In statistics, the method of moments is a method of estimation of population parameters. The same principle is used to derive higher moments like skewness and kurtosis. It starts by expressing the population moments (i.e., the expected values of powers of the random variable under consideration) as functions of the parameters of interest. Those expressions are then set equal to the sample moments. The number of such equations is the same as the number of parameters to be estimated. Those equations are then solved for the parameters of interest. The solutions are estimates of those parameters. The method of moments was first introduced by Karl Pearson in 1895.
Method Suppose that the parameter θ {\displaystyle \theta } = ( θ 1 , θ 2 , … , θ k {\displaystyle \theta _{1},\theta _{2},\dots ,\theta _{k}} ) characterizes the distribution f W ( w ; θ ) {\displaystyle f_{W}(w;\theta )} of the random variable W {\displaystyle W} . Suppose the first k {\displaystyle k} moments of the true distribution (the "population moments") can be expressed as functions of the θ {\displaystyle \theta } s:
m 1 ≡ E [ W ] = g 1 ( θ 1 , θ 2 , … , θ k ) , m 2 ≡ E [ W 2 ] = g 2 ( θ 1 , θ 2 , … , θ k ) , ⋮ m k ≡ E [ W k ] = g k ( θ 1 , θ 2 , … , θ k ) . {\displaystyle {\begin{aligned}m_{1}&\equiv \operatorname {E} [W]=g_{1}(\theta _{1},\theta _{2},\ldots ,\theta _{k}),\\[4pt]m_{2}&\equiv \operatorname {E} [W^{2}]=g_{2}(\theta _{1},\theta _{2},\ldots ,\theta _{k}),\\&\,\,\,\vdots \\m_{k}&\equiv \operatorname {E} [W^{k}]=g_{k}(\theta _{1},\theta _{2},\ldots ,\theta _{k}).\end{aligned}}}
Suppose a sample of size n {\displaystyle n} is drawn, resulting in the values w 1 , … , w n {\displaystyle w_{1},\dots ,w_{n}} . For j = 1 , … , k {\displaystyle j=1,\dots ,k} , let
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