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-- AlexanderFedotov - 04-Dec-2009

## Sample Statistics |

Let have a probability density function

A *raw moment*
(or just a *moment*, or a *crude moment* )
is a moment taken about 0 [1] :

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 *central* moments are expressed via the *raw* moments using
*binomial transform* [2] :

The *characteristic function* associated with the
*probability density function* is defined as a *Fourier transform*
[3] :

A *sample* is a subset of a population [4]

*Power sum* is the sum of th powers of the sample elements:

The th *sample raw moment* is defined as [5]

The *sample mean* [6] is defined by

The th *sample central moment* is defined as [7]

In terms of the population central moments, the expectation values of the first few sample central moments are

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

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

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]

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 :

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:

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 .

[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

Topic revision: r7 - 2010-02-01 - AlexanderFedotov

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