Kurtosis
compute kurtosis of a univariate distribution
kurtosis(
x,
na.rm = FALSE,
method = c("excess", "moment", "fisher", "sample", "sample_excess"),
...
)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 |
... |
arguments to be passed. |
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
Diethelm Wuertz, Matthieu Lestel
Carl Bacon, Practical portfolio performance measurement and attribution, second edition 2008 p.84-85
## 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"))
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