. Consider a random sample X1, X2, . . ., Xn from the shifted exponential pdf
f x; λ; θ ð Þ ¼
λe_λðx_θÞ x _ θ
0 otherwise
(
Taking θ ¼ 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 3.5, in which the variable of interest was time headway in traffic flow and θ ¼ .5 was
the minimum possible time headway.
(a) Obtain the maximum likelihood estimators of θ and λ.
(b) If n ¼ 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.