xn are iid observations from a uniform distribution

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Unformatted text preview: 2, . . . , p Example: Normal Distribution • Suppose X1, ...., Xn are i.i.d. observations from N(a, b2). We want to find the MoM estimators of a and b. n ￿ 1 m1 = Xi n i=1 n ￿ 1 2 m2 = (Xi ) n i=1 Estimated Sample Moments: First two moments of a normal ￿ 2￿ 2 N(a, b2) random variable: µ2 = E X = Var(X ) + (E[X ]) = b2 + a2 MoM estimators are computed based on the following system of equation: µ1 = m1 µ2 = m2 n ￿ 1 a= ˆ Xi n i=1 n ￿ ˆ2 + a2 = 1 and b ˆ (Xi )2 n i=1 µ1 = E [X ] = a 1 a= ˆ n n ￿ i=1 Xi ˆ2 = 1 and b n n ￿ i=1 (Xi ) − 2 ￿ 1 n n ￿ i=1 Xi ￿2 20 Example: Gamma Distribution • • Suppose X1, ...., Xn are i.i.d. observations from a gamma distribution with scale parameter λ and shape parameter α. We want to compute the MoM estimators of λ and α. Fact: If X has a gamma distribution with scale parameter λ and shape parameter α, then α ￿ α α −1 −λx E[X ] = λx e ￿ 2￿ if x ≥ 0, α α2 λ Γ(α) f (x) = E X = 2+ 2 α 0 otherwise. λ λ Var[X ] = 2 λ MoM estimators are computed based on the following system of equation: α ˆ 1 = m1 = ˆ n λ α ˆ α2 ˆ 1 + = m2 = ˆ ˆ n λ2 λ2 n ￿ i=1 n ￿ i=1 Xi (Xi ) 2 ￿n 2 ( i=1 Xi ) α= ˆ ￿n ￿n 2 2= 2−( m2 − m1 n i=1 Xi i=1 Xi ) ￿n n i=1 Xi m1 ˆ= λ ￿n ￿n 2 2= 2−( m2 − m1 n i=1 Xi i=1 Xi ) m2 1 21 Example: Potential Problems with MoM • Suppose X1, ...., Xn are i.i.d. observations from a uniform distribution over the interval [0,α]. We want to find the MoM estimator of α. n ￿ 1 m1 = Xi n i=1 α µ1 = E[X ] = 2 α ˆ = m1 2 ⇔ • • By LLN, as n →∞, the MoM estimator converges to 2 E[X] = α, which seems reasonable Major Flaw: Suppose the data are 0.2, 0.1, and 0.9. Then, MoM gives n ￿ 2 α = 2m1 = ˆ Xi n i=1 0.2 + 0.1 + 0.9 α = 2m1 = 2 × ˆ = 2 × 0.4 = 0.8 3 Our estimate of α is smaller than one of the sample values! • • In many cases, MoM estimators will have undesirable properties. So, MoM should NOT be the method of choice. Use MoM only when more sophisticated methods (like maxim...
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