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I know that this question is a duplicate of this question but it was not completely answered.

It is clear that kurtosis does not give a measure of 'peakedness' of a data distribution. Then which type of parameter will affect and measure the 'peakedness'?

Note: I have found 1 that defines "relative peakedness".

A random variable $X$ at point $a$ is "more peaked" than another random variable $Y$ at point $b$ when: $$P(|X−a|\geq h)\leq P(|Y−b|\geq h) \forall h\geq 0$$

1 Birnbaum, Z. W. "On random variables with comparable peakedness." The Annals of Mathematical Statistics 19.1 (1948): 76-81.

MM Khan
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  • The Birnbaum definition is kind of silly in my view. Let X ~ N(0,2) and Y ~ N(0,1). By the Birnbaum definition, X is "more peaked" than Y. A more rationale definition of peakedness would require the distributions to have a common scale (see my comment below), and then it is unclear what "peakedness" actually means. But I really am curious as to why you care about "peakedness"? What is its relevance? – BigBendRegion Oct 16 '18 at 19:32
  • also check out Google Scholar's list of papers related to Birnbaum (1948): https://scholar.google.com/scholar?q=related:T4yf5avGSqMJ:scholar.google.com/&scioq=&hl=en&as_sdt=0,5 – Felipe G. Nievinski Jun 21 '22 at 21:23

2 Answers2

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I think that the entropy of the probability distribution could be used as an inverse or opposite peakedness measure. Indeed, broad distributions have a high entropy, whereas peaked one have a small one (vanishes for Kroenecker delta). Continuous distributions can even have a negative differential entropy (increasingly thin normal distributions tend towards a Dirac delta with an entropy whose limit is $-\infty$). Note that multipeaked distributions would also have a low entropy if their peaks are thin.

Alexandre
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A suggestion: For continuously twice differentiable unimodal distributions, whose mode is in the interior range of the support (not on the boundary, like the exponential pdf), the second derivative of the distribution of the (suitably) standardized random variable evaluated at its mode might be a useful measure of peakedness. But who cares about "peakedness" anyway?

BigBendRegion
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