Sufficient statistics for a shifted exponential

Question

Response times cannot start before a fixed delay $\theta$ , after which they decay at rate $\lambda$ . You model $X_1, \dots, X_n$ as independent draws from a shifted exponential: density $\lambda e^{-\lambda(x - \theta)}$ for $x \ge \theta$ , and $0$ below $\theta$ . Both the shift $\theta$ and the rate $\lambda$ are unknown.

Find a pair of statistics that are jointly sufficient for $(\theta, \lambda)$ .

Accepted Answer

Two effects, two conditions. The rate \lambda sits in an exponent, and the shift 	heta sits both in the exponent and in a support condition (every point must be at least 	heta). Handle the support condition with an indicator. Factor the joint density. For all points above 	heta, f(x; 	heta, \lambda) = \prod{i=1}^n \lambda e^{-\lambda(xi - 	heta)}\,\mathbf{1}\{xi \ge 	heta\} = \lambda^n \exp\!\Big(-\lambda \sumi xi + n\lambda	heta\Big)\,\mathbf{1}\{\mini xi \ge 	heta\}. Two things to notice. The exponent touches the data only through \sumi xi. And "all xi \ge 	heta" is the same as "th…

Sufficient statistics for a shifted exponential

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