1

The skewness of the difference between two independent random variables, X and Y, is given by:

$$\text{skewness}(X-Y) = \frac{\mu _3(X) - \mu _3(Y)}{\big(\mu _2(X) + \mu _2(Y)\big)^{3/2}}$$

But what is the expression for kurtosis(X - Y)?

user120911
  • 439
  • 1
    The standard deviation is a second order measure of the distribution of a data set. Skewness ( you mention in your question) is third order ... Kurtosis is fourth order ... read more about it here https://en.wikipedia.org/wiki/Kurtosis ... It fluctuates so sensitively that it does not appear to give a sensible measure of anything about the distribution – Donald Splutterwit May 05 '17 at 19:13
  • "It fluctuates so sensitively that it does not appear to give a sensible measure of anything about the distribution"

    But does it follow it has no definite expression?

     – user120911 May 05 '17 at 19:20

1 Answers1

1

If we write $Z=X-E[X]$, $W=Y-E[Y]$, then clearly $\mu_4(X-Y)=\mu_4(Z-W)$, while $E[Z]=E[W]=0$ and so $E[Z^n]=\mu_n$ and likewise for $W$. Then $$ \mu_4(Z-W) = E[(Z-W)^4] = E[Z^4]-4E[Z^3]E[W]+6E[Z^2]E[W^2]-4E[Z]E[W^3]+E[Z^4] \\ = E[Z^4]+6E[Z^2]E[W^2]+E[Z^4] \\ = \mu_4(Z) + \mu_4(W) + 6\mu_2(Z)\mu_2(W), $$ by independence and linearity of expectation. Hence, $$ \operatorname{kurt}{(X-Y)} = \operatorname{kurt}{(Z-W)} = \frac{\mu_4(Z) + \mu_4(W) + 6\mu_2(Z)\mu_2(W)}{(\mu_2(Z)+\mu_2(W))^2} \\ = \frac{\mu_4(X) + \mu_4(Y) + 6\mu_2(X)\mu_2(Y)}{(\mu_2(X)+\mu_2(Y))^2} $$

Chappers
  • 67,606