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Inference Based on Type II Progressively Interval Censored from Inverse Flexible Weibull Distribution Using Different Simulation Methods
W.M. Afify
Abstract
This paper considers the estimation problem for inverse flexible Weibull model, when the lifetimes are collected under type-II progressive interval censoring. The maximum likelihood and the Bayes estimators for the two unknown parameters of the inverse flexible Weibull distribution are derived. Point estimation and confidence intervals based on maximum likelihood and bootstrap method are also proposed. Bayesian estimation for population parameter under type-II progressive interval censoring is studied via Markov Chain Monte Carlo (MCMC) simulation. To illustrate the proposed methods will discuss an example with the real data. Finally, comparing the two techniques through comparisons between the maximum likelihood using Monte Carlo simulation and bootstrap method on the one hand, and comparing them with the Bayes estimators using MCMC study on the other hand.
Keywords: Inverse flexible Weibull distribution, progressive interval type-II censoring, bootstrap-t Algorithm, Bayesian and non-Bayesian approach, MCMC
Copyright © 2016 Science and Education Publishing. All Rights Reserved.Cite this article:
- W.M. Afify. Inference Based on Type II Progressively Interval Censored from Inverse Flexible Weibull Distribution Using Different Simulation Methods. American Journal of Applied Mathematics and Statistics. Vol. 4, No. 4, 2016, pp 126-135. https://pubs.sciepub.com/ajams/4/4/5
- Afify, W.M.. "Inference Based on Type II Progressively Interval Censored from Inverse Flexible Weibull Distribution Using Different Simulation Methods." American Journal of Applied Mathematics and Statistics 4.4 (2016): 126-135.
- Afify, W. (2016). Inference Based on Type II Progressively Interval Censored from Inverse Flexible Weibull Distribution Using Different Simulation Methods. American Journal of Applied Mathematics and Statistics, 4(4), 126-135.
- Afify, W.M.. "Inference Based on Type II Progressively Interval Censored from Inverse Flexible Weibull Distribution Using Different Simulation Methods." American Journal of Applied Mathematics and Statistics 4, no. 4 (2016): 126-135.
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At a glance: Figures
1. Introduction
Assuming that 
units are randomly selected at the beginning of study which will be terminated when there are 
or more failed units. Under a type II progressive interval censored inspection scheme, that trial is terminated after the 
inspection if the total number of failed units is equal to or exceeds
. Also; 
, 
is the predetermined inspection times. Suppose that at the 
inspection, 
failed units Markov chain Monte Carlo (MCMC) are observed and 
units are fixed removed from the test. In other words, 
is the number of failed units between any two successive inspections 
and
, where 
. Thus, 
is random variable with observed value pending on the outcomes of the study. Denote
, the test is terminated when 
and
, for the predetermined integer value
,
.
There is shortage of topics for type II progressive interval censored. For instance, Xiang & Tse [10] treated the number of dropouts as a random variable and discussed a type II progressive interval censoring with random removals for Weibull distributed lifetime data. Ashour and Afify [1] considered the estimations of the parameters of exponentiated Weibull family with type II progressive interval censoring with random removals.
In Bayesian approach, It is too difficult to find integrate over the posterior distribution and the problem is that the integrals are usually impossible to evaluate analytically. But in MCMC technique provided a convenient and efficient way to sample from complex, high-dimensional statistical distributions. Recently, application of the MCMC method to the estimation of parameters or some other vital properties about statistical models is very common. Green et al. [5] using the MCMC method for estimating the three parameters Weibull distribution, and they showed that the MCMC method is better than the ML method, when given a proper prior distribution of the parameters. As a generalization of the two parameters Weibull model, Gupta et al. [6] gave a complete Bayesian analysis of the Weibull extension model using MCMC simulation and complete sample.
Bebbington et al. [2] shown that the flexible Weibull distribution is quite flexible, being able to model various ageing classes of lifetime distributions. So we can say that the flexible Weibull distribution is very important in several basic fields include engineering sciences, reliability, biological, demography and actuarial sciences. Also, El-Gohary et al. [3] introduced Inverse Flexible Weibull Extension Distribution.
A random variable 
is said to have a flexible Weibull distribution with parameters 
if 
then the random variable Y has the inverse flexible Weibull extension distribution, symbolically we write 
. The cumulative distribution function and the probability density function of Y are respectively given by
 | (1) |
 | (2) |
The main aim of this paper is evaluating the estimates the model under Type II progressive interval censored using both of bootstrap-t (Boot-t) and Monte Carlo simulation based on Classical estimation and Metropolis–Hastings algorithms based on Bayes estimation. In addition, we will assume the lifetime model which has inverse flexible Weibull distribution with two scale parameters. We assumed that the both scale parameters 
and 
have gamma prior and they are independently distributed. We will evaluate performance some simulation experiments to see the behavior of the proposed Bayes estimators and compare their performances with the maximum likelihood estimators MLEs.
The rest of the paper is organized as follows. In the next section, bootstrap-t (Boot-t) based on Classical estimation are presented. In Section 3, we cover Bayesian estimation and MCMC technique. To illustrate the behavior of the proposed methods as well as evaluate the statistical performances of these estimates, we performed a real data analysis in section 4 with comparisons among estimators are investigated through Monte Carlo simulations in previous section and conclusions appear.
2. Bootstrap-t (Boot-t) Based on Classical Estimation
Classical estimation (MLEs) of the unknown parameters and approximate confidence intervals are presented. Also, the corresponding parametric bootstrap confidence intervals using Boot-t for the parameters are given in this section.
2.1. Classical EstimationXiang & Tse [10] point out that the 
where 
is random and corresponds to the number of inspections before the termination of the experiment, the joint likelihood function of 
and 
, is given by
 |
where
 |
and
 | (3) |
Note that 
and 
are random variables in equation (3), to ensure that there at least 
failed units at the end of the study, the number of units removed at each inspection time, 
, is restricted to be any integer value between 
and 
, thus, 
would not be affected by 
for all 
.
Xiang & Tse [10] concluded that, the likelihood function under type II progressive censoring may be considered as a special case of equation (4) when all 
's are fixed to be 1 and 
, where 
is the 
ordered survival time. By all previous condition, it reduces to the type II censored if 
for 
and
.
By taking logarithm in (3), the log likelihood function for type II progressive interval censored ignoring the normalized constant can be written as follows
 | (4) |
where 
.
Thus, the maximum likelihood estimates 
and 
can be obtained by maximizing (4) with respect to 
and 
; that is, by simultaneously solving the estimating equations,
 | (5) |
and
 | (6) |
To construct confidence intervals for the unknown parameters we need to compute the asymptotic matrix variance which obtained by inverting the Fisher information matrix 
in which elements are negatives of expected values of the second partial derivatives of the 
. The first and second partial derivatives for 
with respect to 
and the elements of the sample information matrix will be obtained in Appendix. The asymptotic normality of the MLEs can be used to compute the approximate confidence intervals (ACI) for parameters 
and 
. Therefore, (
) 100% confidence intervals for parameters 
and 
become
 |
where 
is percentile of the standard normal distribution with right-tail probability 
.
We can increase information about the population value more than does a point estimate by using a parametric bootstrap interval. We propose to use confidence intervals based on the parametric bootstrap methods using bootstrap-t Algorithm (Boot-t) based on the idea of Hall [7]. The algorithms for estimating the confidence intervals using this method is illustrated as follows
1. Specify the values of 
, 
and 
.
2. Specify the values of 
and 
.
Step 1: set 
and let 
.
Step 2: 
.
• Generate 
as a binomial random variable with parameters 
and
 |
• Calculate 
, where 
progressive schemes.
Step 3: Set 
and 
.
Step 4: If 
and 
, stop else go to step 2.
1. Compute the maximum likelihood estimates of the parameters 
, 
by solving the likelihood equations simultaneously in Appendix.
2. Using 
and 
to generate a bootstrap sample 
. Based on 
compute the bootstrap estimate of 
and 
using likelihood equations respectively, say 
and 
and the following statistics
 |
3. Where 
and 
are obtained using the Fisher information matrix.
4. Repeat Step 4, B boot times.
5. For the 
and 
values obtained in step 4, determine the upper and lower bounds of the 
confidence interval bootstrap (CIB) of 
and 
as follows: let 
be the cumulative distribution function of 
and 
for a given 
, define
 |
3. Bayesian Estimation and MCMC Technique
In this section, we will focus to Bayesian approach using Markov chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are developed.
3.1. Bayesian EstimationIn Bayesian scenario, we need to assume the prior distribution of the unknown model parameters to take into account uncertainty of the parameters. We consider the Bayesian estimation under the assumption that the random variables 
and 
have an independent gamma prior distributions. Assumed that 
~ Gamma(B,A) and 
~ Gamma(D,C), then, the joint prior density of 
and 
can be written as
 | (7) |
Note that when 
, (we call it prior 0) they are the non-informative 
and 
respectively. It follows from (1), (3) and (7) that the joint posterior density function of 
and 
given 
is thus
 | (8) |
The Bayes estimate of any function of
and 
, say 
, is
 | (9) |
By using binomial and exponential series for equation (8), the posterior conditional distribution for 
and 
are
 | (10) |
respectively.
It is not possible to compute (9) analytically. The problem is that the integrals in (9) are usually impossible to evaluate analytically, and the numerical methods may fail. The MCMC method provides an alternative method for parameter estimation. In the following subsections, we propose using the MCMC technique to obtain Bayes estimates of the unknown parameters and construct the corresponding credible intervals.
3.2. MCMC TechniqueComputer simulation of Markov chains in the space of parameter will depend on Markov chain Monte Carlo (MCMC) [see Gilks et al. [4]]. The Markov chains are defined in such a way that the posterior distribution in the given statistical inference problem is the asymptotic distribution. However, the posterior likelihood usually does not have a closed form for a given type II progressively interval censored data. Moreover, a numerical integration cannot be easily applied in this situation. The Metropolis – Hastings algorithm is a very general MCMC method first expansion by Metropolis et al. [9] and later extended by Hastings [8]. It is possible to use these algorithms by implement posterior simulation in essentially any problem which allows point wise evaluation of the prior distribution and likelihood function. It can be used to obtain random samples from any arbitrarily complicated target distribution of any dimension that is known up to a normalizing constant. In fact, Gibbs sampler is just a special case of the M-H algorithm.
Now, we propose the following scheme to generate 
and 
from density functions and in turn obtain the Bayes estimates and the corresponding credible intervals.
0. Start with an 
and 
.
1. Set 
.
2. Generate 
and 
from (10).
3. Set 
4. Repeats Steps 1-3 
times.
5. Obtain the Bayes estimates of 
and 
with respect to the squared error loss function as
 |
 |
6. To compute the credible intervals of 
and 
order 
and 
as 
and 
. Then the 
symmetric credible intervals (SCI) of 
and 
become:
 |
4. Numerical Results
To illustrate the behavior of the proposed methods as well as evaluate the statistical performances of these estimates a numerical illustration is conducted where the performance of the different results obtained in the previous sections can't be compared theoretically. We reanalyze a real data set analyzed by Xiang and Tse [10]. Also, a simulations study is used to compare the performance of the different estimators, different confidence intervals using different parameter values and different schemes. In this section, the numerical study is carried out under type II progressive interval censored with unknown parameters. All of computations were performed using MATHCAD program version 2007.
4.1. Real DataIn the first subsection, we will rely on re-analyzed the real data which was originally analyzed by Xiang and Tse [10]. The data was obtained an experiment which was conducted to assess the toxicity of substance to animals. Forty mice were selected, every week a blood sample was collected from each of them, and the number of mice that showed evidence of toxicity was recorded. During the course of study, some mice which had to be removed from the study because they had developed other diseases, which made them unfit for the study. The data collected in the study are summarized in the following table:
 |
Before computing the MLEs, we get the MLEs of 
and 
from equations 5 and 6 respectively. On the hand, for fixed 
, the MLE of 
can be obtained as function in 
as 
, By Substituting 
in (4) and from other hand, for fixed 
, the MLE of 
can be obtained as function in 
as 
, By Substituting 
in (4); we can plot the profile log likelihood of 
and 
as follows
Download as
It is noted from Figure 1 that the likelihood equations have a unique solution, so we suggest using 
and 
as initial values to start the iteration to obtain the MLEs of 
and 
. The maximum likelihood estimates are 
and 
, and the corresponding 95% confidence intervals are (4.988, 5.734) and (2.084, 2.577) respectively. Based on the bootstrap sample of size 1000, the bootstrap estimates are 
and 
, and the corresponding 95% confidence intervals bootstrap are (5.201, 5.511) and (2.227, 2.670). We assume the non-informative priors, because we have no prior information about the unknown parameters. Based on the MCMC samples of size 10000, the Bayes estimates under the squared error loss function are 
and 
, and the corresponding 95% symmetric credible intervals (5.014, 5.467) and (2.201, 2.470).
The analysis of the previous real data set demonstrates the importance and usefulness of type II progressive Interval censored and inferential procedures based on them. From the previous example, we observed that the predetermined time number of inspection and number of failures plays an important role for the estimation of the unknown parameters and the corresponding confidence intervals. Also, it can be seen that the performance of the different methods for estimation are quite close to each other. However, the MLEs and Bayes estimators under squared error loss function with respect to the non-informative priors are the closest.
4.2. Simulation StudyThe simulation study is conducted by considering different values of sample sizes 
different effective number of failure 
, and by choosing 
in all the cases, also used generate type II progressive interval censored data under each of the four progressive schemes with withdraw probabilities denoted as 
and 
and depend on 
. All the progressive schemes used for the study are defined as follows:
 |
Where censoring in 
is lighter for the all intervals and 
is heavier for the all intervals.While 
are the conventional interval censoring where there are no removals prior to the experiment termination and the censoring in 
only occurs at the left-most and the right-most.
Table 1. The average values (AVE),mean square error (MSE), variance(VAR), bias andlength of 95% ACI (LACI)of the MLEs using Monte Carlo simulation
,mean square error (MSE), variance(VAR), bias andlength of 95% ACI (LACI)of the MLEs using Monte Carlo simulation)
Table 2. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% ACI (LACI) of the MLEs using Monte Carlo simulation
, mean square error (MSE), variance (VAR), bias and length of 95% ACI (LACI) of the MLEs using Monte Carlo simulation)
Table 3. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% ACI (LACI) of the MLEs using Monte Carlo simulation
, mean square error (MSE), variance (VAR), bias and length of 95% ACI (LACI) of the MLEs using Monte Carlo simulation)
Table 4. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95%CIB (LCIB) of the MLEs using Bootstrap method
, mean square error (MSE), variance (VAR), bias and length of 95%CIB (LCIB) of the MLEs using Bootstrap method)
Table 5. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% CIB (LCIB) of the MLEs using Bootstrap method
, mean square error (MSE), variance (VAR), bias and length of 95% CIB (LCIB) of the MLEs using Bootstrap method)
Table 6. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% CIB (LCIB) of the MLEs using Bootstrap method
, mean square error (MSE), variance (VAR), bias and length of 95% CIB (LCIB) of the MLEs using Bootstrap method)
Table 7. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% SCI (LSCI) of the Bayes estimates using MCMC
, mean square error (MSE), variance (VAR), bias and length of 95% SCI (LSCI) of the Bayes estimates using MCMC)
Table 8. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% SCI (LSCI) of the Bayes estimates using MCMC
, mean square error (MSE), variance (VAR), bias and length of 95% SCI (LSCI) of the Bayes estimates using MCMC)
Table 9. The average values (AVE), mean square error (MSE), variance (VAR), bias and length of 95% SCI (LSCI) of the Bayes estimates using MCMC
, mean square error (MSE), variance (VAR), bias and length of 95% SCI (LSCI) of the Bayes estimates using MCMC)
Under a type II progressive interval censored, using the different simulation methods, on the hand, were obtained the unknown parameters dependent on Monte Carlo simulation and Bootstrap method using maximum likelihood estimation and from other hand were used MCMC to obtain the unknown parameters using Bayes estimation.
In general, tables from (1) to (9) show that variance is usually smaller and bias is usually larger in both the estimation methods by using different simulations. The mean-squared error (MSE) associated with both MLE and Bayes estimates of the parameters decrease with increasing the sample size n. Also, it decreases when m is large.
With increasing n, there is an improvement in the value of estimators regardless of the type of estimation and the method of simulation. At each table with increasing m, there is an improvement in the value of the estimators, and also estimators obtained from progressive schemes 
and 
are the best forever. By comparison with the different methods of simulation the worst methods was bootstrap and which fail to give good estimators. Also, the estimators which were obtained from maximum likelihood estimation and Bayes estimation approximately one. In other words, the difference between them was trivial.
Under a type II progressive interval censored inspection scheme, that trial is terminated after the 
inspection if the total number of failed units is equal to or exceeds 
. In the fact, the total number of failure units greater than the value of 
and know 
which refer to estimated value to estimate value which obtained in accordance with the conditions of the experiment.
Table 10. The average values (AVE), mean square error (MSE), variance (VAR), bias and 95% CI of thetotal number of failed unitswith different simulation methods
, mean square error (MSE), variance (VAR), bias and 95% CI of thetotal number of failed unitswith different simulation methods)
We considered the following values: 
and 
and 15, we computed 
using Monte Carlo simulation and Bootstrap method using maximum likelihood estimation and MCMC using Bayes estimation to study of 10000 samples. The results are displayed in Table 10, By compared between simulation methods and both of estimations, which referred to MCMC gave the smallest 
, so, MCMC is the better in economic terms, where decrease of failure units (unobserved) which mean ending the experiment early and therefore is the best estimate. Bootstrap method is the worst, where the experiment ends in late unlike other methods.
References
| [1] | Ashour and Afify (2008), Estimations of the Parameters of Exponentiated Weibull Family with Type II Progressive Interval Censoring with Random Removals. Journal of Applied Sciences Research, 4(11): 1428-1442. | ||
 In article | |||
| [2] | Bebbington, M., Lai, C. D. and Zitikis, R. (2007), A flexible Weibull extension. Reliability Engineering and System Safety, 92, 719-726. | ||
| In article | View Article | ||
| [3] | El-Gohary A,El-Bassiouny, A. H. and El-Morshedy, M. (2015), Inverse Flexible Weibull Extension Distribution. International Journal of Computer Applications (0975-8887), 115 (2) 46-51. | ||
| In article | View Article | ||
| [4] | Gilks W. R., Richardson S. and Spiegelhalter D. J. (1996), Markov chain Monte Carlo in Practices, Chapman and Hall, London. | ||
| In article | |||
| [5] | Green E. J., Roesch F. A., Smith .A .F .M, and Strawderman W. E., (1994), Bayesian estimation for the three-parameter Weibull distribution with tree diameter data. Biometrica (50) 254-269. | ||
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| [7] | Hall, P. (1988), Theoretical comparison of Bootstrap confidence intervals, Annals of Statistics 16, 927-953. | ||
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| [8] | Hastings, W. K. (1970), Monte Carlo sampling methods using markov chains and their applications, Biometrika, 57(1):97-109. | ||
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| [9] | Metropolis N., Rosenbluth A.W., Rosenbluth M. N., Teller A. H. and Teller E., (1953), Equations of state calculations by fast computing machine, Journal of Chemical Physics, (21) 1087-1091. | ||
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| In article | View Article PubMed | ||
Appendix
The asymptotic variance-covariance matrix of the maximum likelihood estimators for parameters 
and 
are given by elements of the inverse of the Fisher information matrix. Unfortunately, the exact mathematical expressions for the above expectations are very difficult to obtain. Therefore, we give the approximate (observed) asymptotic varaince-covariance matrix for the maximum likelihood estimators, which is obtained by dropping the expectation operator E, where
 |
Fisher information matrix and the variance-covariance matrix will be obtained by numerical technique.
From equations (5) and (6), we will determine the second partials by differentiating the first partials as following
Note that: 
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