Well-behaved Statistic - Well-behaved

In order to derive the distribution law of the parameter T, compatible with, the statistic must obey some technical properties. Namely, a statistic s is said to be well-behaved if it satisfies the following three statements:

1. monotonicity. A uniformly monotone relation exists between s and ? for any fixed seed – so as to have a unique solution of (1);
2. well-defined. On each observed s the statistic is well defined for every value of ?, i.e. any sample specification such that has a probability density different from 0 – so as to avoid considering a non-surjective mapping from to, i.e. associating via to a sample a ? that could not generate the sample itself;
3. local sufficiency. constitutes a true T sample for the observed s, so that the same probability distribution can be attributed to each sampled value. Now, is a solution of (1) with the seed . Since the seeds are equally distributed, the sole caveat comes from their independence or, conversely from their dependence on ? itself. This check can be restricted to seeds involved by s, i.e. this drawback can be avoided by requiring that the distribution of is independent of ?. An easy way to check this property is by mapping seed specifications into s specifications. The mapping of course depends on ?, but the distribution of will not depend on ?, if the above seed independence holds – a condition that looks like a local sufficiency of the statistic S.