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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" [1] [2]) 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. [3]
Second Definition: The statistic is monotonic, well-defined, and locally sufficient. [4]
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
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:
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.
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 ?
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.
Distribution | Definition of density function | Sufficient/Well-behaved statistic |
---|---|---|
Uniform discrete | ||
Bernoulli | ||
Binomial | ||
Geometric | ||
Poisson | ||
Uniform continuous | ||
Negative exponential | ||
Pareto | ||
Gaussian | ||
Gamma |
{{
cite web}}
: Cite uses generic title (
help)
![]() | This article is written like a
personal reflection, personal essay, or argumentative essay that states a Wikipedia editor's personal feelings or presents an original argument about a topic. (September 2009) |
![]() | This article includes a list of general
references, but it lacks sufficient corresponding
inline citations. (November 2010) |
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" [1] [2]) 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. [3]
Second Definition: The statistic is monotonic, well-defined, and locally sufficient. [4]
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
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:
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.
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 ?
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.
Distribution | Definition of density function | Sufficient/Well-behaved statistic |
---|---|---|
Uniform discrete | ||
Bernoulli | ||
Binomial | ||
Geometric | ||
Poisson | ||
Uniform continuous | ||
Negative exponential | ||
Pareto | ||
Gaussian | ||
Gamma |
{{
cite web}}
: Cite uses generic title (
help)