Statistical inferences for type-II hybrid censoring data from the alpha power exponential distribution

This paper describes a method for computing estimates for the location parameter μ > 0 and scale parameter λ > 0 with fixed shape parameter α of the alpha power exponential distribution (APED) under type-II hybrid censored (T-IIHC) samples. We compute the maximum likelihood estimations (MLEs) of (μ, λ) by applying the Newton-Raphson method (NRM) and expectation maximization algorithm (EMA). In addition, the estimate hazard functions and reliability are evaluated by applying the invariance property of MLEs. We calculate the Fisher information matrix (FIM) by applying the missing information rule, which is important in finding the asymptotic confidence interval. Finally, the different proposed estimation methods are compared in simulation studies. A simulation example and real data example are analyzed to illustrate our estimation methods.


Introduction
In experiments involving life testing and reliability, complete failure time information may not be achieved for all items. Therefore, the data obtained from this type of experiment are called censored data, for which cost effectiveness and minimization of the total testing time are important. Various censoring schemes that can be used in reliability analysis have been published. The two most frequently considered censoring schemes in the reliability literature are the type-I censoring scheme (T-ICS) and T-IICS. In T-ICS, a test stops at a prefixed time T, whereas in T-IICS, the test stops after a prefixed r test units have failed. Thus, the number of observed failures is random in T-ICS and the duration of the experiment is random in T-IICS.
T-ICS and T-IICS can be combined to form a hybrid censoring scheme (HCS), as first presented by Epstein [1], who studied the properties of the one-parameter exponential distribution. For more details about HCS, the reader is referred to Balakrishnan and Kundu [2]. HCSs are categorized as type-I and type-II. To see this, a type-I hybrid censoring scheme (T-IHC) is defines as follows. If the test is stopped at a random time, T � = min{X r:n , T}, where r and T are 1 À e À lðxÀ mÞ ifa ¼ 1; The corresponding reliability and hazard rate functions are 1À e À lðxÀ mÞ a À 1 ifa 6 ¼ 1; e À lðxÀ mÞ ifa ¼ 1; and HðtÞ ¼ le À lðtÀ mÞ a 1À e À lðtÀ mÞ lna a 1À e À lðtÀ mÞ À 1 ifa 6 ¼ 1; le À lðtÀ mÞ 1 À e À lðtÀ mÞ ifa ¼ 1; where X > 0, μ > 0 and λ > 0. Note that in the rest of this paper, it is assumed that α 6 ¼ 1. For more details about APED see Salah [19].
The primary purpose of the present paper is to propose an estimation method for the parameters and the reliability characteristics for APED using incomplete sample observations obtained by a T-IIHC. To the best of our knowledge, no attempt has been made to estimate these characteristics for APED using a T-IIHC. This study aims to fill this gap using MLEs via EMA to compute and compare the outcomes with those calculated using the NRM. Furthermore, the asymptotic confidence intervals (ACIs) for the APED parameters were computed. The rest of this article is organized as follows. In the following section, the MLEs of the unknown parameters and the failure functions and reliability are discussed. The ACIs and the FIM are presented in Sections 3 and 4, respectively. A real data set is examined for illustrative purposes in Section 5, while conclusions are discussed in Section 6.

Maximum likelihood estimation
MLE is an important and widely used method for fitting statistical models because of its attractive properties, such as asymptotic efficiency, consistency, and asymptotic unbiasedness. Here, we describe the attainment of MLEs of the model parameters based on T-IIHC samples via the NRM and EMA.

Newton-Raphson algorithm
Consider a sample of n units and let T be a preselected experimental time and r a predetermined number of units out of a total of n units. The experiment ends at time T � = max{x k:n , T}. Furthermore, let J be the number of failures that occurred before time T, i.e, J ¼ P n i¼1 IfX i < Tg: Then, under the T-IIHC, we have the following observations: Case 1: x 1:n < x 2:n < . . . < x r:n if x r:n > T, or Case 2: x 1:n < . . . < x r:n < x r+ 1:n < x J:n < T < x J+ 1:n if r � J < n, and x J:n < T < x J+ 1:n , or Case 3: x 1:n < x 2:n < . . . < x n:n < T, if x n:n < T. The likelihood functions of these three different cases are: Hence, l ln a a À 1 e À lðx i À mÞ a 1À e À lðx i À mÞ a À a 1À e À lðx r À mÞ a À 1 l ln a a À 1 e À lðx i À mÞ a 1À e À lðx i À mÞ a À a 1À e À lðTÀ mÞ a À 1 l ln a a À 1 e À lðx i À mÞ a 1À e À lðx i À mÞ : By combining the three likelihood functions, one obtains l ln a a À 1 e À lðx i À mÞ a 1À e À lðx i À mÞ a À a 1À e À lðcÀ mÞ a À 1 where k and c are given by ðJ; TÞ if x n:n < T; J > r; ðn; TÞ if x n:n < T: The log-likelihood function ℓ = ln L(μ, λ) without the constant term can be written as The MLEs of μ and λ are obtained by differentiating Eq (6) with respect to μ and λ. The simultaneous equations are expressed as ðx i À mÞe À lðx i À mÞ À ðn À kÞðc À mÞe À lðcÀ mÞ ln a a e À lðcÀ mÞ À 1 ¼ 0: It is difficult to obtain an analytical solution to these nonlinear equations. Therefore, we can estimate the parameters μ and λ using statistical software or by solving the two simultaneous Eqs (7) and (8) numerically by, for example, NRM with a good initial guess of μ (0) and λ (0) . Utilizing the property of invariance (replacing μ and λ by their ML estimatorsm ML and l ML ), we can obtain the MLE of the reliability and hazard function from Eqs (3) and (4) bŷ

Expectation maximization algorithm
The EMA is a very powerful method for finding MLEs for parametric models when the data are censored; see Dempster et al. [20]. It consists of two iterative steps: (i) the expectation step (E-step) and (ii) the maximization step (M-step). The E-step of each iteration computes only the conditional expectation of the log-likelihood with respect to the incomplete data given the observed data. In the M-step, the parameter value is calculated by maximizing the expected log-likelihood function obtained in the E-step. For more details about the EMA, see McLachlan and Krishnan [21].
Let X = (X 1:n , X 2:n , . . ., X k:n ) denote the observed data and Z = (Z 1 , Z 2 , . . ., Z n−k ) denote the censored data. Here, for a given k, Z 1 , Z 2 , . . ., Z n−k are not observable. The complete data are given by the combination of W = (X, Z).
According to the invariant property of MLEs, the MLEs of the reliability and hazard functions of APED via the EMA, denoted byR EMA ðtÞ andĤ EMA ðtÞ, respectively, can be obtained by replacing μ and λ in Eqs (3) and (4) with their MLE estimateŝ In the following section, we consider interval estimation and the reliability and hazard functions of APED(μ, λ) under type-II hybrid censored data.

Asymptotic confidence intervals
In this subsection, we derive the ACIs of μ, λ, R(t), and H(t). To achieve this aim, we use the bivariate central limit theorem to obtain the asymptotic distribution of the unknown parameters, i.e., μ and λ, and apply the delta method to determine the asymptotic distributions of R(t) and H(t).
One of the characteristics that distinguishes the MLE is its asymptotic variance of the inverse of the Fisher information matrix. Because the MLEs of the parameters are not obtained in a closed form, it is not possible to obtain the Fisher information matrix and construct ACIs. Therefore, the Fisher information is approximated using the observed Fisher information evaluated at the MLE. The ACIs based on the asymptotic normal distribution of the MLEs are approximated as the inverse of the observed Fisher information matrix evaluated at the MLE.

Simulation study
This section presents Monte Carlo simulation study to estimate the performance of the MLEs of μ, λ, R(t), and H(t) achieved by applying the NRM and EMA. The parameter values of μ, λ, and α and sample size n are required for this simulation. In this study, we used parameters values α = 10, μ = 2, and λ = 1, the sample size n was set to 20, 30, and 40, and k was chosen such that the observed data were 70% and 90% censored. A mission time of t = 3.0 was taken for the survival and failure rate functions. Hence, R(t) = 0.6348 and H(t) = 1.1048. For the point estimation methods, we compared the expected values (EVs) and mean squared errors (MSEs) of the estimators for μ, λ, and the reliability and hazard functions, see Zeg et al. [26,27]. For the interval estimation methods, the 95% confidence intervals were compared according to the average length (AL) and coverage probability (CP). For the selected options of (n, k, T), the MLEs of μ, λ, R(t) and H(t) were obtained using the NRM and EMA in 1000 replications. The results are reported in Tables 1 and 2. From these results the following conclusions can be drawn.
(i). When the number of failures k is fixed and sample size n increases, the MSEs and width of the 95% confidence intervals of the MLEs computed using both the EMA and NRM decrease. Therefore, the MLE process performs well in terms of estimating the parameters of APED. Moreover, the expected values are close to the true values. (ii). The parameters estimation values under the EMA algorithm and their respective MSEs are smaller than those computed via the NRM.
(iii). As the sample size n increases, the average length of all intervals decreases. On average, the ACIs obtained via the EMA have a shorter length, and the coverages of the confidence intervals in all cases are close to 95%. (iv). The MSEs and the widths of the confidence intervals of the MLEs estimated by the EMA and NRM decrease as the number of failures (k) increases for a fixed n sample size.

Table 2. Expected Value (EV), Mean Squared Error (MSE), Average Length (AL), and Coverage Probability (CP) of R(t) and H(t) when μ = 2, λ = 1, α = 10 and t = 3 for varying (n, k, T
(v). The MSEs and the length of the ACIs for the parameters, as well as for the reliability and hazard functions, decrease for fixed n and k as T increases.

Simulated data analysis
Here, T-IIHC data with n = 20, k = 15, T = 1.5 were generated from APED with μ = 1.5, λ = 1, and α = 3. The generated data were Conversely, we used the EMA, as described in Sections 2 and 3, and stopped the iterative process when the difference between two consecutive iterations was less than 10 −6 . The MLEs for μ and λ obtained via the EMA require 0.06 s and seven iterations to converge tom EMA ¼

Real data example
In the experiment described in this subsection, one set of real data was used to demonstrate the applicability of the suggested method to real-life applications. The data, which represent the strength of a single carbon fiber and impregnated 1000-carbon fiber tows, measured in GPa, were taken from Bader and Priest [28]. Mahdavi and Kundu [18] reported the data of single carbon fibers tested at a gauge length of 1 mm as For the previous complete data, Mahdavi and Kundu [18] obtained the MLEs of α, μ, and λ, which were found to be 673.8379, 2.247, and 1.1562, respectively. They examined the validity of the APED based on the parametersâ ML ,m ML andl ML , using the Kolmogorov-Smirnov (KS) test. They observed that the KS test results was 0.0925, and the corresponding p-value was 0.7243. Therefore, they concluded the APE model provides a good fit for the data set presented above.
Here, we estimate the values of μ and λ when α is known (α = 673.8379). First, we computed the MLEs of the unknown parameters using the NRM:m ML ¼ 2:247 andl ML ¼ 1:19161. The KS distance between the fitted and empirical CDFs was 0.1117, and the associated p-value was 0.487. Therefore, according to the result of the NRMM, we cannot reject the assumption that the source of the data set is the two-parameter APED. Furthermore, the 95% confidence intervals of μ and λ are (1.89194, 2.60206) and (0.944682, 1.43854). We also computed the KS distance based on the EMA, where the MLEs of μ and λ werê m ML ¼ 2:36522 andl ML ¼ 1:2568. The associated 95% confidence intervals were (2.0511, 2.67933) and (1.00053, 1.513), respectively. The KS distance was 0.0927 and the associated pvalue was 0.7217. Based on the p-value of the KS statistic, the MLEs obtained via EMA also provide a satisfactory estimate of the data set. The empirical survival function and the fitted survival functions are drawn in Fig 1. From the above data set, we artificially created a hybrid censored data set with n = 56, k = 50, and T = 4. Based on the T-IIHC sample, the MLEs obtained via the NRM and EMA for μ, λ, R(t = 4), and H(t = 4) were computed with the associated 95% confidence intervals; see Table 3. According to Table 3, all estimates are satisfactory for this data set.

Conclusion
In this article, statistical inference of T-IIHC data from an APED was described. The MLE method cannot be derived analytically; therefore, the EMA and NRM were conducted to compute the considered parameters. A simulation study was performed to assess the performance of the different schemes for the APED in estimated and real data. In the simulation study, we noted that both the EMA and the NRM produced satisfactory results, but the EMA provided better estimates. Based on the T-IIHC sample, the MLEs obtained via the NRM and EMA for μ, λ, R(t = 4) and H(t = 4) were computed, along with the associated 95% confidence intervals, and we can state that all considered estimates are satisfactory for the real data set. Its deserve to study in future the inferences on APE parameters under a balanced two-sample type-II progressive censoring scheme.