Inference for Inverse Power Lomax Distribution with Progressive First-Failure Censoring

This paper investigates the statistical inference of inverse power Lomax distribution parameters under progressive first-failure censored samples. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals are derived based on the iterative procedure and asymptotic normality theory of MLEs, respectively. Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained using independent gamma priors. For Bayesian computation, Tierney–Kadane’s approximation method is used. In addition, the highest posterior credible intervals of the parameters are constructed based on the importance sampling procedure. A Monte Carlo simulation study is carried out to compare the behavior of various estimates developed in this paper. Finally, a real data set is analyzed for illustration purposes.


Introduction
In the life test of a product, due to the restrictions of test time, cost and other conditions, the complete life test is not generally performed. In these cases, experimenters often use censoring schemes to obtain censored lifetime data. There are many types of censoring schemes, and the most popular censoring schemes are Type-I and Type-II censoring. In Type-I censoring, the test ends at a pre-fixed time, while in Type-II censoring, the test ends when the m-th failure occurs (m is fixed in advance). For the above two censoring schemes, the common disadvantage is that no unit in the test can be removed before the test is terminated. Thus, progressive censoring (PC) was proposed, which has better efficiency in lifetime experiments. Under this censoring scheme, one can remove the test units at various stages of the experiment. For more details, refer to Balakrishnan and Aggarwala [1]. An excellent review of progressive censoring schemes can be found in Ref. [2]. Besides the PC, there is another censoring scheme, namely the first failure censoring scheme. Under this censoring scheme, experimenters group the test units into several sets and then perform all the test units simultaneously until the first failure in each set. The first-failure censoring scheme was studied by Johnson [3], Balasooriya et al. [4], Wu et al. [5] and Wu and Yu [6]. However, this censoring scheme does not allow the removal of units from the test at points other than the final termination point. Wu and Kus [7] combined the advantages of the first failure censoring and progressive censoring to propose mixed censoring, that is, a progressive first-failure censoring (PFFC) scheme. They obtained maximum likelihood estimates (MLEs), interval estimation and expected time on test for the parameters of the Weibull distribution based on the PFFC sample.
The PFFC scheme can be described as follows: suppose that n independent groups with k items within each group are put on a life test at time zero, and the progressive censoring Scheme R = (R 1 , R 2 , . . . , R m ) is fixed in advance. At the first failure time X 1:m:n:k , R 1 groups and the group in which the first failure is observed are randomly removed from the test. Similarly, at the second failure time X 2:m:n:k , R 2 groups and the group in which the second failure is observed are randomly removed from the remaining (n − R 1 − 1) groups. This procedure continues until the mth failure time X m:m:n:k is observed in the remaining groups, and then all the remaining R m groups are removed. It is clear that n = m + R 1 + R 2 + . . . + R m . The observed failure times, X 1:m:n:k < X 2:m:n:k < . . . < X m:m:n:k , are called the PFFC sample with the progressive censoring scheme R = (R 1 , R 2 , . . . , R m ). Here, (m, n, k) must be pre-specified.
The main advantage of the PFFC scheme is that it reduces time where more items are used, but only m out of n × k items are observed. It is observed that if R 1 = R 2 = . . . = R m = 0, the PFFC reduces to first failure censoring; If k = 1, the scheme becomes progressively Type II censoring; when k = 1, R 1 = R 2 = . . . = R m−1 = 0 and R m = n − m, this scheme reduces to Type II censoring scheme. Furthermore, the progressively first-failure censored sample X 1:m:n:k < X 2:m:n:k < . . . < X m:m:n:k can be viewed as a progressively Type-II censored sample from a population with the distribution function 1 − (1 − F(x)) k , which enables us to extend all the results on progressive type-II censored order statistics to progressive first-failure (PFF) censored order statistics.
Because of the flexibility of the PFFC scheme, many scholars have discussed and applied it in reliability studies. Ref. [8] studied statistical inferences of the unknown parameters, the reliability and failure functions of the inverted exponentiated Half-Logistic distribution using PFFC samples. Ref. [9] investigated a competing risks data model under PFFC from a Gompertz distribution using Bayesian and non-Bayesian methods. Ref. [10] considered the estimates of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using PFFC samples. Ref. [11] established different reliability sampling plans using two criteria from a Lognormal distribution based on the PFFC. Some recent studies on the PFFC scheme can be found in Refs. [12][13][14][15][16].
The inverse distributions have a wide range of applications in issues related to econometrics, biological sciences, survey sampling, engineering sciences, medical research and life testing problems. In recent years, some scholars have studied the statistical inference of inverse distribution. For example, Dube et al. [10] studied the MLEs and Bayesian estimators of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using progressively first-failure censored samples. Panahi and Moradi [17] discussed the estimation of the inverted exponentiated Rayleigh distribution based on an adaptive Type II progressive hybrid censored sample. Bantan et al. [18] studied the estimation of the Rényi and q-entropies for inverse Lomax distribution under multiple censored data. An efficient estimation strategy was proposed by using the maximum likelihood and plugging methods. But they did not investigate the statistical inference of the three-parameter inverse power Lomax distribution under the progressive first failure sample. Some other related studies on inverse distribution can be found in Nassar and Abo-Kasem [19], Lee and Cho [20], Xu and Cui [21] and Rashad et al. [22].
In 2019, a new three-parameter lifetime distribution named the inverse power Lomax (IPL) distribution was introduced by Hassan and Abd-Allah [23]. The probability density function (PDF) f (·), cumulative distribution function (CDF) F(·) of the IPL distribution are given, respectively, by where α > 0, β > 0 are shape parameters, and λ > 0 is scale parameter. The IPL is very flexible in analyzing situations with a realized non-monotonic failure rate. Therefore, the IPL model can be used for several practical data modeling and analysis, see Ref. [23]. In order to facilitate engineering applications, Ref. [23] studied some statistical properties for the IPL distribution. The MLEs of the model parameters are obtained based on conventional Type I and Type II censored samples. However, they did not discuss the PFFC scheme. The PFFC scheme is more widely used in survival analysis and the life test.
Since the IPL distribution contains three unknown parameters, it is more complicated to estimate the unknown parameters under progressive censoring. So, to date, there has been no published work on statistical inference for IPL distribution under the PFFC scheme. The main aim of this paper is to focus on the classical and Bayesian inference for IPL distribution under the PFFC scheme.
The rest of this paper is organized as follows: In Section 2, the MLEs and asymptotic confidence intervals of the unknown parameters are derived. Based on Tierney-Kadane's approximation method, Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained in Section 3. In addition, the highest posterior density (HPD) credible intervals of the parameters are constructed by using the importance sampling method. In Section 4, Monte Carlo simulations are carried out to investigate the performances of different point estimates and interval estimates. In Section 5, a real data set has been analyzed for illustrative purposes. The conclusions are given in Section 6.

Maximum Likelihood Estimation
In this section, the MLEs of the parameters for the IPL distribution will be discussed under the PFFC. Let X i = X i:m;n , i = 1, 2, · · · , m be the progressive first-failure censored order statistics from the IPL distribution with the censored scheme R = (R 1 , R 2 , . . . , R m ). Then, using Equations (1) and (2), the likelihood function is given by where C = n(n − R 1 − 1) · · · (n − m + 1 − R 1 − · · · R m−1 ), → x = (x 1 , x 2 , · · · , x m ) and The log-likelihood function is given by Let l(α, β, λ| → x ) = ln L( → x ; α, β, λ). By taking the first partial derivative of log-likelihood function with regard to α, β and λ and equating them to zero, the following results can be obtained.

Bayesian Estimation
In this section, we discuss the Bayesian estimates and corresponding credible intervals of the unknown parameters for the IPL distribution. In order to select the best decision in the decision theory, an appropriate loss function must be specified. Here, we consider both the symmetric and asymmetric loss functions. A very well-known symmetric loss function is the square error loss (SEL). The most commonly used asymmetric loss function is the generalized entropy loss (GEL) function. The SEL and GEL function are, respectively, defined by Here,θ is an estimation of θ, and the constant q denotes how much influence that an error will have. When q < 0, negative errors affect the consequences more seriously. When q > 0, positive errors cause more serious consequences than negative ones.
Under the SEL and GEL function, the Bayesian estimator of θ are, respectively, given byθ The Bayesian analysis requires the choice of appropriate priors for the unknown parameters in addition to the experimental data. Arnold and Press [24] correctly pointed out that there is no clear cut way how to choose prior. Now, we assume the following independent gamma priors for the parameters α, β and λ as therefore, the joint prior distribution of α, β and λ is given by The assumption of independent gamma priors is reasonable [10]. The class of the gamma prior distributions is quite flexible as it can model a variety of prior information. It should be noted that the non-informative priors on the parameters are the special cases of independent gamma priors and can be achieved by approaching hyper-parameters to zero [10].
Based on Equations (3) and (18), the joint posterior distribution of the parameters α, β and λ can be written as: Entropy 2021, 23, 1099 6 of 19 Let g = g(α, β, λ) be a function of α, β and λ, then the posterior mean of g is given by From Equation (20), we observe that the posterior mean of g(α, β, λ) is in the form of ratio of two integrals for which a closed-form solution is not available [10]. Therefore, we use Tierney-Kadane's approximation method to obtain the approximate solution of Equation (20).

Tierney-Kadane's Approximation Method
Tierney and Kadane [25] proposed an alternative method to approximate such a ratio of integrals to derive the Bayesian estimates of unknown parameters. In this subsection, we present the approximate Bayesian estimates of α, β and λ under the SEL and GEL function using Tierney-Kadane's (T-K) method. Although Lindley's approximation [26] plays an important role in the Bayesian analysis, this approximation requires the evaluation of third derivatives of the log-likelihood function, which is very tedious in some situations, such as the present one. Moreover, Lindley's approximation has an error of order O(n −1 ), whereas the T-K approximation has an error of order O(n −2 ).

∂Q ∂α
∂Q ∂β We obtain the Σ from Based on the T-K approximation method, we can derive the Bayesian estimates of the parameters α, β and λ under the different loss functions.
(I) Squared error loss function In order to compute the Bayesian estimator of unknown parameters under the squared error loss function (SELF), we take g(α, β, λ) = α, and accordingly, the function The MLE (α Q * ,β Q * ,λ Q * ) of (α, β, λ) can be obtained by solving the following system of the equations.
Thus, Σ * α can be calculated by Under SELF, the Bayesian estimator of α is given bŷ Similarly, the Bayesian estimators of β and λ under SELF are given, respectively, bŷ (II) General entropy loss function Firstly, we compute the Bayesian estimator of parameter α. In this case, g(α, β, λ) = α −q , then function Q * α (α, β, λ) is given by By solving the following system of the equations, we obtain the maximum likelihood estimator (α Q * ,β Q * ,λ Q * ) of α, β and λ.
Thus, Σ * α can be calculated by The Bayesian estimator of α under the general entropy loss function (GELF) is given bŷ Similarly, the Bayesian estimators of β and λ under GELF are given by, respectively,

The Highest Posterior Density Credible Interval
In the previous subsection, we used the T-K approximation method to obtain Bayesian point estimation of unknown parameters. However, this approximation method cannot determine the Bayesian credible intervals of unknown parameters. The importance sampling method is an effective approach to attain the Bayesian credible interval of unknown parameters. Kundu [27] considered Bayesian estimation for the Marshall-Olkin bivariate Weibull distribution, and the Bayesian estimates and associated credible intervals of the unknown parameters were constructed using the importance sampling method. Maurya et al. [28] derived the HPD credible intervals of unknown parameters in a Burr Type XII distribution using the importance sampling method. Sultana et al. [29] considered the estimation of unknown parameters for two-parameter Kumaraswamy distribution with hybrid censored samples. In the subsection, we use the importance sampling method to obtain the HPD credible intervals of unknown parameters of the inverse power Lomax distribution.
Based on the Equation (19), the joint posterior distribution of the parameters α, β and λ can be rewritten as where x ) are the PDF of the Gamma distribution Ga(m + a 2 , V 2 ) and Ga(αm + a 3 , b 3 ), respectively. To obtain the HPD credible intervals for unknown parameters, the importance sampling method is used and the steps as follows.

Simulation Study
In this section, we evaluate the performance of different estimates developed in this paper by the Monte Carlo simulation study. For the given true values of parameters α, β, λ and different combinations of (n, m, k, R), progressive first-failure censored samples are generated from the IPL distribution by modifying the method introduced by Ref. [30]. The following steps provide the specific generation method.
Step 1: Set the initial values of both group size k and censoring scheme R = (R 1 , R 2 , . . . , R m ).
In the simulation study, the true values of parameters in the IPL distribution are taken as α = 1.5, β = 1, λ = 0.5. For Bayesian estimates, the means of prior distributions are equal to the true values of the parameters, that is, a 1 /b 1 = α, a 2 /b 2 = β, a 3 /b 3 = λ. Therefore, the true values of the hyper-parameters in prior distribution are taken as In each case, we compute the MLEs and Bayesian estimates of the unknown parameters. In the Newton iterative algorithm and importance sampling algorithm, we choose the initial values of α, β and λ as α (0) = 1.4, β (0) = 0.9, λ (0) = 0.4; the value of ε is taken as 10 −5 . All Bayesian points and interval estimates are computed under two different loss functions, SELF and GELF, using the the T-K approximation and importance sampling methods, respectively. In addition, we obtain the average length (AL) of 95% asymptotic confidence and HPD credible intervals and corresponding coverage probability (CP) of the parameters based on the simulation. Here, we use N = 2000 for the importance sampling procedure and use M = 2000 simulated samples in each case.
Extensive computations are performed using R statistical programming language software. The results of ML and Bayesian point estimates using the Monte Carlo simulation are presented in Tables 1-5. From these tables, the following observations can be made:

1.
When n increases but m and k are fixed, the MSEs of MLEs and Bayesian estimates of three parameters decrease. Therefore, we tend to get better estimation results with an increase in sample size.

2.
When m increases but n and k are fixed, the MSEs of MLEs and Bayesian estimates decrease. While when k increases but n and m are fixed, the MSEs of all estimates decrease in most of the cases. 3.
In the case of Bayesian estimates, there is little difference between the MSEs under SELF and GELF, and the estimation effect of GELF is slightly better than SELF in terms of MSE. While under GELF, there is no significant difference in MSEs among the three modes. The estimation effect seems better when q = 1.  Furthermore, the average lengths of 95% asymptotic confidence HPD credible intervals were computed. These results are displayed in Tables 6 and 7. From the obtained results in Tables 6 and 7, the following conclusions can be drawn:

1.
When n increases but m and k are fixed, the average length of asymptotic confidence and HPD credible intervals narrow down. While the average length of 95% asymptotic confidence and HPD credible intervals narrow down when the group size k increases.

2.
When m increases but n and k are fixed, the average length of 95% asymptotic confidence HPD credible intervals narrow down in most of the cases. 3.
The HPD credible intervals are better than asymptotic confidence intervals in respect of average length.

4.
For the CPs of interval for the unknown parameters, the HPD credible intervals are slightly better than asymptotic confidence intervals in almost all cases.

Real Data Analysis
In this section, a real data set is considered to illustrate the proposed method. The data set represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli. This data set was observed and reported by Bjerkedal [31]. The data are listed as follows: 0. The above data set was analyzed by Hassan and Abd-Allah [23] in fitting the IPL distribution (IPLD). The IPLD was compared with Lomax (L), exponentiated Lomax (EL), power Lomax (PL), inverse Weibull (IW), generalized inverse Weibull (GIW) and inverse Lomax (IL) distribution, respectively. The method of maximum likelihood is used to estimate the unknown parameters of the selected models. The following statistics: Akaike information criterion (AIC), the corrected Akaike information criterion (CAIC), Bayesian formation criterion (BIC), the Hannan-Quinn information criterion (HQIC), and Kolmogorov-Smirnov (K-S) statistic was used to compare all the models.
In this section, all computations are performed using R statistical programming language software. Table 8 lists the values of MLEs of the parameters, AIC, CAIC, BIC, HQIC and K-S statistic for the considered models. The plots of the estimated CDFs of the fitted distributions are displayed in Figure 1.

Real Data Analysis
In this section, a real data set is considered to illustrate the proposed method. The data set represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli. This data set was observed and reported by Bjerkedal [31]. The data are listed as follows: 0. The above data set was analyzed by Hassan and Abd-Allah [23] in fitting the IPL distribution (IPLD). The IPLD was compared with Lomax (L), exponentiated Lomax (EL), power Lomax (PL), inverse Weibull (IW), generalized inverse Weibull (GIW) and inverse Lomax (IL) distribution, respectively. The method of maximum likelihood is used to estimate the unknown parameters of the selected models. The following statistics: Akaike information criterion (AIC), the corrected Akaike information criterion (CAIC), Bayesian formation criterion (BIC), the Hannan-Quinn information criterion (HQIC), and Kolmogorov-Smirnov (K-S) statistic was used to compare all the models.
In this section, all computations are performed using R statistical programming language software. Table 8 lists the values of MLEs of the parameters, AIC, CAIC, BIC, HQIC and K-S statistic for the considered models. The plots of the estimated CDFs of the fitted distributions are displayed in Figure 1. From the numerical results in Table 8, it can be seen that the most fitted distribution to these data is IPLD compared to other distributions since the IPLD has the lower statistics. According to the results in Figure 1, it is clear that the IPLD is the most appropriate model for this data set. Therefore, we can perform statistical analysis on this data set.  To analyze this data set under PFF censored samples, we randomly divide the given data into 36 groups with k = 2 independent items within each group. Then the following first-failure censored data are obtained: 0. Next, we generate progressive first-failure censored samples using three different censoring schemes from the above first-failure censored sample with m = 26. The different censoring schemes and the corresponding progressive first-failure censored samples are presented in Table 9. In the different censoring schemes, we calculate the ML and Bayesian estimates of the parameters. For Bayesian estimates, we use non-informative priors as we have no prior information about the parameters. We obtain 95% asymptotic confidence and HPD credible intervals for the parameters. The results of all estimates are listed in Tables 10-12.