Closed form expression for the distribution of the sample kurtosis of Gaussian distribution

Question

Is there a closed-form expression for the distribution of the Sample Kurtosis of data sampled from Gaussian distribution? i.e.,

$P(\hat{K}<a)$ where $\hat{K}$ is the sample kurtosis.

Answer 1

The exact sampling distribution is tricky to derive; there have been the first few moments (dating back to 1929), various approximations (dating back to the early 1960s), and tables, often based on simulation (dating back to the 1960s).

To be more specific:

Fisher (1929) gives moments of the sampling distribution of the skewness and kurtosis in normal samples, and Pearson (1930) (also) gives the first four moments of the sampling distribution of the skewness and kurtosis and proposes tests based on them.

So for example$^*$:

$E(b_2)=\frac{3(n-1)}{n+1}$

$\text{Var}(b_2)=\frac{24n(n-2)(n-3)}{(n+1)^2(n+3)(n+5)}$

The skewness of $b_2$ is $\frac{216}{n}(1-\frac{29}{n}+\frac{519}{n^2}-\frac{7637}{n^3}+\ldots)$

The excess kurtosis of $b_2$ is $\frac{540}{n}-\frac{20196}{n^2}+\frac{470412}{n^3}+\ldots$.

* Beware - the values for the moments and so on depend on the exact definition of the sample kurtosis being used. If you see a different formula for $E(b_2)$ or $\text{Var}(b_2)$, for example, it will generally be because of a slightly different definition of sample kurtosis.

In this case, the above formulas should apply to $b_2=n\frac{\sum_i (X_i-\bar X)^4}{(\sum_i (X_i-\bar X)^2)^2}$.

Pearson (1963) discusses approximating the sampling distribution of kurtosis in normal samples by a Pearson type IV or a Johnson $S_U$ distribution (doubtless the reason the first four moments were given three decades earlier was in large part to make use of the Pearson family possible).

Pearson (1965) gives tables for percentiles of kurtosis for some values of $n$.

D'Agostino and Tietjen (1971) give more extensive tables of percentiles for kurtosis.

D'Agostino and Pearson (1973) give graphs of percentage points of kurtosis which cover a more extensive range of cases again.

Fisher, R. A. (1929),
"Moments and Product Moments of Sampling Distributions,"
Proceedings of the London Mathematical Society, Series 2, 30: 199-238.

Pearson, E.S., (1930)
"A further development of tests for normality,"
Biometrika, 22 (1-2), 239-249.

Pearson, E.S. (1963)
"Some problems arising in approximating to probability distributions, using moments,"
Biometrika, 50, 95-112

Pearson, E.S. (1965)
"Tables of percentage points of $\sqrt{b_1}$ and $b_2$ in normal samples: A rounding off,"
Biometrika, 52, 282-285

D'Agostino, R.B. and Tietjen, G.L. (1971),
"Simulation probability points of $b_2$ for small samples,"
Biometrika, 58, 669-672.

D'Agostino, R.B., and Pearson, E.S. (1973),
"Tests for departure from normality. Empirical results for the distribution of $b_2$ and $\sqrt{b_1}$,"
Biometrika, 60, 613-622.

Answer 2

The Sample Kurtosis from a normal sample, is approximately distributed as a zero-mean normal with variance $\approx 24/n$, where $n$ is the sample size (naturally, the larger $n$ the better the approximation. More complicated expressions for the variance can be found in the wikipedia page). For Gaussian samples of small size (<40), percentiles have been derived in this paper: Lacher, D. A. (1989). Sampling distribution of skewness and kurtosis. Clinical chemistry, 35(2), 330-331.