= Moments =

'''Moments''' are measures of a distribution's shape and density.

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== Errors ==

Models generally assume that individual errors average to zero, i.e. the first moment of errors is zero: ''E[Ŷ - Y] = 0''. Nonetheless, higher order moments are important.

The '''mean square error''' ('''MSE''') is the second moment of the error: ''MSE(θ̂) = E[(θ̂ - E[θ̂])^2^]''. MSE can be decomposed into the variance of the estimator and bias: ''MSE(θ̂) = Var(θ̂) + Bias(θ̂,θ)^2^ = Var(θ̂) + (E[θ̂]-θ)^2^''.

Two important notes:
 * '''Bias''', i.e. ''E[θ̂] - θ'', is ''not'' the same as the first moment of errors.
 * If there is no bias, then MSE ''is'' the variance of the estimator: ''MSE(θ̂) = Var(θ̂)''.



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CategoryRicottone