This preview shows page 1. Sign up to view the full content.
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...
View Full
Document
 '10
 PAAT

Click to edit the document details