Kurtosis
Maximal entropy

Since mean μ {\displaystyle \mu } and covariance Σ {\displaystyle \Sigma } are the first two moments, it is natural to consider extension to higher moments. In fact, by Lagrange multiplier method, for any prescribed first n moments, if there exists some probability distribution of form p (x) ∝ e ∑ i a i x i + ∑ i j b i j x i x j + ⋯ + ∑ i 1 ⋯ i n x i 1 ⋯ x i n {\displaystyle p(x)\propto e^{\sum _{i}a_{i}x_{i}+\sum _{ij}b_{ij}x_{i}x_{j}+\cdots +\sum _{i_{1}\cdots i_{n}}x_{i_{1}}\cdots x_{i_{n}}}} that has the prescribed moments (if it is feasible), then it is the maximal entropy distribution under the given constraints.
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