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Statistics

  

Cumulant

  

compute cumulants

 

Calling Sequence

Parameters

Description

Computation

Data Set Options

Random Variable Options

Examples

References

Compatibility

Calling Sequence

Cumulant(A, n, ds_options)

Cumulant(M, n, ds_options)

Cumulant(X, n, rv_options)

Parameters

A

-

data sample

M

-

Matrix data set

X

-

algebraic; random variable or distribution

n

-

algebraic; order

ds_options

-

(optional) equation(s) of the form option=value where option is one of ignore, or weights; specify options for computing the cumulant of a data set

rv_options

-

(optional) equation of the form numeric=value; specifies options for computing the cumulant of a random variable

Description

• 

The Cumulant function computes the cumulant of order n of the specified random variable or data set.

• 

The first parameter can be a data set (e.g., a Vector), a Matrix data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

• 

The second parameter can be any Maple expression.

Computation

• 

All computations involving data are performed in floating-point; therefore, all data provided must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

• 

By default, all computations involving random variables are performed symbolically (see option numeric below).

• 

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Data Set Options

  

The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

• 

ignore=truefalse -- This option controls how missing data is handled by the Cumulant command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Cumulant command will return undefined. If ignore=true all missing items in A will be ignored. The default value is false.

• 

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight 1.

Random Variable Options

  

The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

• 

numeric=truefalse -- By default, the cumulant is computed symbolically. To compute the cumulant numerically, specify the numeric or numeric = true option.

Examples

withStatistics:

Compute the third cumulant of the beta distribution with parameters 3 and 5.

Cumulant'Β'3,5,3

1768

(1)

Cumulant'Β'3,5,3,numeric

0.001302083333

(2)

Generate a random sample of size 100000 drawn from the above distribution and compute the third cumulant.

ASample'Β'3,5,105:

CumulantA,3

0.00134333467162695

(3)

Create a beta-distributed random variable Y and compute the third cumulant of 1/(Y+2).

YRandomVariable'Β'5,2:

Cumulant1Y+2,3,numeric

0.00001053303

(4)

Verify this using simulation.

CSample1Y+2,105:

CumulantC,3

0.0000106411152316840

(5)

Compute the cumulant of a weighted data set.

Vseqi,i=57..77,undefined:

W2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5:

CumulantV,4,weights=W

HFloatundefined

(6)

CumulantV,4,weights=W,ignore=true

6.34287269040942

(7)

Consider the following Matrix data set.

MMatrix3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007

M:=31130114694415271273683907884642878964844995128007

(8)

We compute the second cumulant of each column.

CumulantM,2

0.55999999999999955998.63999999972.57876120640001108

(9)

References

  

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

Compatibility

• 

The M parameter was introduced in Maple 16.

• 

For more information on Maple 16 changes, see Updates in Maple 16.

See Also

Statistics

Statistics[Computation]

Statistics[DescriptiveStatistics]

Statistics[Distributions]

Statistics[ExpectedValue]

Statistics[RandomVariables]

 


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