure of CL on the basis of the type-II progressive hybrid censored sample with max p-value method.

This article is organized in successive sections as follows, the CL and its properties introduced in Section 2. The ML and Bayes estimators of CL are discussed in Section 3. Testing hypothesis of CL in Section 4 are developed over a lower bound for the CL. In Section 5 numerical example is used to illustrate our purpose. In Section 6 Monte Carlo method is conducted to assess our results. Finally, simple comments are used to describe the numerical results in concluding remarks in Section 7.

2 The lifetime performance index

The quality of products show that the longer lifetime product is a larger-the-better type quality characteristic. The lifetime performance of products can be described with lifetime performance index CL . Then random lifetime T is used to denote the lifetime of Chen product with PDF given by (3). Generally, for products to satisfy customers and be economically profitable the lifetime is required to exceed L unit times. The larger-the-better quality characteristic measured by capability index CL [1]. Under the process mean μ, standard deviation σ and lower specification limit L the CL is defined by

(7) C L = μ − L σ .

The assessment problem of the CL can be defined as the lifetime performance index.

Let T be the lifetime random variable with Chen distribution (3) the random variable Z defined by Z = exp (Ta) − 1, a> 0 is distributed with exponential form with PDF given by

(8) f ( z ) = b exp ⁡ ( − b z ) .

This transformation present some important properties, as follows

  1. The random variable Z has CDF and h(z) described respectively by

    (9) F ( z ) = 1 − exp ⁡ ( − b z ) , z > 0 ,

    and

    (10) h ( z ) = b , b > 0.

  2. The transform lifetime performance index CL of Chen distribution with process mean μ = E ( z ) = 1 b and process standard deviation σ = 1 b is given by

(11) C L = μ − L σ = 1 − b L ,  - ∞ < C L < 1 ,

where L is the lower specification limit.

Transformation z = exp (ta) − 1, a > 0 present one-to-one and strictly increasing of t to z, so data set of t transformed to data set z _ and have the same effect in assessing the CL. Also, this transformation enables easy calculation, the CL > 0 for the value of mean 1 b > L . The equations (10) and (11) show that, the smaller the value of b the smaller the failure rate and the larger the CL , and the larger the value of b the larger the failure rate and the smaller the CL. Hence, the CL is used as accurately description to the lifetime performance of a new product. If the lifetime of a product exceeds the lower specification limit (i.e. Z ≥ L), the product is labeled as a conforming product. Otherwise, the product is labeled as a nonconforming product. The conforming rate (Pr) is calculated as the ratio of conforming products that defined by

(12) P r = P r ( z ≥ L ) = exp ⁡ ( − b L ) = exp ⁡ ( C L − 1 ) , − ∞ < C L < 1.

Hence, we obtained the strictly increasing relationship between Pr and CL, for known b > 0. For example, if a = 0.38, some different values CL and the corresponding value of Pr are summarized in Table 1. Other values of Pr can be calculated by Eq. (12). Since Pr and CL have an one-to-one relationship, therefore, the lifetime performance index can be a flexible and effective tool, in evaluating product quality and estimating the conforming rate Pr.

Table 1

The CL values and the corresponding conforming rate Pr for a = 0.38

CL Pr CL Pr CL Pr CL Pr
−∞ 0.0000 0.10 0.40657 0.45 0.5770 0.80 0.8187
−10 0.0000 0.15 0.4274 0.50 0.6065 0.85 0.8607
−5 0.0025 0.20 0.4493 0.55 0.6376 0.90 0.9048
−2 0.0498 0.25 0.4724 0.60 0.6703 0.95 0.9512
−1 0.1353 0.30 0.4966 0.65 0.7047 0.97 0.9705
−0.05 0.3499 0.35 0.5221 0.70 0.7408 0.99 0.9901
0 0.36788 0.40 0.5488 0.75 0.7788 1.00 1.0000

3 Bayesian and Non-Bayesian estimations of CL

3.1 Maximum likelihood approch in estimation of CL

Let T = (T1;m,n., T2;m,n., .... , Tr;m,n.) be the type-II progressive hybrid censored sample from Chen lifetime distribution with the PDF and CDF given by (3) and (4). For the prior censoring scheme R = {R1, R2, .... , Rm} and prescribed time of the experiment τ the joint likelihood function without normalized constant is given by

(13) L ( b | a , t _ ) = ( a b ) r ∏ i = 1 r t i a − 1 exp ⁡ { ∑ i = 1 r { t i a + b R i + 1 1 − exp t i a } + b δ j × 1 − exp τ a ∑ i = r + 1 m R i + 1 } ,

where δj are given by (2) and

0 < t 1 < t 2 < … < t r < ∞ .

So, the log-likelihood function of (13) is given by

(14) ℓ ( b | a , t _ ) = r log ⁡ ( a b ) + ∑ i = 1 r { a − 1 log ⁡ t i + t i a + b R i + 1 1 − exp t i a } + b δ j 1 − exp τ a ∑ i = r + 1 m ( R i + 1 ) .

For simplicity the value of observed value ti is written instead of ti;m,n.. After calculating the derivatives of (14) for b and equating to zero, we get the ML estimator of b as

(15) b ^ ( a ) = r η + A ,

where

(16) η = ∑ i = 1 r R i + 1 exp t i a − 1 A = δ j exp τ a − 1 ∑ i = r + 1 m R i + 1 .

By using the invariance property of ML estimators see [18], the ML estimator of CL is given by

(17) C ^ L = 1 − b ^ L = 1 − r η + A L .

Theorem 1

Let T1, T2, ...., Tr be an type-II progressive hybrid censored order statistic from two-parameter Chen distribution (3) and z = exp (ta) − 1, with censored scheme R, then

(18) 2 b η ∼ χ 1 − α ( 2 r ) 2 ,

where η given by (16)

Proof: let Yi = bZi , i = 1, .... , r. It can be seen that Y1 < Y2 <... < Yr is an type-II progressive hybrid censored order statistic from standard exponential distribution. Consider the following transformation

(19) ζ 1 = n Y 1 , ζ 2 = n − R 1 − 1 Y 2 − Y 1 , ζ 3 = n − R 1 − R 2 − 2 Y 3 − Y 2 , ⋮ ζ r = ( n − R 1 − R 2 − … − R r − 1 − r + 1 ) Y r − Y r − 1 .

The independent and identically generalized spacings ζ1, ζ2, .... , ζr are distributed as standard exponential distribution, see [19]. Hence,

(20) 2 ∑ i = 1 r ζ i = 2 ∑ i = 1 r ( R i + 1 ) Y i = 2 b ∑ i = 1 r ( R i + 1 ) Z i = 2 b η ,

has a chi-squared distribution with 2r degrees of freedom.

Remark 1: The expectation of Equation (17) of ĈL presented by

(21) E ( C ^ L ) = 1 − E r η + A L .

Hence ĈL is not unbiased estimator of CL. But as r −→ ∞, E(ĈL) −→CL, so ĈL is asymptotic unbiased estimator, also Ĉ L is consistent.

3.2 Bayesian approch in estimation of CL

Consider the gamma prior distribution of the parameter b given by

(22) h ∗ ( b ) ∝ b φ − 1 exp ⁡ ( − ψ b ) ,

where the parameters φ and ψ are obtained from our previous experience. Then the posterior distribution can be obtained from Eqs (13) and (22) as follows

(23) h ( b | t _ ) ∝ b r + φ − 1 exp ⁡ { − b { ∑ i = 1 r { R i + 1 ( exp t i a − 1 ) } + ψ + δ j exp τ a − 1 ∑ i = r + 1 m R i + 1 } } .

By considering a squared-error loss function , (b,b̂) = (b̂ -b)2, hence, the posterior mean is considered as the Bayes estimator of b given by

(24) b ^ B = r + φ η ∗ ,

where

(25) η ∗ = ∑ i = 1 r R i + 1 exp t i a − 1 + B ,

and

(26) B = ψ + δ j exp τ a − 1 ∑ i = r + 1 m R i + 1 .

Hence, the Bayes estimator C ~ L of CL can be written by using (11) and (24) as

(27) C ~ L = E C L | h ( b | t _ ) = 1 − r + φ η ∗ L .

Theorem 2

Let T1, T2, .... , Tr be an type-II progressive hybrid censored order statistic from two-parameter Chen distribution (3) and Z = exp (Ta)−1, with censored scheme R, then

(28) Y = 2 λ η ∗ ∼ χ 1 − α ( 2 ( r + φ ) ) 2 ,

where η * given by (25).

Proof: Let Y=2bη* by using the change of variables see [20], then we obtain that the PDF of Y is given by

(29) f Y ( y ) = h ( 2 b η ∗ | t _ ) | | I y | | = y 2 ( r + φ ) 2 − 1 2 2 ( r + φ ) 2 Γ 2 ( r + φ ) 2 exp − y 2 ,

hence, 2 b η ∗ ∼ χ 2 ( r + φ ) 2

Remark 2: The expectation of C ~ L can be derived as follows

(30) E ( C ~ L ) = 1 − 2 ( r + φ ) b L E 1 2 b η ∗ = 1 − ( r + φ ) λ L ( r + φ ) − 1 .

Hence, estimator C ~ L is not an unbiased estimator of CL. When r −→ ∞, E ( C L ) ⟶ C L , so Beyes estimator C ~ L is asymptotic unbiased estimator, also ĈL is consistent.

4 Confidence interval for CL

In this section, we construct 100(1 − α)% lower bound for CL with ML and Bayes methods to decide hypothesis testing procedure for determining that the lifetime performance index of products meets the predetermined level. For this purpose assume that target value c which the lifetime performance index is larger than it. Then, the null hypothesis H0 : CL < c is constructed against the alternative hypothesis H1 : CL > c. The two methods is described as follows

  1. In the classical approach, from Theorem (1) and the ML estimate ĈL (17) the lower (1 − α) percentile of χ 1 − α ( 2 r ) 2 is given by

    (31) 1 − α = P 2 b η ≤ χ 1 − α ( 2 r ) 2 = P b ≤ χ 1 − α ( 2 r ) 2 2 η = P 1 − C L L ≤ χ 1 − α ( 2 r ) 2 2 η = P C L ≥ 1 + ( 1 − C ^ L ) L χ 1 − α ( 2 r ) 2 2 [ A ( 1 − C ^ L ) − r L ] .

    For known α the 100(1 − α)% lower bound for CL is

    (32) L BML = 1 + ( C ^ L − 1 ) L χ 1 − α ( 2 r ) 2 2 [ A ( 1 − C ^ L ) + r L ] .

  2. In the Bayesian approach, under consideration known α and 2 λ η ∗ ∼ χ 1 − α ( 2 ( r + φ ) ) 2 the 100(1 − α)% one-sided credible interval for CL given by

(33) 1 − α = P 2 b η ∗ ≤ χ 1 − α ( 2 ( r + φ ) ) 2 = P b ≤ χ 1 − α ( 2 ( r + φ ) ) 2 2 η ∗ = P 1 − C L L ≤ χ 1 − α ( 2 ( r + φ ) ) 2 2 η ∗ = P C L ≥ 1 + C ~ L − 1 χ 1 − α ( 2 ( r + φ ) ) 2 2 ( r + φ ) ,

then, the lower bound for CL is given by

(34) L BB = 1 + C ~ L − 1 χ 1 − α ( 2 r ) 2 2 ( r + φ ) ,

where φ is a prior parameters, r is the number of failure and C ~ L denotes the Bayes estimates of CL.

In the following, we employ ^the one-sided hypothesis testing to test if the lifetime performance index reduces to the required level. The testing steps can be described as follows

Step 1. Gini statistic is employed to estimate the shape Chen parameter a as given in [21] and [22], as follows the Gini statistic is defined by

(35) G r = ∑ i = 1 r − 1 i D i + 1 ( r − 1 ) ∑ i = 1 r D i ,

where Di = (r − 1)(Yi;m,n.-Yi−1;m,n.) for i = 2, 3, ...., r and D1 = rY1;m,n. while Ti;m,n. = exp (Ta) − 1. For r > 20, 12 r − 1 ( G r − 0.5 ) tends to the standard normal distribution N(0, 1). Hence, the P- value = P | z | > | 12 r − 1 ( g r − 0.5 ) | where gr is the observed value of Gm and Z has an approximation of N(0, 1). So, by using the maximum P-value method, the optimum value of a is selected and then we suppose a is known.

Step 2. From the observed original data set (t1, t2, ..., tn) present (y1;m,n. , y2;m,n. , ..., ym;m,n.) with transform given by (8).

Step 3. The index value c and the corresponding lower lifetime limit L is determine, then the null hypothesis H0: CL < c against alternative H1: CL > c is constructed.

Step 4. Two values ĈL and C ~ L are calculated.

Step 5. The two lower bound LBML and LBB for CL are calculated.

Step 6. Specify a significance level α.

Step 7. Statistical test measured as, if c ∉[LBB, ∞) or c ∉ [LBML,∞), the null hypothesis is rejected and say that lifetime performance index of product meets the required level.

5 Illustrative examples

In the real life problem, we consider a real data presented in [23]. The data is listed in Table 3 to represent graft survival times in months of 148 renal transplant patients. This data is fitted with Chen distribution see [15]. The observed type-II progressive hybrid censored order statistic is generated from the data in Table 2, with censoring scheme R = {5(5), 0(5), 5(5), 0(5), 4(5), 0(5), 4(5), 0(3), 1(10)}, n = 148, m = 48 and τ = 10 this data are summarized in Table 3. The estimate value of shape parameter a of Chen distribution is calculated by using the concept of Gini statistic with maximum P-value method as given in Table 4. The optimum values a that maximize P-value as given in Table 4 is a=0.38 and P-value = 0.9943 hence the parameter a is known. The observed exponential type-II progressive hybrid censored order statistics obtained with transformation zi,m,n. = exp t i 0.38 −1 are reported also in Table 4.

Table 2

A real data set originally reported in Hand et al. [12]

0.035 0.068 0.100 0.101 0.167 0.168 0.197 0.213 0.233 0.234
0.508 0.508 0.533 0.633 0.767 0.768 0.770 1.066 1.267 1.300
1.600 1.639 1.803 1.867 2.180 2.667 2.967 3.328 3.393 3.700
3.803 4.311 4.867 5.180 6.233 6.367 6.600 6.600 7.180 7.667
7.733 7.800 7.933 7.967 8.016 8.300 8.410 8.607 8.667 8.800
9.100 9.233 10.541 10.607 10.633 10.667 10.869 11.067 11.180 11.443
12.213 12.508 12.533 13.467 13.800 14.267 14.475 14.500 15.213 15.333
15.525 15.533 15.541 15.934 16.200 16.300 16.344 16.600 16.700 16.933
17.033 17.067 17.475 17.667 17.700 17.967 18.115 18.115 18.933 18.934
19.508 19.574 19.733 20.148 20.180 20.900 21.167 21.233 21.600 22.100
22.148 22.180 22.180 22.267 22.300 22.500 22.533 22.867 23.738 24.082,
24.180 24.705 25.213 25.705 29.705 30.443 31.667 31.934 32.180 32.367
32.672 32.705 33.148 33.567 33.770 33.869 34.836 34.869 34.934 35.738
36.180 36.213 39.410 39.433 39.672 40.001 41.733 41.734 42.311 42.869
43.180 43.279 43.902 44.267 44.475 44.900 45.148 46.451
Table 3

The random generated type-II progressive hybrid censored order statistic

ti 0.035 0.068 0.100 0.101 0.167 0.168 0.197 0.213 0.233 0.234
0.508 0.508 0.633 0.768 1.267 1.600 1.639 1.867 3.328 3.393
3.803 4.311 4.867 6.233 6.600 7.180 8.016 8.300 8.667 9.100
zi 0.323 0.433 0.517 0.520 0.660 0.662 0.715 0.743 0.777 0.779
1.166 1.166 1.318 1.471 1.987 2.305 2.342 2.553 3.851 3.908
4.266 4.711 5.200 6.422 6.756 7.290 8.074 8.345 8.698 9.119
Table 4

Numerical values of P -values for Real data given in Table 3

0.255 0.0088 0.330 0.3226 0.405 0.6318 0.480 0.0725
0.260 0.0123 0.335 0.3754 0.410 0.5676 0.485 0.0603
0.265 0.0168 0.340 0.4329 0.415 0.5071 0.490 0.0499
0.270 0.0226 0.345 0.4947 0.42 0.4506 0.495 0.0412
0.275 0.0302 0.350 0.5604 0.425 0.3983 0.500 0.0338
0.280 0.0397 0.355 0.6297 0.430 0.3501 0.505 0.0276
0.285 0.0515 0.360 0.7018 0.435 0.3061 0.510 0.0225
0.290 0.0661 0.365 0.7761 0.440 0.2662 0.515 0.0182
0.295 0.0838 0.370 0.8520 0.445 0.2303 0.520 0.0147
0.300 0.1050 0.375 0.9288 0.450 0.1982 0.525 0.0118
0.305 0.1300 0.455 0.1697 0.530 0.0095
0.310 0.1592 0.385 0.9180 0.460 0.1446 0.535 0.0076
0.315 0.1929 0.390 0.8430 0.465 0.1225 0.540 0.0060
0.320 0.2313 0.395 0.7699 0.470 0.1034 0.545 0.0048
0.325 0.2745 0.400 0.6994 0.475 0.0868 0.550 0.0038

The lower lifetime limit and transformed (limit) Lt and Ly , respectively given by 0.314 and exp 0.3140.38− 1 = 0.90393. For consideration of product managers’ the Pr of operational performances is required to exceed 80%.With results in Table 1, the CL value operational performances are required to exceed 0.80. Hence, the target value of performance index is taken as c = 0.80. The testing hypothesis

H0: CL <0.80 against the alternative H1: CL > 0.80 is constructed.

The Bayes estimates which do not have any prior informations small values are selected to the gamma hyper parameters to reflect vague prior informations. So, φ and ψ are assumed to be the same value 0,0001.Hence the results in the Bayesian and non-Bayesian are conforming. Calculate the 95% one-sided confidence and credible intervals of [LBML,∞) and [LBB,∞) from (32) and (34), respectively.

  1. ML approach the 95% lower bound for CL is obtained as [LBML, ∞)=[0.9675, ∞), since the target value c= 0.80 ∉ [0.9675,∞), so the null hypothesis H0 : CL ≤ 0.8 is rejected.

  2. In Bayesian approch the 95% lower bound for CL is obtained as [LBB, ∞)=[0.9513, ∞), since the target value c= 0.80 ∉ [0.9513,∞), so the null hypothesis H0 : CL ≤ 0.8 is rejected.

Then, in the two approches we can say the lifetime performance index of the products does meet the required level.

6 Simulation study

Over the one sided confidence or credible intervals of CL the simulation study is constructed to study the confidence level (1 - α) of the lifetime performance index CL. In our studying consider confidence level α = 0.05 and for the Chen parameters are chosen to be two sets (a, b) = {(1.2, 1.2), (1.7, 0.2)}. This study is discussed for different size (n, m), different priors (φ, ψ) and different sampling schemes. The lower lifetime limit Lx respected to model parameters is assumed to be 0.2 and 0.4, respectively. This problem is constructed under the following steps

  1. The type-II progressive hybrid censored order statistic (t1, t2, ..., tr) is generated from Chen distribution as proposed in [15]. The observed exponential data (z1, z2, .... , zr) is obtanied by transformation z = exp (ta) − 1.

  2. From Eqs (32) and (34) 95% lower bounds LBML and LBB are calculated.

  3. Put Ω1 = 1 or Ω2 = 1 If CL > LBML or CL > LBB, respectively elsewhere Ω1 = 0 or Ω2 = 0.

  4. For each data generation from Steps 2 and 3 are repeated to 100 times.

  5. The total count TotalCount  Ω 1 100 or TotalCount  Ω 2 100 are used as the estimate of confidence level (1 − ^) respectively.

  6. The last steps are repeated 1000 times, we get ( 1 − α ^ ) ( 1 ) , ( 1 − α ^ ) ( 2 ) , . . . , ( 1 − α ^ ) ( 1000 )

  7. The average empirical confidence level is calculated by

AV ( 1 − α ^ ) = 1 1000 ∑ i = 1 1000 ( 1 − α ^ ) ( i ) ,

with error Er (mean square error) given by

Er ( 1 − α ^ ) = 1 1000 ∑ i = 1 1000 ( 1 − α ^ ) ( i ) − ( 1 − α ) 2 .

The results of simulation study are reported in Tables 5 and 6.

Table 5

AV(Er) of empirical confidence level (1 − α) for CL when α = 0.05 and (a, b) =(1.2, 1.2)

τ n m C.S. MLE Bayes
φ = 1, ψ = 1 φ = 2, ψ = 3 φ = 3, ψ = 2
0.4 30 15 (15, 014) 0.93(0.0010) 0.92(0.0007) 0.97(0.0008) 0.94(0.0009)
(014, 15) 0.95(0.0015) 0.933(0.008) 0.955(0.0008) 0.95(0.0009)
(07,15,07) 0.95(0.0013) 0.95(0.0007) 0.94(0.0009) 0.95(0.0009)
(115) 0.95(0.0012) 0.94(0.0007) 0.94(0.0009) 0.95(0.0010)
0.7 30 15 (15, 014) 0.95(0.0009) 0.94(0.0006) 0.97(0.0006) 0.95(0.0008)
(014, 15) 0.94(0.0020) 0.93(0.0008) 0.96(0.0006) 0.95(0.0007)
(07,15,07 ) 0.95(0.0010) 0.96(0.0007) 0.96(0.0006) 0.95(0.0006)
(115) 0.93(0.0011) 0.95(0.0006) 0.96(0.0007) 0.94(0.0007)
0.4 30 25 (5, 024) 0.94(0.0009) 0.95(0.0005) 0.95(0.0006) 0.95(0.0006)
(024, 5) 0.94(0.0010) 0.94(0.0006) 0.94(0.0007) 0.95(0.0008)
(012,5,012 ) 0.93(0.0011) 0.94(0.0006) 0.96(0.0007) 0.93(0.0007)
((1,04)5 ) 0.93(0.0011) 0.95(0.0007) 0.96(0.0008) 0.96(0.0007)
0.7 30 25 (5, 024) 0.95(0.0008) 0.95(0.0004) 0.94(0.0006) 0.94(0.0006)
(024, 5) 0.94(0.0009) 0.95(0.0005) 0.96(0.0006) 0.94(0.0007)
(012,5,012 ) 0.95(0.0010) 0.97(0.0006) 0.95(0.0006) 0.94(0.0006)
((1,04)5 ) 0.96(0.0010) 0.96(0.0005) 0.95(0.0007) 0.95(0.0006)
0.4 50 25 (25, 024) 0.95(0.0007) 0.96(0.0004) 0.95(0.0005) 0.95(0.0005)
(024, 25) 0.94(0.0007) 0.94(0.0005) 0.95(0.0006) 0.95(0.0006)
(012 ,25,012) 0.94(0.0007) 0.95(0.0005) 0.94(0.0005) 0.95(0.0006)
(125) 0.96(0.0007) 0.95(0.0004) 0.96(0.0005) 0.94(0.0007)
0.7 50 25 (25, 024) 0.94(0.0006) 0.95(0.0004) 0.94(0.0005) 0.95(0.0005)
(024, 25) 0.94(0.0006) 0.94(0.0004) 0.94(0.0005) 0.94(0.0005)
(012 ,25,012) 0.93(0.0007) 0.94(0.0004) 0.94(0.0006) 0.95(0.0006)
(125) 0.96(0.0007) 0.95(0.0004) 0.95(0.0005) 0.96(0.0006)
0.4 50 40 (10, 039) 0.94(0.0005) 0.95(0.0003) 0.95(0.0004) 0.95(0.0004)
(039, 10) 0.94(0.0006) 0.95(0.0005) 0.96(0.0004) 0.94(0.0004)
(019 ,10,020) 0.95(0.0006) 0.95(0.0004) 0.95(0.0004) 0.95(0.0005)
((1,03)10 ) 0.95(0.0005) 0.95(0.0004) 0.96(0.0004) 0.95(0.0004)
0.7 50 40 (10, 039) 0.95(0.0004) 0.95(0.0002) 0.95(0.0003) 0.95(0.0004)
(039, 10) 0.94(0.0005) 0.96(0.0004) 0.94(0.0004) 0.94(0.0004)
(019 ,10,020) 0.94(0.0005) 0.94(0.0004) 0.94(0.0005) 0.94(0.0004)
((1,03)10 ) 0.95(0.0005) 0.95(0.0005) 0.96(0.0004) 0.95(0.0004)
Table 6

AV(Er) of empirical confidence level (1 − α) for CL when α = 0.05 (a, b) = (1.7, 0.2)

τ n m C.S. MLE Bayes
φ = 1, ψ 1 φ = 2, ψ = 3 φ = 3, ψ = 2
0.4 30 15 (15, 014) 0.91(0.0025) 0.93(0.0014) 0.92(0.0017) 0.96(0.0018)
(014, 15) 0.90(0.0030) 0.95(0.0022) 0.94(0.0023) 0.92(0.0019)
(07,15,07) 0.92(0.0027) 0.94(0.0020) 0.97(0.0020) 0.94(0.0021)
(115) 0.91(0.0025) 0.92(0.0019) 0.93(0.0018) 0.96(0.0020)
0.7 30 15 (15, 014) 0.92(0.0023) 0.94(0.0012) 0.95(0.0014) 0.92(0.0013)
(014, 15) 0.93(0.0032) 0.93(0.0019) 0.95(0.0018) 0.94(0.0020)
(07,15,07 ) 0.94(0.0028) 0.93(0.0017) 0.96(0.0015) 0.94(0.0014)
(115) 0.92(0.0024) 0.94(0.0016) 0.97(0.0017) 0.96(0.0017)
0.4 30 25 (5, 024) 0.92(0.0018) 0.94(0.0011) 0.95(0.0012) 0.95(0.0015)
(024, 5) 0.93(0.0019) 0.95(0.0013) 0.95(0.0015) 0.94(0.0015)
(012,5,012 ) 0.92(0.0020) 0.94(0.0015) 0.97(0.0017) 0.94(0.0018)
((1,04)5 ) 0.93(0.0019) 0.94(0.0014) 0.96(0.0018) 0.95(0.0019)
0.7 30 25 (5, 024) 0.93(0.0015) 0.94(0.0012) 0.95(0.0014) 0.92(0.0013)
(024, 5) 0.92(0.0017) 0.93(0.0013) 0.95(0.0018) 0.94(0.0020)
(012,5,012 ) 0.95(0.0016) 0.93(0.0012) 0.96(0.0015) 0.94(0.0014)
((1,04)5 ) 0.92(0.0016) 0.94(0.0013) 0.97(0.0014) 0.96(0.0013)
0.4 50 25 (25, 024) 0.93(0.0013) 0.94(0.0010) 0.95(0.0010) 0.95(0.0009)
(024, 25) 0.93(0.0018) 0.95(0.0012) 0.95(0.0012) 0.94(0.0011)
(012 ,25,012) 0.94(0.0014) 0.93(0.0013) 0.95(0.0011) 0.94(0.0010)
(125) 0.94(0.0015) 0.94(0.0012) 0.94(0.0011) 0.95(0.0010)
0.7 50 25 (25, 024) 0.94(0.0012) 0.94(0.0007) 0.94(0.0007) 0.93(0.0008)
(024, 25) 0.92(0.0014) 0.95(0.0009) 0.95(0.0009) 0.94(0.0009)
(012 ,25,012) 0.94(0.0010) 0.95(0.0008) 0.94(0.0009) 0.95(0.0007)
(125) 0.93(0.0010) 0.94(0.0009) 0.93(0.0009) 0.95(0.0010)
0.4 50 40 (10, 039) 0.94(0.0008) 0.94(0.0005) 0.94(0.0004) 0.96(0.0006)
(039, 10) 0.93(0.0009) 0.95(0.0009) 0.95(0.0006) 0.94(0.0007)
(019 ,10,020) 0.95(0.0008) 0.94(0.0008) 0.96(0.0006) 0.95(0.0007)
((1,03)10 ) 0.93(0.0007) 0.94(0.0009) 0.93(0.0007) 0.96(0.0007)
0.7 50 40 (10, 039) 0.95(0.0006) 0.94(0.0004) 0.94(0.0004) 0.96(0.0004)
(039, 10) 0.94(0.0007) 0.95(0.0005) 0.95(0.0005) 0.94(0.0005)
(019 ,10,020) 0.93(0.0006) 0.94(0.0006) 0.96(0.0004) 0.95(0.0004)
((1,03)10 ) 0.93(0.0007) 0.94(0.0005) 0.93(0.0005) 0.96(0.0004)

7 Conclusions and recommendations

The problem of determine the quality of the industrial products requires some tests on the lifetime of the products. The calculation of some statistical measures which are used to prove whether the product quality meets the required level. Capability index CL which can be used in measuring process performance and potential capability. This paper, consider this problem when the lifetime of products follows Chen distribution and the experiment runing under general hybrid censoring scheme. Then, we constructed the ML and Bayes stimators for CL and developed a testing procedure for the lifetime performance index of the products with Chen distribution. We reported some comments observed from numerical results as follows.

  1. Form Tables 5 and 6 all empirical results are close to confidence level (1 - α).

  2. The value of Er is smaller over all of the average empirical confidence levels.

  3. The results in Bayesian approach are smaller than the ML approach through Er for one-sided credible and confidence intervals.

  4. The results are improved in terms of AV and (Er) when τ is increasing, this means that when τ → ∞.

Acknowledgement

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (G-189-363-39). The authors, therefore, acknowledge with thanks DSR for technical and financial support. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

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Received: 2019-05-16
Accepted: 2019-09-03
Published Online: 2019-11-21

© 2019 M. M. Badr et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2019-0062
Keywords for this article
Chen distribution; Process capability indices; Hybrid progressive censoring; Bayesian estimation; Hypothesis tests
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Non-equilibrium Phase Transitions in 2D Small-World Networks: Competing Dynamics
  3. Harmonic waves solution in dual-phase-lag magneto-thermoelasticity
  4. Multiplicative topological indices of honeycomb derived networks
  5. Zagreb Polynomials and redefined Zagreb indices of nanostar dendrimers
  6. Solar concentrators manufacture and automation
  7. Idea of multi cohesive areas - foundation, current status and perspective
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  9. An application of Nwogu’s Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves
  10. Competing Risks Model with Partially Step-Stress Accelerate Life Tests in Analyses Lifetime Chen Data under Type-II Censoring Scheme
  11. Group velocity mismatch at ultrashort electromagnetic pulse propagation in nonlinear metamaterials
  12. Investigating the impact of dissolved natural gas on the flow characteristics of multicomponent fluid in pipelines
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  20. Integrability, exact solutions and nonlinear dynamics of a nonisospectral integral-differential system
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  25. Understanding quantum mechanics: a review and synthesis in precise language
  26. Plane Wave Reflection in a Compressible Half Space with Initial Stress
  27. Evaluation of the realism of a full-color reflection H2 analog hologram recorded on ultra-fine-grain silver-halide material
  28. Graph cutting and its application to biological data
  29. Time fractional modified KdV-type equations: Lie symmetries, exact solutions and conservation laws
  30. Exact solutions of equal-width equation and its conservation laws
  31. MHD and Slip Effect on Two-immiscible Third Grade Fluid on Thin Film Flow over a Vertical Moving Belt
  32. Vibration Analysis of a Three-Layered FGM Cylindrical Shell Including the Effect Of Ring Support
  33. Hybrid censoring samples in assessment the lifetime performance index of Chen distributed products
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  110. Erratum
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