CODEWITHSUNDEEP
kurtosis
Jinny
Jinny

Reputation: 1

Estimate the probability density of a given value if it belongs to a highly peaked multivariate dataset with high kurtosis (>100)

I have a dataset that have multiple variables with each of them heavily centered around zero to form a high peak. The kurtosis of each variable is more than 100.

What I want to estimate is the probability density of any given value if it belongs to the dataset. The most accessible distribution function I found currently is the multivariant Gaussian distribution. However, since my dataset is not is a normal shape and I am worried that it is inaccurate estimate the probability density using this function.

Does anyone have any good suggestions on which function to use to for this purpose?

Upvotes: 0

Views: 45

Answers (1)

BigBendRegion
BigBendRegion

Reputation: 220

You are repeating a common incorrect interpretation of kurtosis, namely, "peakedness," which contributes the confusion about what distribution to use.

Kurtosis does not measure "peakedness" at all. You can have a distribution with a perfectly flat peak, with a V-shaped peak, with a trimodal peak, with a wavy peak, or with any shape peak whatsoever, that has infinite kurtosis. And you can have a distribution with infinite peak than has negative (excess) kurtosis.

Instead, kurtosis is a measure of the tails (outlier potential) of the distribution, not the peak. The only reason people think that there is a "high peak" when there is high kurtosis is that the outliers stretch the horizontal scale of the histogram, making the data appear concentrated in a narrow vertical strip. But if you zoom in on the bulk of the data in that strip, the peak can have any shape whatsoever. Further, if you compare the height of your histogram of standardized data with the height of a corresponding standard normal, either can be higher, no matter what your data show. The "height" mythology was debunked around 1945 by Kaplansky.

For your data, you do not need a "peaked" distribution. Instead, you need a distribution that allows such extreme values as you have observed. Examples include mixture distributions, lognormal distributions, t distributions with small degrees of freedom, or multivariate versions of such, if that's what you need.

References:

Westfall, P.H. (2014). Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195.

(A simplified discussion of the above paper is given in the talk section of the Wikipedia entry on kurtosis.)

Upvotes: 0

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