Spotting a sufficient statistic by factoring

Question

You have independent waiting times X1, \dots, Xn from an exponential distribution with rate \lambda. Using the factorization idea, find a one-number summary of the data that carries all the information about \lambda (a sufficient statistic).

Accepted Answer

What "sufficient" means, in words. A statistic is sufficient for a parameter if, once you know its value, the raw data have nothing more to tell you about that parameter. You could throw the data away and keep only the summary without losing anything relevant to \lambda. The factorization test. The Fisher-Neyman theorem gives an easy check: write the joint density as a product of two pieces, f(x; \lambda) = g\big(T(x), \lambda\big) \cdot h(x), where the first piece touches the parameter and depends on the data only through some summary T(x), and the second piece does not involve the parameter…

Spotting a sufficient statistic by factoring

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