Conditions for a Well-Behaved Statistic: First Definition
More formally the conditions can be expressed in this way.
is a statistic for
that is a function of the sample,
. For
to be well-behaved we require:
: Condition 1
differentiable in
, and the derivative satisfies:
: Condition 2
Conditions for a Well-Behaved Statistic: Second Definition
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:
- monotonicity. A uniformly monotone relation exists between s and ? for any fixed seed
– 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
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; - 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.
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 domain where the variable takes specifications. Facing a sample
, given a sampling mechanism
, with
scalar, for the random variable X, we have

The sampling mechanism
, of the statistic s, as a function ? of
with specifications in
, has an explaining function defined by the master equation:

for suitable seeds
and parameter ?
Example
For instance, for both the Bernoulli distribution with parameter p and the exponential distribution with parameter ? the statistic
is well-behaved. The satisfaction of the above three properties is straightforward when looking at both explaining functions:
if
, 0 otherwise in the case of the Bernoulli random variable, and
for the Exponential random variable, giving rise to statistics

and

Vice versa, in the case of X following a continuous uniform distribution on
the same statistics do not meet the second requirement. For instance, the observed sample
gives
. But the explaining function of this X is
. Hence a master equation
would produce with a U sample
and a solution
. This conflicts with the observed sample since the first observed value should result greater than the right extreme of the X range. The statistic
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.
Common distribution laws together with related sufficient and well-behaved statistics.Distribution | Definition of density function | Sufficient/Well-behaved statistic |
---|
Uniform discrete |  |  |
Bernoulli |  |  |
Binomial |  |  |
Geometric |  |  |
Poisson |  |  |
Uniform continuous |  |  |
Negative exponential |  |  |
Pareto |  |  |
Gaussian |  |  |
Gamma |  |  |
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