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MomentEvaluate
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MomentEvaluate

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MomentEvaluate[mexpr,dist]

evaluates formal moments in the moment expression mexpr on the distribution dist.

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MomentEvaluate[mexpr,list]

evaluates formal moments and formal sample moments in mexpr on the data list.

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MomentEvaluate[mexpr,dist,list]

evaluates formal moments on the distribution dist and formal sample moments on the data list.

Details

  • A moment expression is an expression involving formal moments and formal sample moments.
  • A formal moment is an expression of the form:
  • Moment[r]formal r^(th) moment
    CentralMoment[r]formal r^(th) central moment
    FactorialMoment[r]formal r^(th) factorial moment
    Cumulant[r]formal r^(th) cumulant
  • A formal sample moment is an expression of the form:
  • PowerSymmetricPolynomial[r]formal r^(th) power symmetric polynomial
    AugmentedSymmetricPolynomial[{r1,r2,…}]formal {r1,r2,…} augmented symmetric polynomial
  • For a sample moment expression PowerSymmetricPolynomial[0] is taken to be the length of the list of data.
  • MomentEvaluate[mexpr,…,n] assumes that n is taken to be the length of the list of data.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Evaluate formal moments for a univariate distribution:

In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-b5k4e6
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b0kaa1amqc-fnlqys
Out[3]=3

Evaluate formal moments for a multivariate distribution:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-c426bo
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-dpoy4m
Out[2]=2

Evaluate sample formal moments for data:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-gicn7x
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-nuzv75
Out[2]=2

Evaluate formal moments for data:

In[3]:=3
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https://wolfram.com/xid/0b0kaa1amqc-hn4ad
Out[3]=3

Scope  (6)Survey of the scope of standard use cases

Evaluate mixed univariate formal moment polynomial for a distribution:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-bcpnt1
Out[1]=1

Evaluate mixed multivariate formal moment polynomial for a distribution:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-jkvcy3
Out[1]=1

Evaluate polynomial in formal moments for data:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-jirjc0
In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-hdekd4
Out[2]=2

Compare with direct evaluation:

In[3]:=3
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https://wolfram.com/xid/0b0kaa1amqc-cnjsgl
Out[3]=3

Evaluate formal sample polynomial for data:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-gsjom
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-bcixw
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b0kaa1amqc-nex81
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0b0kaa1amqc-bv27dv
Out[4]=4

Evaluate formal sample polynomial for data with n being the sample size:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-q06dj
Out[1]=1

Evaluate an expression containing both formal moments and formal sample moments:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-bwyn04
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-bco5c0
In[3]:=3
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https://wolfram.com/xid/0b0kaa1amqc-95onm
Out[3]=3

Alternatively:

In[4]:=4
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https://wolfram.com/xid/0b0kaa1amqc-3bpi6
Out[4]=4

Generalizations & Extensions  (1)Generalized and extended use cases

Compute mean, variance, skewness, and excess kurtosis expressed in terms of Cumulant:

In[5]:=5
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https://wolfram.com/xid/0b0kaa1amqc-k4tg1x
Out[5]=5

Compare with direct evaluation:

In[6]:=6
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https://wolfram.com/xid/0b0kaa1amqc-twy63
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0b0kaa1amqc-rv114
Out[7]=7

Applications  (2)Sample problems that can be solved with this function

Find expectation of estimator on a sample from Bernoulli distribution:

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-fh61e2

Express the expectation of the estimator in terms of formal moments:

In[4]:=4
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https://wolfram.com/xid/0b0kaa1amqc-g3akce
Out[4]=4

Expectation of the estimator for a sample from Bernoulli distribution:

In[15]:=15
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https://wolfram.com/xid/0b0kaa1amqc-mtznm0
Out[15]=15

Variance of the sample estimator:

In[17]:=17
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https://wolfram.com/xid/0b0kaa1amqc-cs73qw
Out[17]=17

Construct sample and unbiased estimators for :

In[1]:=1
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https://wolfram.com/xid/0b0kaa1amqc-482za
In[2]:=2
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https://wolfram.com/xid/0b0kaa1amqc-wrdn

Accumulate statistics of these estimators on the same data:

In[3]:=3
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https://wolfram.com/xid/0b0kaa1amqc-jlsq38
In[4]:=4
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https://wolfram.com/xid/0b0kaa1amqc-cbxss4
In[5]:=5
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https://wolfram.com/xid/0b0kaa1amqc-gvfudq
In[6]:=6
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https://wolfram.com/xid/0b0kaa1amqc-gjwwji
In[7]:=7
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https://wolfram.com/xid/0b0kaa1amqc-o7yxsa
Out[7]=7

Compare the means of these statistics with population cumulant:

In[8]:=8
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https://wolfram.com/xid/0b0kaa1amqc-c1c68l
Out[8]=8

Find sampling population expectation of estimators for distribution dist:

In[9]:=9
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https://wolfram.com/xid/0b0kaa1amqc-os986l
Out[9]=9
In[10]:=10
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https://wolfram.com/xid/0b0kaa1amqc-c0ovi2
Out[10]=10

Find sampling population variance of estimators for distribution dist:

In[11]:=11
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https://wolfram.com/xid/0b0kaa1amqc-yd9dh
Out[11]=11
In[12]:=12
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https://wolfram.com/xid/0b0kaa1amqc-bpf7sl
Out[12]=12

Numerically evaluate expected variances for sample sizes used:

In[13]:=13
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https://wolfram.com/xid/0b0kaa1amqc-ly1drk
Out[13]=13

Compare to sample values:

In[14]:=14
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https://wolfram.com/xid/0b0kaa1amqc-5k4c
Out[14]=14

Properties & Relations  (1)Properties of the function, and connections to other functions

MomentEvaluate effectively evaluates a moment expression by evaluating its constituents:

In[13]:=13
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https://wolfram.com/xid/0b0kaa1amqc-stml3
In[14]:=14
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https://wolfram.com/xid/0b0kaa1amqc-ff0bp2
Out[14]=14
In[15]:=15
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https://wolfram.com/xid/0b0kaa1amqc-mggrqq
Out[15]=15
In[16]:=16
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https://wolfram.com/xid/0b0kaa1amqc-snx16
Out[16]=16
Wolfram Research (2010), MomentEvaluate, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentEvaluate.html.
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Wolfram Research (2010), MomentEvaluate, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentEvaluate.html.

Text

Wolfram Research (2010), MomentEvaluate, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentEvaluate.html.

✖
Wolfram Research (2010), MomentEvaluate, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentEvaluate.html.

CMS

Wolfram Language. 2010. "MomentEvaluate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MomentEvaluate.html.

✖
Wolfram Language. 2010. "MomentEvaluate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MomentEvaluate.html.

APA

Wolfram Language. (2010). MomentEvaluate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MomentEvaluate.html

✖
Wolfram Language. (2010). MomentEvaluate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MomentEvaluate.html

BibTeX

@misc{reference.wolfram_2025_momentevaluate, author="Wolfram Research", title="{MomentEvaluate}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentEvaluate.html}", note=[Accessed: 10-May-2025 ]}

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@misc{reference.wolfram_2025_momentevaluate, author="Wolfram Research", title="{MomentEvaluate}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentEvaluate.html}", note=[Accessed: 10-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_momentevaluate, organization={Wolfram Research}, title={MomentEvaluate}, year={2010}, url={https://reference.wolfram.com/language/ref/MomentEvaluate.html}, note=[Accessed: 10-May-2025 ]}

✖
@online{reference.wolfram_2025_momentevaluate, organization={Wolfram Research}, title={MomentEvaluate}, year={2010}, url={https://reference.wolfram.com/language/ref/MomentEvaluate.html}, note=[Accessed: 10-May-2025 ]}