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stat231 method_of_moments

# stat231 method_of_moments - The Method of Moments...

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Unformatted text preview: The Method of Moments Deﬁnitions: • The k th moment of a random variable is µk = E [Y k ]. • Let X1, . . . , Xn be a random sample from a population. n 1 The k th sample moment is mk = Xik . n i=1 Method of Moments: Let t be the number of parameters of the population that are to be estimated. Choose as estimates those values of the parameters that are solutions of the equations µ1 = m 1 . . µt = m t 1 The Method of Moments Example: Let X1, . . . , Xn be a random sample from a uniform distribution on (0, θ). Use the method of moments to ﬁnd an estimator of θ. 2 The Method of Moments Example: Let X1, . . . , Xn be a random sample from a uniform distribution on (α, β ). Recall, the ﬁrst two moments for this uniform distribution are: β 1 E [X ] = x dx β−α α 1 = β−α α+β = 2 β β 2 α2 − 2 2 E [X ] = α 2 x2 1 dx β−α β 3 α3 − 3 3 1 = β−α α2 + αβ + β 2 = 3 3 Use the method of moments to ﬁnd estimators of α and β. 4 ...
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stat231 method_of_moments - The Method of Moments...

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