Interval Estimation of the Unknown Exponential Parameter Based on Time Truncated Data

Let $X_{1},\ldots,X_{n}$ be a random sample from an exponential distribution with parameter $\lambda$, then it has the following probability density function (PDF)

Here $\lambda\in(0,\infty)$ is the natural parameter space. Since for $\lambda$ = 0, $f(x;\lambda)$ as defined in (1) is not a proper PDF, we are not including the point 0 in the parameter space. Suppose $n$ items are tested and their ordered failure times are denoted by $x_{1:n}<x_{2:n}<\ldots<x_{n:n}$. If the experiment is stopped at a prefixed time $T$ then it results in a simple type-I censoring case. Let there be $D$ failures in $[0,T]$. Then $x_{1:n}<x_{2:n}<\ldots<x_{D:n}\leq T$ and $T<x_{D+1:n}<\ldots x_{n:n}$, although $x_{D+1:n},\ldots,x_{n:n}$ are not observed. Here $D$ is a random variable that can take the values $0,1,\ldots,n$. The main aim of this note is to draw inference on $\lambda$, based on $D$ observations.

This is an old problem and Bartholomew (1963) was the first to consider it. He considered the following form of the exponential PDF;

He considered $[0,\infty)$ as the parameter space of $\theta$. In this case the maximum likelihood estimator (MLE) of $\theta$ does not exist when $D=0$. Hence all the inferences related to $\theta$ are based on the conditional argument that $D\geq 1$. Later on a series of papers starting with Chen and Bhattacharyya (1988) and then continuing with Gupta and Kundu (1998) and Childs et al. (2003) considered type-I hybrid censoring case for the model (2), and obtained the exact inference on $\theta$ based on the conditional argument that $D\geq 1$.

In case of type-I censoring, $\displaystyle P(D=0)=exp(-n\lambda T)$ and this probability can be quite high for small value of $n\lambda T$. The natural question here is whether it is possible to draw any inference on $\lambda$, when $D=0$. The second aim of the study is to determine if there exists any significant difference between the conditional and unconditional inference. For example, in this paper it has been shown that it is possible to construct an exact 100$(1-\alpha)\%$ confidence interval of $\lambda$ even when $D$ = 0. Following the approach of Chen and Bhattacharyya (1998), an exact 100$(1-\alpha)\%$ confidence interval of $\lambda$ also can be obtained based on the conditional MLE of $\lambda$, conditioning on the event that $D\geq 1$. The question is whether the lengths of the confidence intervals based on the two different approaches are significantly different or not. We perform some simulation experiments to compare the confidence intervals based on the two different methods. We further obtain the Bayes estimate and the associated credible interval of the unknown parameter based on the non-informative prior. Finally the results are extended for the two-parameter exponential, Weibull and generalized exponential distributions also.

Rest of the paper is organized as follows. In Section 2, we provide two different constructions of confidence intervals and the Bayesian credible intervals. The simulation results for confidence and credible intervals of the parameter $\lambda$ are provided in Section 3. In Section 4, we extend the results for the two parameter exponential, Weibull and generalized exponential distributions, and finally we conclude the paper in Section 5.

In this section we proceed to construct the confidence and credible intervals of the parameter of interest $\lambda$, based on a new estimator of $\lambda$ and the posterior distribution of $\lambda$, respectively.

In this section we present some simulation results to compare the performances of the two confidence intervals and the corresponding credible intervals. We compare the performances in terms of the average lengths and the coverage percentages. Different values of $n$, $\lambda$ and $T$ are taken. We consider $n$ = 5, 10, 15, 20, $\lambda$ = 0.5, 1.0, 2.0 and $T$ = 1, 2. For Bayesian inference we have taken $a=b$ = 0.001. All the results are based on 5,000 replications. The results are reported in Tables 1 to 3.

The following points have been revealed from these simulation experiments. In all these cases it is observed that biases and mean squared errors (MSEs) decrease as sample size increases, $T$ increases or $\lambda$ increases as expected. Now comparing the confidence intervals and credible intervals in Tables 1 to 3 it is clear that for small sample sizes, small $\lambda$ and small $T$ values, the confidence intervals based on unconditional distribution performs better that the credible intervals based on non-informative priors in terms of the coverage percentages. The coverage percentages of the confidence intervals based on unconditional distribution are very close to the nominal value (95%) in all cases. Although, as expected for large sample sizes they are almost equal. The confidence intervals based on the conditional distribution do not perform very well when the sample size is very small and $\lambda$ is also small, although when the sample size is not very small it performs well. Performances of the two confidence intervals match as expected when sample size is large.

In this section we consider some of the related two-parameter models, and construct an exact 100$(1-\alpha)\%$ confidence set of the two parameters, when $n$ items are tested and there is no observed failure during $[0,T]$.

Here $\beta>0$ and $\lambda>0$. In this case $P_{\{\lambda,\beta\}}(D=0)=e^{-n\lambda T^{\beta}}$. A 100$(1-\alpha)\%$ confidence set of $(\lambda,\beta)$, can be obtained as $A=\{(\lambda,\beta);\lambda>0,\beta>0,P_{\{\lambda,\beta\}}(D=0)\geq(1-\alpha)\}$. Hence,

Just for illustrative purposes, taking $n$ = 5 and $T$ = 0.5, we provide the joint 100$(1-\alpha)\%$ confidence set as the shaded region of (i) $(\lambda,\mu)$ for two-parameter exponential distribution, (ii) $(\beta,\lambda)$ for two-parameter Weibull distribution and (iii) $(\beta,\lambda)$ for two-parameter generalized exponential distribution, in Figure 2 when $D$ = 0.

In this paper we have considered the time truncated exponential distribution and provide the exact confidence interval of the unknown scale parameter based on the unconditional argument. All the existing results are based on the conditional argument and it does not provide any statistical inference of the unknown parameter when there is no observation. In this paper we have provided the inference on the scale parameter even when there is no observation. Simulation experiments are performed and it is observed that the proposed method works quite well even when the sample size is very small. The joint confidence sets have been provided for two-parameter exponential, Weibull and generalized exponential distributions also for time truncated case when there is no observation during the time period of the experiment. The results can be extended to the competing risks model also, as considered in Kundu, Kannan and Balakrishnan (2004).