# kurtosis {PerformanceAnalytics}

### Description

compute kurtosis of a univariate distribution

### Usage

kurtosis(x, na.rm = FALSE, method = c("excess", "moment", "fisher", "sample", "sample_excess"), ...)

### Arguments

- x
- a numeric vector or object.
- na.rm
- a logical. Should missing values be removed?
- method
- a character string which specifies the method of computation. These are either
`"moment"`

,`"fisher"`

, or`"excess"`

. If`"excess"`

is selected, then the value of the kurtosis is computed by the`"moment"`

method and a value of 3 will be subtracted. The`"moment"`

method is based on the definitions of kurtosis for distributions; these forms should be used when resampling (bootstrap or jackknife). The`"fisher"`

method correspond to the usual "unbiased" definition of sample variance, although in the case of kurtosis exact unbiasedness is not possible. The`"sample"`

method gives the sample kurtosis of the distribution. - ...
- arguments to be passed.

### Details

This function was ported from the RMetrics package fUtilities to eliminate a dependency on fUtilties being loaded every time. This function is identical except for the addition of `checkData`

and additional labeling.

kurtosis(moment) = sum((x-mean(x))^4/var(x)^2)/length(x) kurtosis(excess) = sum((x-mean(x))^4/var(x)^2)/length(x) - 3 kurtosis(sample) = sum(((x-mean(x))/var(x))^4)*n*(n+1)/((n-1)*(n-2)*(n-3)) kurtosis (fisher) = ((n+1)*(n-1)*((sum(x^4)/n)/(sum(x^2)/n)^2 - (3*(n-1))/(n+1)))/((n-2)*(n-3)) kurtosis(sample excess) = sum(((x-mean(x))/var(x))^4)*n*(n+1)/((n-1)*(n-2)*(n-3)) - 3*(n-1)^2/((n-2)*(n-3))

where n is the number of return, \overline{r} is the mean of the return distribution, σ_P is its standard deviation and σ_{S_P} is its sample standard deviation

### References

Carl Bacon, *Practical portfolio performance measurement and attribution*, second edition 2008 p.84-85

### See Also

`skewness`

.

### Examples

## mean - ## var - # Mean, Variance: r = rnorm(100) mean(r) var(r) ## kurtosis - kurtosis(r) data(managers) kurtosis(managers[,1:8]) data(portfolio_bacon) print(kurtosis(portfolio_bacon[,1], method="sample")) #expected 3.03 print(kurtosis(portfolio_bacon[,1], method="sample_excess")) #expected -0.41 print(kurtosis(managers['1996'], method="sample")) print(kurtosis(managers['1996',1], method="sample"))

Documentation reproduced from package PerformanceAnalytics, version 1.4.3541. License: GPL-2 | GPL-3