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.
Contents
- Conditions for a Well Behaved Statistic First Definition
- Conditions for a Well Behaved Statistic Second Definition
- Algorithmic inference
- Example
- References
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: Please see below for definition drawn from Apolloni et al.
Conditions for a Well-Behaved Statistic: First Definition
More formally the conditions can be expressed in this way.
Conditions for a Well-Behaved Statistic: Second Definition
In order to derive the distribution law of the parameter T, compatible with
- monotonicity. A uniformly monotone relation exists between s and ? for any fixed seed
{ z 1 , … , z m } – so as to have a unique solution of (1); - well-defined. On each observed s the statistic is well defined for every value of ?, i.e. any sample specification
{ x 1 , … , x m } ∈ X m ρ ( x 1 , … , x m ) = s has a probability density different from 0 – so as to avoid considering a non-surjective mapping fromX m S , i.e. associating vias to a sample{ x 1 , … , x m } a ? that could not generate the sample itself; - local sufficiency.
{ θ ˘ 1 , … , θ ˘ N } constitutes a true T sample for the observed s, so that the same probability distribution can be attributed to each sampled value. Now,θ ˘ j = h − 1 ( s , z ˘ 1 j , … , z ˘ m j ) is a solution of (1) with the seed{ z ˘ 1 j , … , z ˘ m j } . 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{ Z 1 , … , Z m | S = s } is independent of ?. An easy way to check this property is by mapping seed specifications intox i { X 1 , … , X m | S = 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) their corresponding realizations and with gothic letters (such as
The sampling mechanism
for suitable seeds
Example
For instance, for both the Bernoulli distribution with parameter p and the exponential distribution with parameter ? the statistic
and
Vice versa, in the case of X following a continuous uniform distribution on
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.