--
AlexanderFedotov - 04-Dec-2009
1 Population quantities
Let
have a probability density function
1.1 Moments = Raw Moments = Crude Moments. Mean.
A
raw moment
(or just a
moment, or a
crude moment )
is a moment taken about 0 [
1] :
By definition,
the first moment is the
mean of the distribution,
,
1.2 Central moments and their relations with the raw moment
A
central moment
is a moment taken about the
mean [
2] :
The
central moments are expressed via the
raw moments using
binomial transform [
2] :
In particular:
The
raw moments are expressed via the
central moments using
inverse binomial transform [
1]
(note
and
):
In particular:
1.3 Cumulants and their expressions via moments
The
characteristic function associated with the
probability density function is defined as a
Fourier transform
[
3] :
The cumulants
are then defined by
They can be expressed through
raw moments :
or in terms of
central moments :
where
is the
mean and
is the
variance.
2 Sample quantities
A
sample is a subset of a population [
4]
where
is the size of the sample.
2.1 Power sums
Power sum is the sum of
th powers of the sample elements:
2.2 Sample raw moments
The
th
sample raw moment is defined as [
5]
They are related to the
power sums (2.1.1) by
They are unbiased estimators of the population
raw moments (1.1.1) :
2.3 Sample mean
The
sample mean [
6] is defined by
It is equal to the sample
first moment (2.2.1),
It is an unbiased estimator for the population mean
(1.1.2)
2.4 Sample central moments
The
th
sample central moment is defined as [
7]
where
is the sample
mean .
The first few sample central moments are related to power sums
by
In terms of the population central moments, the expectation values of the first few sample central moments are
2.5 k-Statistics
The
th
k-statistic is [
8]
the unique symmetric unbiased estimator of the cumulant
(see. e.g. eq.(1.3.4)), i.e.,
is defined so that
In addition, the variance
is a minimum compared to all other unbiased estimators of
.
The k-statistics can be given in terms of the power sums (2.1.1)
Alternatively, they can be expressed via the sample mean
(2.3.1)
and central moments
(2.4.1) by
The variances of the first few k-statistics are given by
An
unbiased estimator for
is given by
In the special case of a
normal parent population,
an
unbiased estimator for
is given by
References
[1] Weisstein, Eric W. "Raw Moment." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/RawMoment.html
[2] Weisstein, Eric W. "Central Moment."
http://mathworld.wolfram.com/CentralMoment.html
[3] Weisstein, Eric W. "Cumulant."
http://mathworld.wolfram.com/Cumulant.html
[4] Weisstein, Eric W. "Sample."
http://mathworld.wolfram.com/Sample.html
[5] Weisstein, Eric W. "Sample Raw Moment."
http://mathworld.wolfram.com/SampleRawMoment.html
[6] Weisstein, Eric W. "Sample Mean."
http://mathworld.wolfram.com/SampleMean.html
[7] Weisstein, Eric W. "Sample Central Moment."
http://mathworld.wolfram.com/SampleCentralMoment.html
[8] Weisstein, Eric W. "k-Statistic."
http://mathworld.wolfram.com/k-Statistic.html