ABSTRACT
In this chapter, we focus on an exponential distribution, referred to as Expo(θ), having the following probability density function:
f(x; θ) =
{ 1
θ exp (−x/θ) if x > 0
0 if x ≤ 0. (8.1.1)
Here, θ is the unknown mean of this distribution, 0 < θ <∞. This distribution has been used in many reliability and lifetesting ex-
periments to describe, for example, a failure time of complex equipment, vacuum tubes and small electrical components. In these contexts, θ is interpreted as the mean time to failure (MTTF). One is referred to Johnson and Kotz (1970), Bain (1978), Lawless and Singhal (1980), Grubbs (1971), Basu (1991) and other sources. This distribution has also been used extensively to model survival times, especially under random censoring. See, for example, Aras (1987,1989), Gardiner and Susarla (1983,1984,1991) and Gardiner et al. (1986). For overviews on reliability theory, one may refer to Lomnicki (1973),
Barlow and Proschan (1975) and Ansell and Phillips (1989). The articles of Tong (1977), Brown (1977), Beg and Singh (1979) and Beg (1980) will provide added perspectives. A volume edited by Balakrishnan and Basu (1995) presented a wide spectrum of statistical methodologies with exponential distributions. Suppose that we have recorded n independent observations X1, . . . ,Xn,
each following Expo(θ) distribution from (8.1.1). Then, the MTTF parameter θ is customarily estimated by the maximum likelihood estimator (MLE). The MLE coincides with the uniformly minimum variance unbiased estimator (UMVUE), namely the sample mean Xn ≡ n−1
∑n i=1Xi.
