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Parameter Estimation and Bayesian Prediction in the Log-logistic Distribution under Type-II Censored Data

Wassim Abou Ghaida*, Ayman Baklizi

Foundation Program Unit, University of Doha for Science and Technology, Doha, Qatar

Abstract: We consider Bayesian inference and point prediction in log-logistic distribution based on type-II censored data. We assume that the scale and shape parameters have independent gamma priors. The Bayes estimators cannot be obtained in closed form; therefore, we use Metropolis Hasting algorithm to approximate the Bayes estimates of the unknown scale and shape parameters. We compare the performance of the Bayes estimator with the Maximum Likelihood estimators. In addition, we obtained the Bayesian credibility intervals and compared them with the Wald intervals. Moreover, we derived Bayesian point and interval predictors for future observation and investigated their performance using simulation techniques. And finally, we analysed real data set for illustration purposes.

Keywords: Bayes estimates, Maximum Likelihood estimators, Metropolis Hasting algorithm, Predictive density, Bayesian prediction, Prediction interval

Received September 25, 2025; Revised October 27, 2025; Accepted November 04, 2025

Citation: Wassim Abou Ghaida, Ayman Baklizi, Parameter Estimation and Bayesian Prediction in the Log-logistic Distribution under Type-II Censored Data. International Journal of Business and Risk Management, 6 (1). 11-18. January 2025. doi:10.12691/ijbrm-6-1-1

1. Introduction

Prediction is a natural phenomenon in real life situations [1]. It can be summarized as the process of obtaining inferences about unknown future observations based on current informative samples from available data. These future observations play an important role in reliability and survival analysis.

Prediction of future failures of certain units or products can help companies minimise financial loss and maximize the profit through designing and developing new strategic plans.

Several authors have studied the prediction from a classical and Bayesian approach. [2] has studied the prediction of future observations for generalized exponential distributions based on hybrid censored data. [3] has considered Bayesian estimation and prediction of Burr type XII distribution for progressive sampled data. [4] has predicted observables from a Bayesian approach. [5] has dealt with the Bayesian prediction of Inverse Weibull distribution for censored data. [6] discussed Bayesian prediction of type-II censored Weibull data. [7] have done an extensive classical approach on future failures prediction for log-logistic distribution from hybrid censored data.

Censoring is very common in reliability and survival analysis. Censoring takes place when the lifetimes are only known for a specific number of items or units being studied.

See [8]. Type-I and type-II censoring schemes arethe most popular schemes. The two types of censoring can be summarized as follows: Let's consider a scenario where we subject n units to life testing. In type-I censoring, we predetermine the time T when the experiment will be terminated, and then record the random number of failures that occur until that point. While in type-II censoring, we predetermine and fix the number of failures, and then observe the random time at which the experiment is terminated. [9]. For the purpose of this paper, we will focus only on type-II censored data.

The log-logistic distribution is a right skewed parametric distribution that falls under the scale-shape family and has a heavy tail. Its cumulative distribution function could be expressed in closed form. It could be considered as a substitute to lognormal distribution. The importance of the log-logistic distribution is that it has different applications in reliability and survival analysis, see [10], and it has a great mathematical traceability when the data comes from censored observations [11]. It can be used as a statistical model in actuarial science and economy which was previously known as Fisk distribution [12]. The probability density function (PDF) of the log-logistic distribution is given by:

[picture] | (1)

The cumulative distribution function (CDF) is given by:

[picture] | (2)

Although, previous studies have been conducted on the estimation of the log-logistic distribution parameters and point prediction using frequentist approach, but not much work has been done from a Bayesian approach.

The rest of the paper is constructed as follows. In section 2, the likelihood equations are obtained. In section 3, the Bayesian estimation is presented. In section 4, Bayesian point prediction is discussed. A simulation study is presented in section 5, in section 6, a real data analysis was performed, and finally a conclusion is drawn in section 7.

2. Likelihood Equations

The likelihood function using type-II censored samples, can be developed as follows: Consider a sample from model (1). Based on type-II censoring scheme, the experiment stops after the number of failures has been obtained. If represent the corresponding ordered failure times of the units, then thelikelihood of the observed and censored units can be represented by and respectively.

Hence, which is :

[picture] | (3)

The log-likelihood function

[picture] | (4)

The maximum likelihood estimators (MLEs) of the parameters and are the roots of the partial derivate equations.

[picture] | (5)

[picture] | (6)

It is not possible to solve equations (6) and (7) explicitly. Hence, Newton Raphson method (NS) or EM algorithm can be employed to solve the above equations numerically.

3. Bayesian Estimation

In the following section we will provide the Bayes estimates of the parameters for type-II censored samples. Similar to [13], we assume that the prior distributions of the parameters and are independent gamma priors, gamma (and gamma ( respectively. Given that the hyper parameters a b c, and d are known and positive.

If is an observed sample from model (1), then we combine the likelihood of the observed data, and to get:

Where (7)

The posterior distribution of given the data is:

[picture] | (8)

Hence, the Bayes estimates of function, with respect to the squared error loss function (SELF) is:

[picture] | (9)

Since cannot be computed analytically, therefore we use Metropolis Hasting algorithm to evaluate the Bayes estimates of and, see [14].

The Metropolis Hasting algorithm is demonstrated below; we will simulate samples or draws from the target posterior distribution assuming gamma candidate distributions for both parameters and.

3.1. The Metropolis Hasting Algorithm

According to [15], the Metropolis Hasting algorithm (M-H) was developed by Metropolis [16] and generalized by Hasting [17].

The M-H algorithm is a Markov Chain Monte Carlo method that is used to generate samples from a probability distribution from which sampling directly might be unattainable. The M-H algorithm can be summarized as follows:

Suppose we want to draw samples from some multivariate distribution where a parameter vector and c is a normalizing constant.

The algorithm provides a way of sampling from without knowing the value of. It will generate a sequence .. of draws of the parameter vector, which forms a Markov chain.

Algorithm:

Let

Choose an arbitrary starting point;

1. Set

2. Draw from a bivariate gamma candidate distribution with independent components

3. Compute the acceptance probability

Where is the pdf of the gamma candidate distribution

4. Draw from uniform distribution

5. If, accept the candidate values and set Otherwise,

Reject the candidate values and set

6. Set

7. Repeat the process from step 2 to step 6 for .. times

8. Calculate the Bayes estimates of and using and

9. Arrange the values of, ,…, and, ,…, in order

And denote the ordered values by, , …, and, , …, respectively.

The symmetric credible intervals of and are and

4. Bayesian Point Prediction

In this section, we will discuss the prediction of for based on the observed sample). Following [2] and [7], The conditional density of:

where (10)

For model (1),

[picture] | (11)

The predictive density of given is given by

[picture] | (12)

Where is the conditional density of given and is the joint posterior density of and.

The Bayesian predictor (BP) of under SELF is:

[picture] | (13)

It is clear that it is not possible to compute (14) explicitly. Therefore, we use the Metropolis-Hasting algorithm to obtain the Bayes predictor. Posterior samples will be generated from the posterior distribution and will be used to attain the Bayes predictor and the prediction intervals.

If are the generated samples, then the simulation consistent estimator of is

[picture] | (14)

Replacing (15) in (14) then the BP of can be approximated by:

[picture] | (15)

[picture] | (16)

Where (17)

And (18)

[picture] | (19)

The general formula for the CDF of the order statistic of a sample of size n is

[picture] | (20)

The CDF of the conditional distribution of / is equivalent to the distribution of the order statistic from a sample of size n – m from the population with density , therefore, the CDF of given is given by:

It follows that:

[picture] | (21)

Where is the Bayes predictive density. This can be approximated using Metropolis-Hastings algorithm as:

[picture] | (22)

[picture] | (23)

[picture] | (24)

The lower and upper bounds and of the Bayesian prediction interval for can be found by solving the system of equations:

[picture] | (25)

[picture] | (26)

5. Simulation

Here, we investigate the Bias and MSE of the Bayes estimates, MLEs, and the Bayesian point predictor. We construct 90% and 95% Wald, and Bayesian intervals for the parameters, Bayesian prediction intervals, lower error rate, upper error rate, total error rate, and the expected length of the intervals.

We perform a simulation study using the Metropolis Hasting algorithm to obtain posterior samples. We use different sample size (and different number of observed values. The parameters are taken as

We assumed gamma priors with hyper parameters. Different predictor values are obtained

1000 samples are generated. If is the interval of the parameter estimator obtained from the iteration, then:

and,

where is the perimeter estimator and:

.

In a similar fashion, , and are obtained for the Bayesian predictor.

If is the predictor value of, obtained from the iteration, and then: The Bias and MSE are calculated as follows:

Bias and MSE

The point prediction and prediction intervals are listed in Table 1, the bias and MSE are listed in Table 2, The error rates and expected lengths are listed in Table 3, Table 4, Table 5, and Table 6.

Table 1. Point Prediction and Prediction Intervals

Table 2. Bias and MSE of the MLEs â b^ and the Bayesian Estimators ãb⁻ respectively

Table 3. Error Rates and Expected Lengths of 95% Bayes Intervals for the Parameters

Table 4. Error Rates and Expected Lengths of 95% Wald Intervals for the Parameters

Table 5. Error Rates and Expected Lengths of 90% Bayesian Intervals for the Parameters

Table 6. Error Rates and Expected Lengths of 90% Wald Intervals for the Parameters

• From Table 1, we observe that for fixed values of and , the MSE slightly increases as increases.

• and of the prediction intervals attain the nominal error rate.

• From Table 2, we observe that both Bias and MSE perform well for both the MLE and the Bayesian estimators. However, the bias of the MLEs is slightly smaller than that of the Bayes estimators.

• From Table 3 and Table 4, we observed that of the 95% Bayes interval of is slightly less than that of the 95% Wald interval of while of the 95% Bayes interval of is slightly greater than that of the 95% Wald interval of and of both intervals for and respectively are almost the same.

• of the Bayes intervals are close of the Wald intervals.

• From Table 5 and Table 6, we observe that the error rates of both of the Bayes intervals and Wald intervals attain the nominal error rate.

6. Real Data Analysis

For illustration purposes, we analyse one real data which have been recently considered by [18]. The data consist of 128 patients with bladder cancer and the monthly remission times are shown:

0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 0.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.

Based on type-II sampling scheme and using m=20, the observed sample is:

0.08 0.20 0.40 0.50 0.51 0.66 0.81 0.90 1.05 1.19 1.26 1.35 1.40 1.46 1.76 2.02 2.02 2.07 2.09 2.23.

The MLEs of are and the corresponding Wald intervals are listed in Table 7.

Based on the above sample, and using the Metropolis Hasting algorithm, we generate 10000 samples from the posterior distribution, we obtain the Bayes estimates, Bayes intervals, and we predict different values of and the prediction intervals for.

In order to obtain the Bayes estimates of the parameters and and the Bayes intervals, we take the average of 1000 Bayes estimates with the corresponding Bayes intervals.

Note that for Bayesian estimation and prediction, we assume two different priors:

Prior 1: and Prior 2:.

We take the prior means to be equal to 1 (true value of the parameters).

The Bayes estimates and the corresponding Bayesian intervals are listed in Table 8, and Bayesian prediction and the prediction intervals are listed in Table 9.

From Table 7 and Table 8, we observe that the Bayes estimates of under prior 1 is close to the Bayes estimate obtained under prior 2. Similarly, this holds true for. Moreover, the Wald intervals are narrower than the Bayes intervals.

Table 7. Wald Intervals for the Parameters

Table 8. Bayes Estimates and Bayes Intervals

Table 9. Bayesian Prediction and Prediction Intervals

From Table 9, we observe that the point predictors are close to the exact values under both priors, the prediction intervals contain the exact values, and they get wider as gets bigger. Moreover, as expected, the 90% prediction intervals are quite shorter than the 95% prediction intervals.

7. Conclusion

In this paper, Bayesian parameter estimation, Bayesian point prediction and prediction intervals of type-II censored data of the log-logistic distribution are considered. Classical and Bayesian intervals are compared. Independent gamma priors are assumed and Metropolis Hasting algorithm is used to obtain posterior samples. Based on the simulation study, the Bayesian point predictors performed well and the prediction intervals attained nominal error rate for all sample sizes greater or equal to 20, and the expected lengths of the Bayes intervals of the parameters, are close to the expected lengths of the Wald intervals of the parameters.

Although in this paper, Bayesian point prediction and prediction intervals have been obtained using type-II censoring scheme, other censoring schemes such as type-I censoring scheme might be considered. Further investigation is required in this direction.

Compliance with Ethical Standards

• The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

• The authors have used a pre-existing data of 128 patients which have been previously considered by [18].

Informed Consent: N/A

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