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== Description == The first raw moment is the mean: ''μ = E[X]''. For discrete variables, this is calculated as ''Σ x P(x=X)''; for continuous variables, as ''∫ x f(x) dx'' The second central moment is the [[Statistics/Variance|variance]]: ''σ^2^ = E[(X - E[X])^2^] = E[(X - μ)^2^] = E(X^2^) - (E[X])^2^'' The derivation of this for discrete variables is: * ''Σ (x - μ)^2^ P(x=X)'' * ''Σ (x^2^ - 2μx + μ^2^) P(x=X)'' * ''Σ [x^2^ P(x=X)] - 2μ Σ [x P(x=X)] + μ^2^ Σ [P(x=X)]'' * ''[E[X^2^]] - 2μ [μ] + μ^2^ [1]'' * ''E[X^2^] - 2μ^2^ + μ^2^'' * ''E[X^2^] - μ^2^'' * ''E[X^2^] - (E[X])^2^'' The derivation of this for continuous variables is: * ''∫ (x - μ)^2^ f(x) dx'' * ''∫ (x^2^ - 2μx + μ^2^) f(x) dx'' * ''∫ [x^2^ f(x) dx] - 2μ ∫ [x f(x) dx] + μ^2^ ∫ [f(x) dx]'' * ''[E[X^2^]] - 2μ [μ] + μ^2^ [1]'' * ''E[X^2^] - 2μ^2^ + μ^2^'' * ''E[X^2^] - μ^2^'' * ''E[X^2^] - (E[X])^2^'' Through these derivations, it can be easily proven that (1) constants added to a variable do not affect variance, and (2) constant multipliers applied to a variable scale variance by their square. This is succinctly summarized as ''Var(aX + b) = a^2^ Var(X)'' The third central moment, skewness, measures lopsidedness of a distribution. The fourth central moment, kurtosis, measures the heaviness of the tails on a distribution. ---- |
Moments
Moments are measures of a distribution's shape and density.
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).