Although the term well-behaved statistic often seems to be used in the scientific literature in somewhat the same way as is well-behaved in mathematics (that is, to mean "non-pathological") it can also be assigned precise mathematical meaning, and in more than one way. In the former case, the meaning of this term will vary from context to context. In the latter case, the mathematical conditions can be used to derive classes of combinations of distributions with statistics which are well-behaved in each sense.

First Definition: The variance of a well-behaved statistical estimator is finite and one condition on its mean is that it is differentiable in the parameter being estimated.

Second Definition: The statistic is monotonic, well-defined, and locally sufficient.

Conditions for a Well-Behaved Statistic: First Definition

to be well-behaved we require:

, and the derivative satisfies:

Conditions for a Well-Behaved Statistic: Second Definition

In order to derive the distribution law of the parameter T, 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 – so as to have a unique solution of (1);

2. well-defined. On each observed s 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. 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 will not depend on ?, if the above seed independence holds – a condition that looks like a local sufficiency of the statistic S.

The remainder of the present article is mainly concerned with the context of data mining procedures applied to statistical inference and, in particular, to the group of computationally intensive procedure that have been called algorithmic inference.

Algorithmic inference

In algorithmic inference, the property of a statistic that is of most relevance is the pivoting step which allows to transference of probability-considerations from the sample distribution to the distribution of the parameters representing the population distribution in such a way that the conclusion of this statistical inference step is compatible with the sample actually observed.

By default, capital letters (such as U, X) will denote random variables and small letters (u, x, given a sampling mechanism scalar, for the random variable X, of the statistic s , has an explaining function defined by the master equation:

and parameter ?

Example

For instance, for both the Bernoulli distribution with parameter p and the exponential distribution, 0 otherwise in the case of the Bernoulli random variable for the Exponential random variable, giving rise to statistics

and

Vice versa, in the case of X following a continuous uniform distribution. But the explaining function of this X would produce with a U. This conflicts with the observed sample since the first observed value should result greater than the right extreme of the X is well-behaved in this case.

Analogously, for a random variable X following the Pareto distribution with parameters K and A (see Pareto example for more detail of this case),

and

can be used as joint statistics for these parameters.

As a general statement that holds under weak conditions, sufficient statistics are well-behaved with respect to the related parameters. The table below gives sufficient / Well-behaved statistics for the parameters of some of the most commonly used probability distributions.

References