Sample moment

Let ${𝑋 𝑗} 𝑛 𝑗 = 1$ be identically and independently distributed real random variables. The $𝑘$ th sample moment is #m/def/prob

𝑀 𝑘 = 1 𝑛 𝑛 \sum 𝑗 = 1 (𝑋 𝑗) 𝑘

Note that $𝔼 [𝑀 𝑘] = 𝜇 𝑘$ whence $𝑀 𝑘$ is an estimator of the $𝑘$ th moment by the Law of large numbers. The first sample moment, $𝑀 1 = ―― 𝑋 𝑛$ is the 𝜇-estimator.

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