$𝜇$ -estimator

For a random sample ${𝑋 𝑗} 𝑛 𝑗 = 1$ , of identically and independently distributed random variables with mean $𝜇$ and variance $𝜎 2$ , the first Sample moment

―― 𝑋 𝑛 = 𝑀 1 = 1 𝑛 𝑛 \sum 𝑗 = 0 𝑋 𝑗

estimates the Expectation of the population since

𝔼 [―― 𝑋 𝑛] = 𝔼 [𝑋] V a r (―― 𝑋 𝑛) = 𝜎 2 𝑛

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.

#state/tidy | #SemBr | #lang/en

𝜇-estimator

$𝜇$ -estimator