A Bayesian Estimation Procedure for One-Parameter Exponential Survival Distributions Facilitated by an Inverted Gamma Prior

A Bayesian Estimation Procedure for One-Parameter Exponential Survival Distributions Facilitated by an Inverted Gamma Prior

John Effiong Usen^*; Emmanuel Emmanuel Asuk; Godswill Itobo Egede

University of Cross River State, Calabar, Cross River State, Nigeria

*Corresponding Author

DOI: https://doi.org/10.51584/IJRIAS.2025.100500079

Received: 05 May 2025; Accepted: 09 May 2025; Published: 14 June 2025

ABSTRACT

A Bayesian estimation procedure for one-parameter exponential survival distributions facilitated by an inverted gamma prior was performed in this study using data obtained from data obtained from the University of Calabar Teaching Hospital (UCTH). The exponential survival distributions are just one amongst a number of distributions adopted for tackling problems in survival analysis, and may occur either in one parameter or two parameters under uncensored or censored conditions. The review of literature exposed the absence of studies addressing the need for an alternative procedure that carefully considered the peculiarities of the exponential survival distribution; hence, this study aimed to address this gap. Based on this, a Bayesian estimation technique was employed to estimate only the parameter of the one-parameter exponential survival distribution under uncensored and censored circumstances, with the survivorship and hazard functions deduced, thereafter. The results obtained showed that the parameter of the exponential survival distribution (λ) existed both for the one-parameter uncensored and censored cases of the exponential survival distribution, with known expressions. Both the MLE and Bayesian estimation results were simulated using real-life data, and the results showed near-convergence of the MLE and Bayesian estimation for unit values of the parameters of the inverted gamma prior used in the study.

Keywords: Bayesian Estimation Procedure, One-Parameter Exponential Survival Distributions, Inverted Gamma Prior

INTRODUCTION

Background of the study

In industries, the interest of process engineers may often be the estimation of the time available until a machine fails, while the interest of specialists in the clinical sciences may be the estimation of the time available: until a tumor reoccurs, until a cardiovascular death after some treatment intervention, until AIDS overwhelms an HIV infected person, etc. [1; 2]. Without the adoption of appropriate statistical know-how, these professionals may rely solely on intuition or experience for estimating such time [2]. The need to estimate such times non-intuitively, with high level of precision, has birthed the concept of survival analysis in statistics; and this technique has proven to be efficient [1; 2].

The area of survival analysis uses a host of techniques and statistical routines [3; 4]. As a statistical concept, survival analysis estimates the expected time duration until one or more events (such as death in biological organisms, and failure in mechanical systems) happen [5; 6]. Since its initiation in the early parts of 1940s, as a very useful bio-statistical tool, several statistical distributions like: exponential, Weibull, log-normal, gamma, generalized gamma, and log-logistic distributions, have all been adopted for studies on survival analysis, with the exponential survival distribution being the most commonly used distribution out of the lot [14; 15; 16; 17; 18]. More so, all these used distributions, as evidenced in literature, have had their parameters estimated via the general maximum likelihood estimation (MLE) routine, and this has no doubt produced formidable results; and with the parameter estimates for these distributions, their survival and hazard functions have been obtained [19; 20; 20; 22].

In recent times, to allow for advancements in statistical theory along the direction of exploring alternative estimation routines to the MLE, researchers have begun to attempt the adoption of Bayesian routine for achieving all (and even more) of what the MLE has been used to achieve with the afore-mentioned survival distributions [23; 24; 25; 26]. But one encountered problem, among several, has remained the nature of prior (informative or non-informative), and the suitability of prior (be it the: gamma, inverted gamma, Rayleigh, etc.) to be used [28; 29; 31]. In this study, an informative prior (the inverted gamma distribution) has been adopted, not only for the aim of obtaining Bayesian estimates of the one-parameter exponential distribution, but also for making inference about their survivorship and hazard functions under censored and uncensored circumstances.

Statement of the problem

One of the most commonly used distributions in survival analysis is the exponential survival distribution, be it for the one-parameter case or two-parameter case (with or without covariates). As evidenced in literature, Maximum Likelihood Estimation (MLE) has been used a lot for estimating parameters of one-parameter and two-parameter exponential survival distributions, and for obtaining their survival and hazard functions. And, although the MLE procedure has remained very efficient for achieving this, its usage yet restricts survival analysts to just one choice of estimation procedure. The absence of an attempt, in literature, to proffer the use of the Bayesian alternative to the MLE for achieving the same results, with a suitable informative prior, limits statistical theoretical advancement, going forward. Thus, this study is a foremost attempt in this regard.

Aim and objectives of the study

This study aims to perform a Bayesian estimation for one-parameter exponential survival distributions facilitated by an inverted gamma prior. In line with achieving the stated aim, the objectives of the study are to: (i) review the MLE procedure for obtaining the survivorship and hazard functions under (uncensored and censored conditions), (ii) adopt Bayesian estimation procedure for obtaining the survivorship and hazard functions (under uncensored and censored conditions), and (iii) simulate the MLE and Bayesian alternative on real-life data.

MATERIALS AND METHODS

MLE for data with right-censored observations

Suppose that persons were followed to death or censored in a study. Let ,   be the survival times observed from the  individuals, with  exact times and  right-censored times. Assume that the survival times follow a distribution with density function , and survivorship function , where  denotes unknown  parameters  in the distribution. If the survival time is discrete (that is, it is observed at discrete time only), then  represents the probability of observing , and  represents the probability that the survival or event time is greater than  [7]. In other words,  and  represent the information that can be obtained respectively from an observed uncensored survival time and observed right-censored survival time [8]. Therefore, the product  represents the joint probability of observing the uncensored survival times, and  represents the joint probability of those right-censored survival times. The product of these two probabilities, denoted by ,

represents the joint probability of observing ,  [7].

A similar interpretation applies to continuous survival , called the likelihood function of , which can also be interpreted as a measure of the likelihood of observing a specific set of survival times, namely: , , given a specific set of parameters  [7; 9]. The method of the MLE is to find an estimator of  that maximizes  (or which is most likely to have produced the observed data , ) [8].

Taking the logarithm of , and denoting it by , gives:

Then the MLE  is a  in the set of  that maximizes :

It is clear that  is a solution of the following simultaneous equations, which are obtained by taking the derivative of  with respect to each :

To obtain the MLE , one can use a numerical method. A commonly used numerical method is the Newton-Raphson iterative procedure, which can be summarized as follows [10; 12].

1. Let the initial values  be zero; that is, let

2. The changes for  at each subsequent step, denoted by , is obtained by taking the second derivative of the log-likelihood function:

3. Using , the value of  at  step is

The iteration terminates at, say, the  step if , where  is a given precision, usually a very small value,  or  [11]. Then the MLE  is defined as

The estimated covariance matrix of the MLE  is given by

One of the good properties of a MLE is that if  is the MLE of , then  is the MLE of  if  is a finite function and need not be one-to-one [10; 30].

The estimated  confidence interval for any parameter  is

where  is the  diagonal element of  and  is the  percentile point of the standard normal distribution . For a finite function  of , the estimated  confidence interval for  is its respective range  on the confidence interval in equation (6) [10; 30], that is,

In case  is monotone in , the estimated  confidence interval for  is

MLE for data with right-, left-, and interval-censored observations

If the survival times  observed for the  persons consist of uncensored left-, right-, and interval-censored observations, then the estimation procedures are similar [10; 30]. Assume that the survival times follow a distribution with density function  and the survivorship function , where  denotes all unknown parameters of the distribution. Then the log-likelihood function is given by:

where the first sum is over the uncensored observations, the second sum is over the right-censored observations, the third sum is over the left-censored observations, and the last sum is over the interval-censored observations, with  as the lower end of a censoring interval [10; 30]. The other steps for obtaining the MLE  of  are similar to the steps shown in section (3.1) by substituting for the log-likelihood function defined in equation (1) with the log-likelihood function in equation (9).

MLE on the one-parameter exponential survival distribution

The one-parameter exponential distribution has the following function;

survivorship function;

and hazard function;

where , . Obviously, the exponential distribution is characterized by one parameter, . The estimation of  by maximum likelihood methods for data without censored observations will be given first followed by the case with censored observations.

Maximum likelihood estimation of  for data without censored observations

Suppose that there are  persons in the study and everyone is followed to death or failure. Let  be the exact survival times of the  people. The likelihood function, using (10) and (1), is:

and the log-likelihood function is

From equation (2), the MLE of  is

Since the mean  of the exponential distribution is  and a MLE is invariant under a one-to-one transformation, the MLE of  is

It can be shown that  has an exact chi-square distribution with  degrees of freedom. Since  and  an exact  confidence interval for  is

where  is the  percentage point of the chi-square distribution with  degrees of freedom; that is,  [10; 12]. When  is large (say, ),  is approximately normally distributed with mean  and variance  [10; 12]. Thus, an approximate  confidence interval for  is

where  is the  percentage point, , of the standard normal distribution [10; 12]. Since  has an exact chi-square distribution with  degrees of freedom, an exact  confidence interval for the mean survival time is

Maximum likelihood estimation of  for data with censored observations

We first consider singly censored and then progressively censored data. Suppose that without loss of generality, the study or experiment begins at time zero with a total of  subjects. Survival times are recorded and the data become available when the subjects die one after the other in such a way that the shortest time comes first, the second shortest time comes second, and so on [12; 27]. Suppose that the investigator has decided to terminate the study after  out of the  subjects have and to sacrifice the remaining  subjects at that time. Then the survival times for the  subjects are

where a superscript plus indicates a sacrificed subject, and thus  is a censored observation. In this case,  and  are fixed values and all of the  censored observations are equal [12; 27].

The likelihood function, using equations (1), (10) and (11), is

and from equation (2), the MLE of  is

The mean survival time  can then be estimated by

It has been shown that  has a chi-square distribution with  degrees of freedom [12; 27]. The mean and variance of  are  and  respectively. The  confidence interval for  is

When  is large, the distribution of  is approximately normal with mean  and variance  [12; 27]. An approximate  confidence interval for  is then

It has also been shown that  has a chi-square distribution with  degrees of freedom [12; 27]. Thus, a  confidence interval for  is

They also develop test procedures for the hypothesis  against the alternative . One of their rules of action is to accept  if  and reject  if , where  and  is the significance level [12]. Or if the estimated mean survival time calculated from equation (20) is greater than , the hypothesis  is rejected at the  level of significance [27].

A slightly different situation may arise in laboratory experiments. Instead of terminating the study after the  death, the experimenter may stop after a period of time , which may be six months or a year [27]. If we denote the number of deaths between 0 and  as , the survival data may look as follows:

Mathematical derivations of the MLE of  and  are exactly the same and equation (19) can still be used [27].

Progressively censored data come more frequently from clinical studies where patients are entered at different times and the study lasts a predetermined period of time [13]. Suppose that the study begins at time 0 and terminates at time  and there are a total of  people entered [12; 27]. Let  be the number of patients who die before or at time  and  the number of patients who are lost to follow-up during the study period or remain alive at time . The data look as follows: . Ordering the  uncensored observations according to their magnitude, we have:

The likelihood function, using equations (1), (10) and (11), is

and the log-likelihood function is

and from equation (2), the MLE of the parameter  is

Consequently,

is the MLE of the mean survival time [12; 27]. The sum of all of the observations, censored and uncensored, divided by the number of uncensored observations, gives the MLE of the mean survival time [12; 27]. To overcome the mathematical difficulties arising when all of the observations are censored . [7] defines

In practice, this estimate has little value.

The distribution of  for large  is approximately normal with mean  and variance:

where  is the time that the  person is under observations. In other words,   is computed from the time  person enters the study to the end of the study [12; 27]. If the observations times  are not known, the following quick estimate of  can be used [7].

Thus, an approximate  confidence interval for  is, by (6),

The distribution of  is approximately normal with mean  and variance:

Again, a quick estimate is

An approximate  confidence interval for  is then, by (6),

Informative and non-informative priors

Let  be a prior distribution for the parameter  of a distribution. Then, according to [16; 17],

1. The distribution is an informative prior, if it biases the parameter towards particular values (i.e., an informative prior expresses specific, definite information about a variable);
2. The distribution is a non-informative prior, if it does not influence the posterior hyperparameters (i.e., an uninformative prior, or diffuse prior, expresses unclear or general information about a variable).

The inverted gamma prior

If a random variable  has the gamma distribution , then the random variable  has the inverted gamma distribution  with the density function [16; 17].

For , we obtain

And,

The Bayesian estimation procedure

Let  be a random sample from the density . Before taking the sample, the distribution of ,  (called a prior distribution), is assumed known. The objective in Bayesian estimation of the parameter  is to determine the distribution  (known as the posterior distribution) after taking the sample.

Let us consider the conditional distribution

Substituting for equation (34) in equation (35) gives,

But 

Therefore,

Putting equation (37) into equation (36) gives

Since we are taking a random sample of this distribution

Hence, equation (38) becomes:

The above equation (39) gives  as the posterior Bayes distribution with respect to the prior distribution .

Hence,

is called the posterior Bayes estimator with respect to the prior distribution ; where  is any function of .

An outline of the research estimation procedure

The proposed Bayesian alternative will be implemented with the procedure below.

Step 1: Determine an appropriate prior .

In this case, the appropriate prior for the one-parameter exponential distribution is the inverted gamma distribution with parameters  and . That is,

Step 2: Obtain the Bayesian estimate of  for data without censored observations.

Here the procedure involved requires that we obtain  thus:

where,  denotes the likelihood function. That is,

Step 3: Obtain the Bayesian estimate of  for data with censored observations.

Here the procedure involved requires that we obtain  thus:

where,  denotes the likelihood function. That is,

RESULTS

Bayesian estimation for the one-parameter uncensored case

Theorem 1

If  is an uncensored exponential random variable with parameter , and the prior density of  is the inverted gamma with parameters  and , then the estimate of  provided by the posterior is , given by:

Proof 1:

Now, ;  is the posterior distribution of the parameter . Let  be the likelihood function. Then,

Hence,

Let 

Since the mean  of the exponential distribution is  and Bayesian estimation is invariant under one-to-one transformation, then the Bayesian estimate of  is given by:

Therefore, the following axiom is established:

Axiom 1:

(a) The estimate of the survivorship function is given as:

(b) The estimate of the hazard function is given as:

Bayesian estimation for the one-parameter censored case

Theorem 2:

If  is a censored exponential random variable with parameter , and the prior density of  is the inverted gamma with parameters  and , then the estimate of  provided by the posterior is , given by:

Proof 2:

Now, ;  is the posterior distribution of the parameter . Let  be the likelihood function. Then,

Hence,

Let 

Again, since the mean  of the exponential distribution is  and Bayesian estimation is invariant under one-to-one transformation, then the Bayesian estimate of  is given by:

Therefore, the following axiom is established:

Axiom 2:

(a) The estimate of the survivorship function is given as:

(b) The estimate of the hazard function is given as:

Simulation

One-parameter exponential distribution without censored observations

Given the remission times in months for 734 HIV/AIDS patients (Appendix 1). Assuming that remission duration follows the exponential distribution, we can estimate the parameter  as follows.

Simulating the one-parameter uncensored case, via the MLE, the relapse rate, , is

 per week

The mean remission time  is then  weeks.

Using the analytical procedures given above, confidence intervals for  and  can also be obtained.

A 95% confidence interval for the relapse rate , following  is approximately

A 95% confidence interval for the mean remission time following  is

Once the parameter  is estimated, other estimates can be obtained. For example, the probability of staying in remission for at least 20 months, can be estimated from:

However, simulating the one-parameter uncensored case by Bayesian alternative, we have a Bayesian estimate of the relapse rate, , at ; , gave results consistent with those which were obtained from the uncensored case (via existing method). And this is based on the convergence of the values. That is,

One-parameter exponential distribution with censored observations

A study is carried out on 312 AIDS patients. The study is terminated after half (156) of the AIDS are dead with the other half sacrificed at that time. The survival data of the 312 AIDS patients are shown in Appendix 2. Assuming that the failure of these AIDS patients follows an exponential distribution, the survival rate  and mean survival time  via the MLE method are estimated, respectively, according to

And

By

 per week

and  weeks. A 95% confidence interval for  by  is

A 95% confidence interval for  following  is

The probability of surviving a given time for the patients can be estimated from . For example, the probability that a patient exposed to AIDS will survive longer than 122 months is

The probability of dying in 122 months is then .

However, assuming that the failure of these AIDS patients follows an exponential distribution, the survival rate  and mean survival time  via the Bayesian method are estimated, respectively, at ;  according to

Hence,  per week and  weeks. A 95% confidence interval for  by  is

A 95% confidence interval for  following  is

Similarly, the probability of surviving a given time for the patients can be estimated from . For example, the probability that a patient exposed to AIDS will survive longer than 122 months will now be:

The probability of dying in 122 months is then .

DISCUSSION

The results of this study are summarized in Table 1 and Table 2. These two tables respectively show the parameter estimates of  using the maximum likelihood estimation and Bayesian estimation procedures under uncensored and censored circumstances. Here, Table 1 and Table 2 show that the parameter estimates of  exists and are non-zero. Consequently, this implies the existence of the survivorship and hazard functions.

For the one-parameter exponential distribution without censored observations, we note that at ; , using the Bayesian approach gave results consistent with those which were obtained from the uncensored case through the MLE procedure as confirmed by the convergence of the , , interval and  values.

For the one-parameter exponential distribution with censored observations, we see that at ; , using the Bayesian approach gave results with significant divergence from those which were obtained from the uncensored case based on the MLE method as confirmed by the divergence of the , , and interval  values.

CONCLUSION

In conclusion, this study has attempted a Bayesian estimation alternative to the MLE for estimating parameters of exponential survival distributions (in one parameter and two parameters) under uncensored and censored scenarios. Two axioms were deduced on what the survivorship and hazard functions are, for each of the estimated parameters. The results obtained from the one-parameter cases showed that known parameter estimates of , as well as the survivorship and hazard function, exists, and are similar to the results obtained via the MLE under certain values of the parameters of the prior used (in this case, the inverted gamma distribution).

Table 1 Maximum likelihood estimates for one-parameter exponential survival distribution

ET		Maximum Likelihood
Sur. Dist.	Case	One parameter
	U
	C

ET = Estimation Technique; Sur. Dist. = Survival Distribution; U = Uncensored; C = Censored

Table 2 Bayesian estimates for one-parameter exponential survival distribution

ET		Bayesian
Sur. Dist.	Case	One parameter
	U
	C

ET = Estimation Technique; Sur. Dist. = Survival Distribution; U = Uncensored; C = Censored

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APPENDICES

Appendix 1

Uncensored Data for Remission Times of HIV Patients

S/N = Serial Numbers; RT = Remission Times

Data Obtained from University of Calabar Teaching Hospital (UCTH)

S/N = Serial Numbers; RT = Remission Times

Data Obtained from University of Calabar Teaching Hospital (UCTH)

S/N = Serial Numbers; RT = Remission Times

Data Obtained from University of Calabar Teaching Hospital (UCTH)

Appendix 2

Censored Data for AIDS Patients

S/N = Serial Numbers; ST = Survival Times

Data Obtained from University of Calabar Teaching Hospital (UCTH)

S/N = Serial Numbers; ST = Survival Times

Data Obtained from University of Calabar Teaching Hospital (UCTH)