--
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. Variance. Relations with the raw moments
A
central moment
is a moment taken about the
mean [
2] :
By definition the second central moment
is the
variance
which is usually denoted as
[
10]:
The square root of the variance is called [
10]
the
standard deviation :
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
Since the
is defined to be an unbiased estimator for the
, one has
, and then eqs.(1.3.3) give for the
expectation values :
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
2.6 Sample variance
The sample variance is the second central moment
(2.4.1):
where
is the sample mean (2.3.1) .
It is commonly written as [
9]
or sometimes
The square root of the sample variance is called
the
sample standard deviation [
10]
Another widespread definition for the sample standard deviation is
where
By definition the
statistic is an alternative name
for the
statistic defined above, see eq.(2.5.4b):
According to eq.(2.4.3b), the expectation values for the variance and
the
statistic are
Surely, the eq.(2.6.7) agrees with the eq.(2.5.5b).
The variances are (eq.(4) in [
11])
Eq.(23) in [
11] gives a usefull expression for the
:
2.7 Unbiased estimators for and
According to eq.(2.5.5a-c) the unbiased estimators for
,
and
are given by
,
and (see eq.(2.5.4c))
,
respectively.
It is also important to have estimators for
and
.
Eqs. (2.4.3d) and (2.6.10) can be combined into a matrix equation
Solving this system of equations relative to
and
gives
Removing the averaging brackets around
and
at the right hand side of the equations, one gets the estimators
at the left hand side:
2.8 Unbiased estimator for
Replacing
and
in the expression (2.6.9)
for
by their estimators (2.7.4-5), gives
the estimator for
:
As
, eq.(2.6.6), the same result (2.8.1)
can be obtained using formula (2.5.7) by plugging into it the
expressions (2.5.4b,c) for
.
References
[1] Weisstein, Eric W. "Raw Moment." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/RawMoment.html
,
pdf
[2] Weisstein, Eric W. "Central Moment."
http://mathworld.wolfram.com/CentralMoment.html
,
pdf
[3] Weisstein, Eric W. "Cumulant."
http://mathworld.wolfram.com/Cumulant.html
,
pdf
[4] Weisstein, Eric W. "Sample."
http://mathworld.wolfram.com/Sample.html
,
pdf
[5] Weisstein, Eric W. "Sample Raw Moment."
http://mathworld.wolfram.com/SampleRawMoment.html
,
pdf
[6] Weisstein, Eric W. "Sample Mean."
http://mathworld.wolfram.com/SampleMean.html
,
pdf
[7] Weisstein, Eric W. "Sample Central Moment."
http://mathworld.wolfram.com/SampleCentralMoment.html
,
pdf
[8] Weisstein, Eric W. "k-Statistic."
http://mathworld.wolfram.com/k-Statistic.html
,
pdf
[9] Weisstein, Eric W. "Sample Variance."
http://mathworld.wolfram.com/SampleVariance.html
,
pdf
[10] Weisstein, Eric W. "Standard Deviation."
http://mathworld.wolfram.com/StandardDeviation.html
,
pdf
[11] Weisstein, Eric W. "Sample Variance Distribution."
http://mathworld.wolfram.com/SampleVarianceDistribution.html
,
pdf