An exponential rate

Topic: Estimation · more questions on this topic
‹ Bias of the two variance estimators A Poisson mean ›

Let X_1,\ldots,X_n be a random sample from the exponential distribution with rate \lambda, density \lambda e^{-\lambda x} for x>0.

1. Find the maximum likelihood estimator of \lambda, and check that it is a maximum.

2. Find the method-of-moments estimator of \lambda.

3. Use invariance to find the maximum likelihood estimator of the median lifetime (\log 2)/\lambda.

Worked answer (click to show)

1. The log-likelihood is

\log L(\lambda)=n\log\lambda-\lambda\sum_{i=1}^nx_i.

Its derivative n/\lambda-\sum x_i is zero at \lambda=n/\sum x_i=1/\overline x. The second derivative, -n/\lambda^2, is negative, so this is a maximum. The estimator is \widehat\lambda=1/\overline X.

2. \mathbb{E} X=1/\lambda. Setting 1/\lambda=\overline x gives the same estimator, 1/\overline X.

3. The median lifetime m solves 1-e^{-\lambda m}=1/2, so m=(\log2)/\lambda. By invariance its maximum likelihood estimator is (\log2)/\widehat\lambda=\overline X\log2.