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A sufficient statistic $T(X)$ for $\theta$ is a function of the observations whose conditional distribution of $X$ given $T(X)$ does not depend on $\theta$, whenever that conditioning event is possible. It retains all parameter dependence of the likelihood.

In the discrete case the Fisher-Neyman factorization theorem provides a direct criterion: $f(x\mid \theta )=g(T(x),\theta )h(x)$, where $h$ is independent of the parameter. To justify sufficiency, for a value $t$ with positive probability, sum this expression over its fiber and divide:The parameter-dependent factor cancels. Conversely, parameter-independent conditional probabilities $h_t(x)$ give the factorization $f(x\mid \theta )={\mathbb{P}}_{\theta }(T=t)h_t(x)$ with $t=T(x)$. Thus the criterion is both necessary and sufficient, with conditional laws only required on fibers of positive probability.

For the shifted exponential sample the likelihood isIt factors through $T=X_{(1)}={\min }_i{X}_i$, so the sample minimum is a sufficient statistic. Writing $Y_i=X_i-\theta$, independence gives for $t\ge 0$Therefore $T-\theta$ is exponential with rate $n\lambda$. Its mean is $(n\lambda {)}^{-1}$ and its variance is $(n\lambda {)}^{-2}$. The unbiased endpoint estimator for a shifted exponential sample is consequentlyAlthough the parameter is nonnegative, this unbiased estimator can be negative. Truncating it at zero would change its expectation and lose the requested property of an unbiased estimator.