Statistical Inference to the Parameter of the Inverse Power Ishita Distribution under Progressive Type-II Censored Data with Application to COVID-19 Data

Elnagar, Kariema A.; Ramadan, Dina A.; El-Desouky, Beih S.

doi:https://doi.org/10.1155/2022/7704167

Journal of Mathematics

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Volume 2022 | Article ID 7704167 | https://doi.org/10.1155/2022/7704167

Statistical Inference to the Parameter of the Inverse Power Ishita Distribution under Progressive Type-II Censored Data with Application to COVID-19 Data

Kariema A. Elnagar,¹Dina A. Ramadan,¹and Beih S. El-Desouky¹

Academic Editor: Niansheng Tang

Received27 May 2022

Revised22 Jun 2022

Accepted01 Aug 2022

Published20 Sept 2022

Abstract

The goal of the article is the inference about the parameters of the inverse power ishita distribution (IPID) using progressively type-II censored (Prog–II–C) samples. For IPID parameters, maximum likelihood and Bayesian estimates were obtained. Two bootstrap “confidence intervals” (CIs) are also proposed in addition to “approximate confidence intervals” (ACIs). In addition, Bayesian estimates for “squared error loss” (SEL) and LINEX loss functions are provided. The Gibbs within Metropolis–Hasting samplers process is used to provide Bayes estimators of unknown parameters also “credible intervals” (CRIs) of them by using the “Markov Chain Monte Carlo” (MCMC) technique. Then, an application of the suggested approaches is considered a set of real-life data this data set COVID-19 data from France of 51 days recorded from 1 January to 20 February 2021 formed of mortality rate. To evaluate the quality of the proposed estimators, a simulation study is conducted.

1. Introduction

Statistical researchers in the fields of industrial and mechanical engineering are spent so much time looking at component and unit failure, which provide the foundation of operational systems. Their research involves watching operating units until they fail, recording their lifetimes, applying statistical inference methods to analyse obtained data and then predicting reliability and risk functions for the entire system using the data. However, because some experimental units are costly and have a high level of reliability. Following a specified censoring scheme, units are excluded from the experiment, and the censored sample is the final sample obtained by Klein et al. [1]. One of the main motivations for filtering is to reduce the total test time and the accompanying test. The life test is considered to be complete when all of the items under test are monitored until failure. In most real conditions, the data available is incomplete. As a result, whether due to test time or budget limits, censoring is critical in lifetime data analysis. The most common censoring techniques are type-I and type-II censored. Following that, a progressive censoring scheme is suggested. As a result, the researchers can release the unused units of various points throughout the test, making it more flexible and realistic.

1.1. Progressive Type-II Censored

A progressive type-II censoring strategy is a useful method for removing a particular fraction of at-risk participants from an experiment at each of many ordered failure times. The following is an example of the Prog –II– C scheme: On the life test, the tester allocates independent and identical units. Let us suppose at the start of the experiment, there are units to test. The lifetime test is ended when the unit fails, assuming there are units to test when the experiment begins. After the first failure, time is recorded, and units were selected randomly as from reminder survival units. As a result, when the second failure occurs, time is recorded, with the remnants survival units units were chosen at random. This experiment ends when the failure occurs, at time , and as shown in Figure 1.

As special cases (if and with ) and the conventional type-II right censoring scheme if and with . Many authors have devoted for a Prog–II–C method with various failure time breakdowns. For example, Balakrishnan and Aggarwala [2], Alshenawy [3], Balakrishnan [4], Ahmed [5], and Almetwally et al. [6]. Mann et al. [7] and Meeker et al. [8] looked into the characteristics of progressively censored order statistics and it offered an outline of several inferential approach expansions depend on Prog-I and Prog-II right censored samples, as well as identifying some for future research challenges. Balakrishnan and Cramer [9] give a thorough survey of the literature on progressive censoring, as well as specifics on this progressive censoring technique and its various uses. According to Tse et al. [10], the amount of patients falls out of a clinical trial at each stage is random and not possible predicted. With each failure, the pattern of removal become more random. See Balakrishnan and Sandhu [11] for further details on the gradually censored samples. As lifetime distributions are exponential, log normal, or Weibull, Aggarwala and Balakrishnan [12] has studied inference for Prog–II–C cases. Balakrishnan [11] and Aggarwala [12] have devised a method for simulating general Prog–II–C samples drawn from continuous or uniform distributions. Eryilmaz and Bairamov [13]and Montanari et al. [14] also have been researching parameter estimation from various lifetime distributions using Prog–II–C. Other authors, include Balakrishnan et al. [9], Mousa et al. [15], and Mousa et al. [16], have been researching the estimate of parameters from various lifetime distributions depending on Prog–II–C. In Prog-II right censored order statistics, Salemi et al. [17] recently investigated A-optimal and D-optimal censoring strategies, Qin et al. [18] offer a novel spacing-based test statistic for determining whether general Prog–II–C data are from an exponential distribution and parameters of a log-logistic distribution under the Prog–II–C sample, when Maiti et al. [19] given access to a managed step-wise sample of the second type, there was also a look at the generalized Rayleigh distribution. The IPID is a generalized of the Ishita distribution, which was presented by Elnagar et al. [20] is shown to be a good model for the Covid-19 and glass fiber data failure times. They also started at the maximum likelihood of the unknown parameters estimators, besides their asymptotic confidence intervals, using complete data. The random variable of the IPI distribution, has a probability density function (PDF):and cumulative distribution function (CDF) is

The IPID is a generalized of the ishita distribution, which was presented by Shanker and Shukla [21] and proven to be adaptable to a variety of applications.

Modeling the COVID-19 is the driving force behind the new distribution. The COVID-19 of France was employed of 51 days that is recorded from 1 January to 20 February 2021 as real data for evaluating model techniques. COVID-19 data on new cases or deaths are discrete data (count). For France, we have been using the COVID-19 daily mortality rate. The daily mortality rate is continuous data.

Let ; be a result of a Prog–II–C sample lifetime test that included units derived from a IPI distribution and its censoring scheme is . Balakrishnan and Aggarwala [2] gives the joint PDF of a Prog-II.as .

Balakrishnan et al. [4] have excellent provides a review essay a recent report on progressive censoring schemes. This article is organized as follows: Section 2, introduced MLEs of and . We survey the confidence intervals for the unknown parameters in Section 3. Two parametric bootstrap approaches are used to constructed unknown parameters in Section 4. In Section 5 different loss functions, such as SEL and LINEX loss functions, Bayes estimates for unknown parameters were introduced. In Section 6, a real data set was studied. Section 7 introduces a simulation study to evaluate the accuracy of the various estimators generated in this study. At the end of this paper, conclusions are given in Section 8.

2. Maximum-Likelihood Estimation

The maximum likelihood estimation method (MLE) is used to study the problem of estimating IPID parameters under Prog–II–C data.

Let be a Prog–II–C sample from the IPI distribution with PDF (1) and parameters and . The likelihood function has the following:

On Prog–II–C samples, the log-likelihood function of and parameters as a result of

Computing the first partial derivatives of (5) in relation to and and solving this nonlinear system of equations and , the maximum likelihood estimates ( and ) are obtained. Accordingly, results inand

Since Equations (6) and (7) cannot be solved analytically, so to obtain the estimations, a numerical method like Newton–Raphson method should be employed. In Ahmed [5], the algorithm is given in detail.

3. Asymptotic Confidence Intervals

The elements of the Fisher information matrix’s inverse give us the approximate variances and covariances of the MLEs, and ; , As also . So, precise asymptotic forms for these equations are tough to comprehend get. The Fisher information matrix is then used, we will use the expectation to establish confidence intervals (CIs) for the parameters, which is obtained through inference. Consequently, the information matrix that has been seen is

As a result, the asymptotic variance-covariance matrix for MLEs derived through converting observed information matrix or similar to the asymptotic variance-covariance matrix

It is evident that is approximately distributed as multivariate normal with mean and covariance matrix given specific symmetry requirements, see Bebbington et al. [22]. As a result. The convergent confidence intervals (ACIs) for and may be calculated as follows:as is the right-tail probability percentile standard normal distribution for See Lawless [23].

4. Bootstrap Confidence Intervals

Different bootstrap confidence intervals are suggested here. The percentile bootstrap (Bp) and bootstrap-t (Bt) confidence intervals are used in this study. Efron [24] and Hall [25] have more information on bootstrap confidence intervals.

4.1. Parametric Percentile Bootstrap (Boot-P)

The formula for calculating the parametric percentile bootstrap confidence interval for model parameters is as follows:(1)Calculate the MLE of and by maximizing Equations (6) and (7).(2)Create bootstrap samples with and to get the bootstrap estimate of and , say where be or , from the bootstrap sample.(3)To get , repeat step 2 times, where , .(4)Assign to in ascending order.

Gives a two-sided bootstrap-p for the unknown parameters and .

4.2. Parametric Bootstrap-t Boot-t

The percentile bootstrap-t confidence interval for model parameters can be calculated using the following algorithm:(1)Steps 1 and 2 in boot-p are the same(2)Using the asymptotic variance-covariance matrix (9), the variance-covariance matrix and get the t-statistic of as (3)NBoot times repeat steps 2 and 3 to get (4) the ordered sequences are sorted in ascending order as

Calculates a two-sided bootstrap-t CI for the unknown parameters and .

5. Bayes Estimation

The Bayesian study of the IPID under Prog–II–C is discussed in this section. Under the SEL function and LINEX loss functions, we obtain Bayes estimators. When the parameters and are independent and follow the following gamma prior distributions according to Zellner [26].where the hyper-parameters and , are assumed to be known and chosen to reflect the prior belief about the unknown parameters. The posterior distribution is derived from the likelihood function equation (4) and the prior distribution (13), and with the outcomes posterior distribution of and denoted as .

5.1. Loss Function

We need to design an asymmetric loss function to make statistical Bayesian inference more practical and relevant. The loss function, as defined by Press and James [27], is a real-valued function that satisfies all feasible estimates and parameters. We must adopt an asymmetric loss function as well as create statistical Bayesian inference extra sensible and relevant. According to Press and James [27], the loss function is a real-valued function that satisfy all possible estimations and parameters.

5.1.1. Squared Error Loss Function (SEL)

The SEL function is

Then, for any function of and , the Bayes estimate is under the SEL function iswhere

5.1.2. Linear Exponential (LINEX) Loss Function

The LINEX loss function for a parameter is proposed according to Varian and Hall [28] given by the following equation:

Hence, the bayes estimate of a function under LINEX loss function illustrated by the following equation:

It should be noted that the ratio of multiple integrals in (17) and (19) cannot be expressed explicitly. To produce samples from the joint posterior density function in (14), the MCMC approach is used. We use the Gibbs within Metropolis–Hasting samplers process to add the MCMC approach. The joint posterior distribution expressed as follows:

The conditional posterior densities of and represented as follows:and

Because the conditional posteriors of and in the previous equations do not qualify for a specified distribution, the Metropolis–Hasting sampler must be used to apply the MCMC approach. Tierney and Luke [29] proposed the Metropolis–Hastings algorithm within Gibbs sampling, which generates the posterior samples as follows:(1)Begin with the first suggestion (2)Specify (3)Produce and by using M-H algorithm below using Equations (23) and (24) with the normal indicated distribution and the inverse Fisher information matrix is used to calculate , where and (i)Create a is suggestion to and by (ii)Calculate the probabilities of approvaland(iii)Create a and from a Uniform distribution(iv)If , confess the proposition and Specify else Specify (v)If , confess the proposition and Specify else Specify (4)Calculate and (5)Specify (6)Steps from 3 to 6 should be reiterated N times(7)To evaluate the CRIs of and of , , and as then the CRIs of is

For sufficiently large , the samples chosen are . SEL function is accustomed to get the approximate Bayes estimates of and as follows:and

And, the estimates for the mentioned parameters under LINEX loss function areand

6. Real-Life Data Application

In this section, the proposed estimation methods are applied to the parameters of the IPI model under a Prog–II–C scheme using a real data set from the COVID-19 data from France, which was analyses by Almetwally [6]. The considered COVID-19 data belong to France of 51 days that is recorded from 1 January to 20 February 2021. This data formed of mortality rate. A Prog–II–C sample of size simulated randomly from the sample of size with censoring scheme . For the data, we calculated the Kolmogorov–Smirnov (K–S) distance (D) between the fitted and empirical distribution functions, with K–S = 0.0999069 and p-value = 0.688698.The MLEs of parameters depend on Prog-II failure data in Table 1 are and respectively, and are shown in Table 2. The mean of 1000 Boot-p and Boot-t samples of the lifetime parameters also given in Table 2 through using the algorithm defined in Section 4 of the bootstrap methods. For various values of the LINEX loss function’s shape parameter , the parameters and , Table 3 calculated and shewed the Bayes estimates related to both SEL and LINEX functions; the ACIs and CRIs of the parameters and have calculated. The LINEX loss function becomes symmetric as going to zero, as is well known. In Table 2, the outputs of the SEL and LINEX loss functions are identical at , confirming accuracy of presented s. Figure 2 indicate simulation numbers of the parameters and for COVID-19 data.

(a)

(b)

7. Simulated Data

Simulation experiments are carried out utilizing 1000 (Prog –II–C) samples per one simulation in order to compare parameter estimators for the IPI distribution. IPID is used to create (Prog –II– C) samples, with initial values and . The comparison of the different approaches of the producing and estimators, has been taken into account in the computation of their mean square error (MSE), for , as MSE , as simulated samples numbers. The CIs generated through utilizing asymptotic distributions of the MLEs and CRIs are compared using another criterion. The average confidence interval lengths (ACLs) and coverage probability (CP) are used to make comparisons between them. In this study the progressive schemes that have considered: Scheme I: , for Scheme II: , for and Scheme III: , for

Tables 4 and 5 illustrate the estimated parameters’ results and their MSE and Table 6 shows outcomes of ACL and CP of CIs. From the results, the following notices can be observed:(1)Tables 4 and 5 show that as sample size increase, MSEs decrease, with Bayes estimates having the smallest MSEs for and . As a result, Bayes estimates performance MLE approaches in all situations studied.(2)In the sense that MSEs are smaller, Bayes estimates under LINEX with better results estimates.(3)Scheme I outperforms schemes II and III in terms of MSEs when the sample size is fixed at and the effective size is fixed at .(4)Adapted from Table 6. It can be seen that the CRIs produce more accurate results. Then, the ACIs for different sample sizes, observed failures, and schemes.

8. Conclusion

We discuss an iterative technique for deriving MLEs from an inverse power ishita lifetime distribution using progressively Type-II censored samples. Another estimation approach, based on Bayes estimates, is also considered. The Bayes estimates were produced using loss functions; however, they are not available in explicit form. Propose that the Bayes estimators and associated IPI distribution CIs be computed using the MCMC technique. Derive the posterior summaries of interest, such as credible intervals for the parameters, in a simple way using typical MCMC simulation methods for a Bayesian study of the model. The findings of this study used to the estimate problem of the IPID using complete data, which are usually progressive type-II censoring samples. The proposed methods are demonstrated with numerical examples [30, 31].

Data Availability

The Covid-19 data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

J. P. Klein and M. L. Moeschberger, Survival Analysis: Techniques for Censored and Truncated Data, Springer, New York, pp. 3–5, 2003.
N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications, Springer Science and Business Media, 2000.
R. Alshenawy, “A new one parameter distribution: properties and estimation with applications to complete and type ii censored data,” Journal of Taibah University for Science, vol. 14, no. 1, pp. 11–18, 2020.
View at: Publisher Site | Google Scholar
N. Balakrishnan, “Progressive censoring methodology: an appraisal,” Test, vol. 16, no. 2, pp. 211–259, 2007.
View at: Publisher Site | Google Scholar
E. A. Ahmed, “Estimation of some lifetime parameters of generalized gompertz distribution under progressively type-ii censored data,” Applied Mathematical Modelling, vol. 39, no. 18, pp. 5567–5578, 2015.
View at: Publisher Site | Google Scholar
E. M. Almetwally, H. M. Almongy, and A. E. s. Mubarak, “Bayesian and maximum likelihood estimation for the weibull generalized exponential distribution parameters using progressive censoring schemes,” Pakistan Journal of Statistics and Operation Research, pp. 853–868, 2018.
View at: Publisher Site | Google Scholar
R. Mann, R. Schafer, and N. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data, John Wiley & Sons, New York, 573 pages, 1974, Research supported by the U. S. Air Force and Rockwell International Corp.
W. Q. Meeker and L. A. Escobar, Spacings, Exceedances and Concomitants in Progressive Type Ii Censoring Scheme, Statistical Methods for Reliability Data, 1998.
N. Balakrishnan and E. Cramer, The Art of Progressive Censoring, Statistics for industry and technology, 2014.
S. K. Tse, C. Yang, and H. K. Yuen, “Statistical analysis of weibull distributed lifetime data under type ii progressive censoring with binomial removals,” Journal of Applied Statistics, vol. 27, no. 8, pp. 1033–1043, 2000.
View at: Publisher Site | Google Scholar
N. Balakrishnan and R. A. Sandhu, “A simple simulational algorithm for generating progressive type ii censored samples,” The American Statistician, vol. 49, no. 2, pp. 229-230, 1995.
View at: Publisher Site | Google Scholar
R. Aggarwala and N. Balakrishnan, “Some properties of progressive censored order statistics from arbitrary and uniform distributions with applications to inference and simulation,” Journal of Statistical Planning and Inference, vol. 70, no. 1, pp. 35–49, 1998.
View at: Publisher Site | Google Scholar
I. Bairamov and S. Eryilmaz, “Spacings, exceedances and concomitants in progressive type ii censoring scheme,” Journal of Statistical Planning and Inference, vol. 136, no. 3, pp. 527–536, 2006.
View at: Publisher Site | Google Scholar
G. Montanari, G. Mazzanti, M. Cacciari, and J. Fothergill, “Optimum estimators for the weibull distribution from censored test data progressively-censored tests [breakdown statistics],” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 5, no. 2, pp. 157–164, 1998.
View at: Publisher Site | Google Scholar
M. Mousa and Z. Jaheen, “Statistical inference for the burr model based on progressively censored data,” Computers & Mathematics with Applications, vol. 43, pp. 1441–1449, 2002.
View at: Google Scholar
M. A. M. A. Mousa and S. A. Al-Sagheer, “Bayesian prediction for progressively type-ii censored data from the Rayleigh model,” Communications in Statistics - Theory and Methods, vol. 34, no. 12, pp. 2353–2361, 2005.
View at: Publisher Site | Google Scholar
U. H. Salemi, S. Rezaei, and S. Nadarajah, “A-optimal and d-optimal censoring plans in progressively type-ii right censored order statistics,” Statistical Papers, vol. 60, no. 4, pp. 1349–1367, 2019.
View at: Publisher Site | Google Scholar
X. Qin, J. Yu, and W. Gui, “Goodness-of-fit test for exponentiality based on spacings for general progressive type-ii censored data,” Journal of Applied Statistics, vol. 49, no. 3, pp. 599–620, 2022.
View at: Publisher Site | Google Scholar
K. Maiti and S. Kayal, “Estimation of parameters and reliability characteristics for a generalized Rayleigh distribution under progressive type-ii censored sample,” Communications in Statistics - Simulation and Computation, vol. 50, no. 11, pp. 3669–3698, 2021.
View at: Publisher Site | Google Scholar
K. Elnagar, D. Ramadan, and B. El-Desouky, 2022, The Inverse Power Ishita Distribution: Properties and Applications.
R. Shanker and R. Shanker, “A simulation study on ishita distribution,” Biometrics & Biostatistics International Journal, vol. 6, no. 4, 2017.
View at: Publisher Site | Google Scholar
M. Bebbington, C. D. Lai, and R. Zitikis, “Bathtube-type curves in reliability and beyond,” Australian & New Zealand Journal of Statistics, vol. 49, no. 3, pp. 251–265, 2007.
View at: Publisher Site | Google Scholar
J. Lawless, Statistical Models and Methods for Lifetime Data, wiley, New York, 1982.
B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans, Society for industrial and applied mathematics, 1982.
P. Hall, Theoretical Comparison of Bootstrap Confidence Intervals, The Annals of Statistics, pp. 927–953, 1988.
A. Zellner, “Bayesian estimation and prediction using asymmetric loss functions,” Journal of the American Statistical Association, vol. 81, no. 394, pp. 446–451, 1986.
View at: Publisher Site | Google Scholar
S. Press, Subjective and Objective Bayesian Statics, Wiley Series in Probability and Statistics, 2009.
R. Varian, A Bayesian Approach to Real Estate Assessment Studies in Bayesian Econometric and Statistics, Honor of Leonard J Savage, pp. 195–208, 1975.
L. Tierney, “Markov chains for exploring posterior distributions,” The Annals of Statistics, vol. 22, no. 4, pp. 1701–1728, 1994.
View at: Publisher Site | Google Scholar
D. Lindely, “Approximate bayesian methods,” Trabajos de Estadistica y de Investigacion Operativa, vol. 31, no. 1, pp. 223–245, 1986.
View at: Google Scholar
H. A. Howlader and A. M. Hossain, “Bayesian survival estimation of pareto distribution of the second kind based on failure-censored data,” Computational Statistics & Data Analysis, vol. 38, no. 3, pp. 301–314, 2002.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Kariema A. Elnagar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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