Estimator

𝜇-estimator

For a random sample {𝑋𝑗}𝑛𝑗=1, of identically and independently distributed random variables with mean 𝜇 and variance 𝜎2, the first Sample moment

――𝑋𝑛=𝑀1=1𝑛𝑛∑𝑗=0𝑋𝑗

estimates the Expectation of the population since

𝔼⁡[――𝑋𝑛]=𝔼⁡[𝑋]Var⁡(――𝑋𝑛)=𝜎2𝑛

This confirms the intuition that the larger a sample, the closer its mean will get to the actual population mean. The distribution of the 𝜇-estimator is described by the Central limits theorem.

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