Chapter 6.2, Problem 29E

Consider a random sample X₁, X₂,…, X_n from the shifted exponential pdf

$f(x;\lambda \text{, }\theta )\text{ = }\{\begin{array}{l}\lambda e^{-\lambda (x-\theta )}x\ge \theta \\ 0\text{ otherwise}\end{array}$

Taking θ = 0 gives the pdf of the exponential distribution considered previously (with positive density to the right of zero). An example of the shifted exponential distribution appeared in Example 4.5, in which the variable of interest was time headway in traffic flow and θ = .5 was the minimum possible time headway.

1. a. Obtain the maximum likelihood estimators of θ and λ.
2. b. If n = 10 time headway observations are made, resulting in the values 3.11, .64, 2.55, 2.20, 5.44, 3.42, 10.39, 8.93, 17.82, and 1.30, calculate the estimates of θ and λ.