https://orcid.org/0000-0003-4721-3153
Figures
Abstract
This article considers the estimation of the stress-strength reliability parameter, θ = P(X < Y), when both the stress (X) and the strength (Y) are dependent random variables from a Bivariate Lomax distribution based on a progressive type II censored sample. The maximum likelihood, the method of moments and the Bayes estimators are all derived. Bayesian estimators are obtained for both symmetric and asymmetric loss functions, via squared error and Linex loss functions, respectively. Since there is no closed form for the Bayes estimators, Lindley’s approximation is utilized to derive the Bayes estimators under these loss functions. An extensive simulation study is conducted to gauge the performance of the proposed estimators based on three criteria, namely, relative bias, mean squared error, and Pitman nearness probability. A real data application is provided to illustrate the performance of our proposed estimators through bootstrap analysis.
Citation: Helu A, Samawi H (2022) Inference on P(X < Y) in Bivariate Lomax model based on progressive type II censoring. PLoS ONE 17(5): e0267981. https://doi.org/10.1371/journal.pone.0267981
Editor: Bryan C. Daniels, Arizona State University, UNITED STATES
Received: November 25, 2021; Accepted: April 19, 2022; Published: May 12, 2022
Copyright: © 2022 Helu, Samawi. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
It has always been the main objective of manufacturers to provide reliable products. For their products to remain desired and thus profitable, they strive to create high-quality and long-lasting products. In order to accomplish this, it is necessary to know the failure time distributions of products that are obtained by performing life testing experiments on them before they are released into the market.
In reliability studies, a sample of size n is subjected to a test for observing their failure times. The data recorded is then used to establish a time-to-failure distribution. This may be impractical, costly and time consuming since the experimenter may need to terminate the study before recording the failure times for all the subjects under consideration due to time constraints and facilities restrictions. In addition, some functioning tests’ subjects may need to be removed so they can be used in another test or to collect degradation related information about failure time data which is typically the case when the test subjects are expensive such as medical equipment. Moreover, in some cases, the failure is planned and predicted, but it does not occur due to operator flaw, equipment malfunction, test irregularity, etc. Samples that result from such situations are called censored samples.
The two most common types of censoring schemes in the literature are type-I and type-II censoring schemes, in which the test ceases at a pre-specified time, or after a predetermined number of failures. However, these censoring schemes do not allow intermediate removal of active units during the experiment other than the final termination point. Therefore, the focus in the last few years has been on progressive censoring.
Progressive type-II censoring is a generalization of type-II censoring. It allows researchers to remove subjects before the final termination point if certain circumstances arise, such as losing contact with the subjects. Under this type of censoring, n independent items are placed simultaneously on a life testing experiment, and only m (< n) failures are completely observed. The censoring occurs progressively in m stages as follows: When the first failure is observed, a random sample of size R1 is immediately drawn and removed from the (n − 1) survivals, hence, leaving n − 1 − R1 survival items. Then after the failure of the second item, the sample becomes n − 2 − R1, in which another sample of size R2 is randomly selected and removed from the remaining survival units. Continue with this process until m failures are observed, and all the remaining n − m − R1 − ⋯ −Rm−1( = Rm) surviving units are removed from the experiment. It is assumed that the lifetimes of these n units are independent and identically distributed with a common distribution function F. Moreover, n, m, and the censoring scheme R = (R1, R2, …, Rm) are all pre-specified. Note that if R1 = R2 = ⋯ = Rm−1 = 0, then Rm = n−m which corresponds to type-II censoring. If R1 = R2 = ⋯ = Rm = 0, then m = n which represents the complete data set. For a comprehensive literature review on progressive censoring, readers may refer to [1].
Considerable attention has been directed towards progressive type II censoring. This is largely due to the availability of the high-speed computing resources, which makes it feasible for simulation studies as well as a practical method of gathering lifetime data for both researchers and practitioners [2].
A vast number of researchers dedicated so much of their work to study the stress-strength model. [3] provided an interesting connection between the stress-strength empirical estimate and the classical Man-Whitney statistic. More works have followed to provide point and interval estimation of θ using different approaches. For example [4] provided a comprehensive review of the development of the stress-strength reliability and its applications until the year 2003. Recently, [5] studied the estimation of θ when X and Y are two independent generalized Pareto distributions with different parameters. [6] studied the reliability of the stress-strength model when X and Y are independent Poisson random variables. Whereas, [7] considered the problem of estimating θ when X and Y are distributed as Lindley with different shape parameters. [8] derived a point and interval estimation of θ using maximum likelihood, parametric and nonparametric bootstrap methods when X and Y are independent power Lindley random variables. [9] extended the work of [8] and developed a Bayesian inference on θ. In addition, [10] derived asymptotic confidence intervals for θ when X and Y are two independent generalized Pareto random variables with the same scale parameter.
However, all aforementioned results are based on the assumption that the strength (X) and the stress (Y) are independent random variables. Still little attention has been paid to the actual situation in which X and Y are dependent random variables. Recently, different estimators of θ have been proposed assuming several bivariate distribution families of (X, Y) for describing the joint behavior of strength and stress variables. Among others, [11] assumed that X and Y are marginally distributed as a skewed scale mixture of normal distributions and constructed the corresponding joint distribution using a Gaussian copula. [12] constructed the joint distribution of (X, Y) using a Farlie-Gumbel-Morgenstern copula. [13] derived Bayesian inference of θ when strength and stress are dependent random variables with a bivariate Rayleigh distribution. [14] considered the problem of estimating θ when X and Y are dependent random variables with a bivariate underlying distribution using kernel estimation and bivariate ranked set sampling. Recently, [15] investigated the asymptotic properties of the kernel density estimator based on progressive type II censoring and their application to hazard function estimation. [16] used classical and Bayesian estimation methods to derive estimates of θ when X and Y are dependent random variables from Bivariate Lomax distribution. [17] discussed the kernel based estimation of θ when X and Y are dependent random variables under bivariate progressive type II censored sample and provided asymptotic properties of the kernel estimators of θ based on progressive type-II censoring.
Little work has been done to study the reliability in case of bivariate random variables, and as far as we know, none is done to study the reliability in case of bivariate data based on progressive type II censoring.
The article describes the estimation of the reliability θ = P(X < Y) when X and Y are dependent random variables based on progressively type II censored samples from the Bivariate Lomax distribution using classical and Bayesian approaches. For the classical approach we propose using the method of moments (MOM) and the Maximum likelihood estimators (MLE). While for the Bayesian approach, we derive the estimates based on symmetric and asymmetric loss functions. It is observed that the Bayes estimates cannot be obtained in explicit form, so instead of using numerical techniques, approximation method such as Lindley’s approximation is applied. To compare the performance of the proposed estimators we used real life data. In addition, we used the bootstrap approach to calculate the bootstrap estimates, standard error, and lower and upper confidence interval limits.
The layout of this paper is as follows. In Section 2, we briefly describe the Bivariate Lomax model. The classical analysis is provided in Section 3. In Section 4, we discuss the Bayesian inference. Simulation studies which are conducted to assess the accuracy of the proposed methods are presented in Section 5. In Section 6, the analysis of real data set is presented for illustrative purposes. Finally, conclusions are presented in Section 7.
2. Bivariate Lomax model
This model was proposed by [18], considering a two component system where the component lifetimes X and Y are conditionally independently exponentially distributed with failure rates ηλ1 and ηλ2 respectively, where the parameter η represents the environment effect and λ1 and λ2 are the original failure rates. Then,
(1)
where, G(η) is the cumulative distribution function (cdf) of the environment parameter η. In order to find the unconditional distribution of (X, Y), we assign the gamma distribution gη(c, b) as the distribution of η, with probability density function (pdf)
(2)
Therefore, the joint density function of (X, Y) is given by
(3)
Let and
, hence (3) is reduced to
(4)
which is known as the Bivariate Lomax distribution. It can be shown that the conditional and the joint cumulative survival function of (X, Y), the marginal of X and the marginal of Y are defined as follows:
(5)
(6)
(7)
(8)
The quantity of interest is the parameter θ = P(X < Y) which is derived as
(9)
3. Classical estimation procedures
3.1. Maximum likelihood estimator for θ
Using the definition given by [19]. Let Xobs and Yobs be the observed data such that
and XCen and YCen represent the censored data defined as
where X(i) and Y(i) are 1 × Ri vectors with
and
. By combining (Xobs, Yobs) and (XCen, YCen) we get the the complete data (X, Y), where
and
. The joint density of (X,Y) is given by
(10)
The joint density of (Xobs,Yobs) can be written as
(11)
Let (Xi:m:n, Y[i:m:n]), i = 1, 2, …, m be m pairs of independent random samples from the Bivariate Lomax distribution defined in (4) with progressive censoring scheme R = (R1, …, Rm) where Ri ≥ 0,
. Then, the associated likelihood function L and its corresponding log likelihood function l are given by:
(12)
and,
(13)
Then, the MLEs of α1, α2 and c, denoted by ,
and
respectively, are the solutions of the following loglikelihood equations
(14)
(15)
and,
(16)
Note that there is no explicit solution to (14), (15) and (16). Hence, numerical methods are applied to obtain the MLEs of α1, α2 and c. Once
,
are obtained, the MLE of θ denoted by
is obtained using the invariance property of MLEs as
(17)
3.2. Method of moments estimator for θ
Based on censoring samples, the method of moments is less discussed in the literature due to the complicated construction of the suitable moment equations. [20] used the method of moments to estimate the shape and scale parameters for the Weibull distribution under type I censoring. [20] proposed a systematic and unifiable moment estimator of the exponential distribution parameters based on type I and type II censored samples using the concept of “Mean Residual Lifetime” (MRL).
In the shadow of [21], we will introduce the method of moments estimator based on a progressively type II censored sample as follows:
Let X1:m:n, X2:m:n, …, Xm:m:n be a progressively type II censored sample from a distribution with cdf F(x) and censoring scheme R = (R1, R2, …, Rm) and set
(18)
such that,
(19)
with,
(20)
Theorem: For general lifetime distribution, the moment estimating equation is E(X) = U, where U as in Eq (18).
Proof:
Define
(21)
where, g(Xi:m:n) = E(Xji|Xji > Xi:m:n) and Xji represents the jth removed object after the ith failure and Ii is defined as follows
(22)
Hence, for in Eq (21)
(23)
Considering the following relation:
(25)
therefore, we get
(26)
It suffices to show that,
(28)
which implies,
(29)
Thus, we get the required result
(32)
Hence, the proof of Theorem 1 is complete.
Now, let (Xi:m:n, Y[i:m:n]), i = 1, 2, …, m be m pairs of independent random samples from the Bivariate Lomax distribution in (4). Using Eqs (7) and (8) the cdf’s of X and Y are given below
(33)
and the Mean Residual Lifetime function of the univariate Lomax distribution is
Using (32), the moment estimators of α1 and α2, (denoted by and
), for the univariate Lomax lifetime model under progressive type II censoring are obtained as follows:
Therefore, the suggested moment estimator for θ can be written as
(34)
Notice that for large values of c, Eq (34) is reduced to
(35)
4. Bayes estimator of θ under progressive type II censoring
In this section, we derive the Bayes estimate of θ based on Squared and Linex loss functions. A commonly used loss function is the squared error loss function (SEL)
(36)
The Bayes estimate under Eq (36) is the posterior mean, given by The SEL is widely employed in Bayesian inference due to its computational simplicity.
is a symmetric loss function under which overestimation and underestimation have equal weights. However, this is not a good criterion from a practical point of view. For example [22], states that in the disaster of the space shuttle, Challenger, the management may have overestimated the average life or reliability of the solid fuel rocket booster. In estimating reliability and failure rate functions, an overestimation causes more damage than underestimation. To resolve such a situation, asymmetrical loss functions are more appropriate. [23] introduced the Linex loss function (Linear- Exponential) in response to the criticism of the SEL. The Linex loss function has been widely used by several authors such as [24]. The Linex loss function is defined as follows:
(37)
The magnitude of λ reflects the degree of symmetry while the sign of λ reflects the direction of symmetry. [25] obtained the Bayesian estimator under (Linex) loss function by minimizing the posterior expected loss as follows:
(38)
provided that Eπ(e−λθ) exists and is finite. The Linex loss function is suitable for situations where overestimation may lead to serious consequences, and it is known for its flexibility and popularity in estimating the location parameter.
Suppose that X and Y are dependent random variables from the Bivariate Lomax distribution with survival function given by (6). We are interested in using the Bayesian method to estimate To conduct the Bayesian method, we need to construct prior distributions on the parameters. It is assumed that the parameters α1 and α2 have independent gamma priors with known and non-negative shape parameters a1 and a2 and the same scale parameter (without loss of generality the scale parameter = 1) hence,
(39)
Following the approach of [26], we assume a Gamma(a1 + a2, 1) distribution for S = α1 + α2 with pdf
(40)
Then, given S, has a Beta(a1, a2) prior as follows:
(41)
The joint prior of of θ and S is given by
(42)
By combining (13) and (42) the joint density function of θ and S is given by
(43)
where,
and (xi:m:n, y[i:m:n]) represents the i th progressively type II censored ordered pairs from Bivariate Lomax distribution. Therefore, the Bayes estimators of any function of θ and S, say w(θ, S), are the posterior expected values. Let w(θ, S) be a function of θ and S, then the expected value of w(θ, S) is given by
(44)
where l = ln L and ρ(θ, S) = ln π(θ, S).
It can be noticed that is in the form of a ratio of two integrals which cannot be simplified to a closed form. Hence Lindley’s approximation method is applied to obtain the Bayes estimator of θ, see [27]. Then Eq (44) is reduced to the following numerical expression:
(45)
where
and
are the MLEs of θ and S respectively and
,
,
,
and
. Other expressions can be defined similarly (see the S1 Appendix).
In this paper we are interested in estimating θ, thus w(θ, S) will be considered as a function of θ only and wS = wSθ = wθS = wSS = 0. Hence, Eq (45) is reduced to
(46)
4.1. Bayes estimate of θ
- Approximate Bayes estimate of θ under squared error loss function.
If w(θ, S) = θ, wθ = 1, wθθ = 0. Then Lindley’s approximation of the Bayes estimator is
(47)
- Approximate Bayes estimate of θ under Linex loss function.
If w(θ, S) = e−λθ, wθ = −λe−λθ, wθθ = λ2 e−λθ. Then
and hence, the Bayes estimate
is obtained by
(48)
5. Simulation study
The purpose of the simulation study is to compare the performance of the classical estimates (MLE and MOM) and the Bayesian estimates based on symmetric and asymmetric loss functions using independent gamma priors for α1 and α2 as provided in Eq (39). Progressively censored samples are randomly generated from Bivariate Lomax as follows.
- Values of α1 and α2 are generated from π1(α1) and π2(α2) as given in Eq (39) with specified parameters a1 and a2. The resulting values of α1 and α2 are considered to be the true values that will be used to generate a bivariate progressive type-II censored sample.
- For given m, n, c, (R1, R2, …, Rm) and and the resulted values of α1 and α2 above, we
- (a). generate m independent univariate Lomax (α1, c) random variables X1, X2, …, Xm.
- (b). generate m progressively type II censored samples Yi|Xi, i = 1, 2, …, m using the algorithm provided by Balakrishnan and Cramer (2014). Finally, (xi:m:n, y[i:m:n]), i = 1, 2, …, m is the required progressively type-II censored sample of size m from the Bivariate Lomax distribution with parameters (α1, α2, c)
- The above steps are repeated 5000 times, for each of the following values of a1( = 1, 2, 3, 8, 9), a2( = 1, 2, 6, 9), c( = 30), λ( = −8, 8), n = 70, 75, 120, m = 25, 50, 100 and nine different censoring schemes (R1, R2, …, Rm) as given in Table 1. For the censoring scheme we will follow [28] notations. For instance, if n = 15, m = 10, then the censoring scheme (2, 3, 0*8) means that after the first failure, 2 items are removed at random from the remaining 14 items, then after the second failure, 3 items are removed at random from the remaining 11 items, then the next 8 failure times are observed.
- In each case, the classical estimators (MLE, and MOM) and the Bayesian estimators based on symmetric and asymmetric loss functions using independent gamma priors for α1 and α2 are computed. We obtain the MLEs of α1, α2 and c by solving the nonlinear Eqs (14), (15) and (16) using the Newton-Raphson algorithm which is implemented in SAS/IML.
The three criteria used for comparing all the above estimators are the Absolute Relative Bias (ARBias), Mean Squared Error (MSE) and Pitman Nearness (PN) probability. Suppose is the estimate of θ, for the i th simulated data set, then the ARBias, MSE and PN are computed as follows:
- (i).
,
- (ii).
,
- (iii).
,
and we say that outperforms
if PN > 0.5.
All the computations are performed using SAS/IML. Results are summarized in Tables 1–5 provided at the end of this section as follows.
- Table 1 shows 9 cases of different censoring schemes.
- Tables 2 and 3 present ARBias and MSE values for the proposed estimators.
- Tables 4 and 5 display the PN probability of the estimators of θ relative to each other for large m.
5.1. Results of the simulation
Throughout this subsection, we will refer to the schemes in which (n − m) items are removed at the time of the first failure by 1st type censoring, namely schemes 1, 4, and 7. We will refer to schemes 2, 5, and 8 where (n − m) items are removed at the time of the m th failure by the 2nd type censoring. Any schemes that come in between these two extremes will be called 3rd type censoring, namely schemes 3, 6, and 9 (see Table 1).
A summary of the results is provided below.
- From Tables 2 and 3, it is observed that the estimates obtained based on large sample sizes have smaller ARBias and MSEs as expected.
- Comparing different censoring schemes and for different theoretical values of θ, we noticed that the Bayes estimates yield similar MSEs. These MSEs improve significantly as the effective size (m) increases.
- It is interesting to note that when the censored data are of the 1st type and the theoretical value of θ < 0.5,
outperforms the Bayes estimates as well as
in terms of MSE and ARBias values.
- On the other hand, when the censored data are of the 1st type and the theoretical value of θ > 0.5, the
performs steadily better than the classical estimators as well as
in terms of ARBias and MSE values, except when θ = 0.9,
has slightly smaller MSEs than
. However, as m increases,
and
compete quite well with
as seen from the MSE values.
- It is worth mentioning that the Linex loss function with λ = −8 tends to produce higher ARBias and MSEs. Moreover, λ = 8 is a preferred choice for the Linex case. In general, as m and θ increase, MSE tends to decrease.
- We further observed that the method of moments estimators show superior performance compared to the MLE and Bayes estimates. In fact,
produces smaller MSEs and smaller ARBias as well as higher PN values (see Tables 4 and 5) when censored data are of the 2nd and 3rd types.
- In Tables 4 and 5, we have presented the PN values for all estimates based on 3 types of censoring; we can clearly see that
-
outperforms MLE and Bayes estimates when data are of the 2nd and 3rd types.
-
outperforms all other estimates when data are of the 1st type and θ > 0.5.
-
outperforms all other estimates when data are of the 1st type and θ < 0.5.
- It is worth mentioning that
is always superior to
for all values of θ, when censored data are of the 1st type.
-
6. Real life data
To illustrate the proposed estimators of θ under a Bivariate Lomax distribution, we used the American Football League data from the matches on three consecutive weekends in 1986. It was first published in the ‘Washington Post’ and was proposed by [29]. The validity of the exponential model is checked using Kolmogrov-Smirnov (K-S), Anderson-Darling (A-D), and Chi-square tests. In this bivariate data set (X, Y), the variable X represents the game time to the first points scored by kicking the ball between goal posts, while the variable Y represents the game time by moving the ball into the end zone. The times are given in minutes and seconds and are reported in Table 6.
We fit the exponential distribution for X with failure rate 0.1102, we observed that K-S = 0.17379 with Pvalue = 0.14023, A-D = 1.7151 and chi-square = 2.9102 with a corresponding Pvalue = 0.40569. While for Y we fit the exponential distribution with failure rate 0.07449, and we observed that K-S = 0.14201 with Pvalue = 0.3332, A-D = 0.80191 and chi-square = 3.0078 with a corresponding Pvalue = 0.55652. This indicates that the Exponential model provides a good fit to the above two data sets. Figs (1) and (2) give the histograms of the two data-sets and the plots of the fitted densities. The QQ plots for X and Y given in Figs (3) and (4) suggest that Exponential is very suitable for these data sets.
Next, we construct the Bivariate Lomax distribution to the data by using Eqs (1)–(3) and ηλ1 = 0.1102, ηλ2 = 0.07449, where η is any positive value. For simplicity of calculations, we choose η = 1. We fit the Bivariate Lomax distribution to the data and obtained the MLE of θ as We modify the data to make it progressively censored data with three different censoring schemes given in Tables 7 and 8.
The parametric bootstrap percentile method is used to compute the bootstrap estimates (BootEst) and their corresponding standard error (StdErr). A 95% confidence interval is calculated and reported in terms of (LowerCI, UpperCI). The output of the bootstrap analysis is summarized in Table 9.
According to the different proposed censoring schemes, it can be noticed that the bootstrap estimates are close to the true value of θ under each case of the proposed censoring schemes, especially for ,
and
. Also, scheme 3 provides the smallest standard error and the shortest confidence intervals.
7. Conclusions and recommendations
The importance of drawing inferences about θ = P(X < Y) arises naturally in many disciplines, including but not limited to: quality control, reliability, psychology, and medical applications, particularly in screening tests to verify diseased from non-diseased patients using the area under the receiver operating characteristic (ROC) curves, where θ is interpreted as an index of accuracy (Samawi et al. 2016). Consequently, it is obvious that studying reliability measures of the type θ = P(X < Y) is essential to several areas of scientific research. Therefore, it is of interest to find a reliable estimate of θ.
Progressive censoring has received a great deal of attention from many researchers, and this is due to its advantages in reducing the cost and time of the tests and to saving some active items for other tests.
In this paper, the authors have considered the estimation of the stress-strength reliability when the stress and the strength are dependent random variables following a Bivariate Lomax distribution under progressively type II censoring.The authors have derived the MLE, the method of moments, and the Bayes estimators for θ. Squared error and Linex loss functions are used for deriving the Bayes estimators assuming suitable priors on the unknown parameters based on progressive type II censored samples from a Bivariate Lomax. The Bayes estimates of θ do not have explicit form. Therefore, the authors have used Lindley’s approximation method.
An intensive simulation study has been conducted to evaluate the performance of the proposed estimators. From the simulation study, it has been noticed that progressive censoring of the 2nd and 3rd type could provide the method of moments estimator with significantly smaller biases (ARBias) and MSEs for estimating θ. The 2nd type censoring scheme, namely R = (0, 0, …, 0, n−m) is easy to operate in practice; besides it has the merit of good performance. On the other hand, the progressive censoring of the 1st type provides with the smallest bias and smallest MSEs only when the theoretical value of θ is less than 0.5. However, when the theoretical value of θ is greater than 0.5,
outperforms the other estimators in most cases. It is observed that, based on ARBias, MSEs and PN values, the Bayes estimates have similar performances. In addition, by increasing the effective size m, expected improvements are observed in the performance of all estimators.
In practice, for the system to work, we need to consider the case when θ > 0.5, thus, we recommend using the method of moments if we are using 2nd and 3rd type censoring and if we are using 1st type censoring.
Acknowledgments
The authors thank Professor Richard Noren from Old Dominion University for his valuable comments.
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