Feb. 10th, 2005

STAT 543 Homework 4 Solution

1. Problem 2.1.2 Consider n systems with failure times X 1 ,..., X n assumed to be independent and identically distributed with exponential, Σ(λ ) , distributions. (a) Find the method of moments estimate of λ based on the first moment. (b) Find the method of moments estimate of λ based on the second moment. (c) Combine your answers to (a) and (b) to get a method of moment estimate of λ based on the first two moments. (d) Find the method of moments estimate of the probability P( X 1 ≥ 1) that one system will last at least a month. Solution: Since X 1 ,..., X n i.i.d Ε(λ ) , then ∼

f ( x1 ) = λ e − λ x1 , x > 0

, and

E ( X 1 ) = ∫ xλe − λ x dx =

0 ∞

Γ(2)

λ

=

1

λ

2

, and

E ( X 12 ) = ∫ x 2 λ e − λ x dx =

. λ λ2 (a) The method of moments estimate of λ based on the second moment is 1 n λ= = n x ∑ xi 2

Γ(3)

=

(b) The method of moments estimate of λ based on the second moment is 2n λ= n ∑ i=1 xi2

i =1

(c) From λ 2 = 2λ 2 − λ 2 = µ 2 − µ12 , the method of moments estimate of λ based on the first two moments is 1 1 . λ= = 1 n 2 1 n 2 1 n ∑ xi − x 2 n ∑ i=1 xi − ( n ∑ i =1 xi )2 n i =1 (d) Since P( X 1 ≥ 1) = λ ∫ e− λ x dx = e− λ , then the method of moments estimate 1 ∞

of P( X 1 ≥ 1) that one system will last at least a month is ∑ xi − i −x P ( X 1 ≥ 1) = e = e n .

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STAT 543 Homework 4 Solution

Feb. 10th, 2005

Problem 2.1.3 Suppose that i.i.d . X 1 ,..., X n have a beta, β (α1 , α 2 ) distribution. Find the method of moments estimates of α = (α1 , α 2 ) based on the first two moments. Hint: See Problem B.2.5. Solution: Known that for Beta (α1 , α 2 ) distribution, there exist E( X ) =

α1 α1 + α 2

, and

E ( X 2 ) = (α +α )(α +α +1) 1 2 1 2

α1(α1+1)

(µ2 − µ1 )µ1 ⎧ ⎧ µ = α1 1 ⎪ α1 = µ 2 − µ ⎪ α1 + α 2 2 ⎪ ⎪ 1 ⇒⎨ ⎨ α1(α1 + 1) ⎪ µ2 = ⎪α = (µ1 − µ2 )(µ1 − 1) (α1 + α 2 )(α1 + α 2 + 1) ⎪ ⎪ 2 µ 12 − µ2 ⎩ ⎩ 1 n 1 n 2 Then, plug in ∑ x i , and ∑ x i for µ1 and µ2 , we get the method of moment estimate n 1 n 1 for α = (α1, α 2 ) .as 1 1 1 ⎧ ( ∑ i x i2 − ∑ i x i ) ∑ i x i ⎪ n n n ⎪ α1 = 1 1 2 ( ∑ i x i ) − ∑ i x i2 ⎪ ⎪ n n . ⎨ 1 1 1 ⎪ ( ∑ i x i 1 − ∑ i x i2 )( ∑ i x i − 1) n n ⎪α = n 1 1 ⎪ 2 2 ( ∑ i x i ) − ∑ i x i2 ⎪ n n ⎩

, then by

Problem 2.1.11 In Example 2.1.2 with X ∼ Γ (α , λ ) , find the method of moments estimate based ˆ ˆ on µ1 and µ3 . Hint: See Problem B.2.4. Solution:

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STAT 543 Homework 4 Solution For gamma (α , λ ) distribution, f (x ) =

Feb. 10th, 2005

α ⎧ Ex = µ1 = ⎪ λ ⎪ ⇒⎨ Γ (α + 3 ) α (α + 1)(α + 2) ⎪Ex 3 = µ3 = = ⎪ Γ (α ) λ 3 λ3 ⎩ ⇒ −3µ13 + µ16 + 8 µ13 µ3 α = 2(µ13 − µ3 )

λ α α −1 − λx x e ,x > 0 Γ (α )

2 −3µ1 + µ14 + 8 µ1 µ3 2(µ13 − µ3 ) 1 n 1 n 3 By plug in ∑ x i and ∑ x i for µ1 and µ3 , we get the MOM estimates for α and λ as n 1 n 1

, and λ =

⎛1 n ⎞ ⎛1 n ⎞ ⎛1 n ⎞ 1 n −3 ⎜ ∑ x i ⎟ + ⎜ ∑ x i ⎟ + 8 ⎜ ∑ x i ⎟ ( ∑ x i3 ) ⎝n 1 ⎠ ⎝n 1 ⎠ ⎝n 1 ⎠ n 1 ˆ α = 3 1 n ⎛1 n ⎞ 2(⎜ ∑ x i ⎟ − ∑ x i3 ) n 1 ⎝n 1 ⎠ , and

3

6

3

⎛1 n ⎞ ⎛1 n ⎞ ⎛ 1 n ⎞⎛ 1 n 3 ⎞ −3 ⎜ ∑ x i ⎟ + ⎜ ∑ x i ⎟ + 8 ⎜ ∑ x i ⎟ ⎜ ∑ x i ⎟ ⎝n 1 ⎠ ⎝n 1 ⎠ ⎝n 1 ⎠⎝n 1 ⎠ ˆ= λ . 3 ⎛1 n ⎞ ⎛1 n 3⎞ 2(⎜ ∑ x i ⎟ − ⎜ ∑ x i ⎟) ⎝n 1 ⎠ ⎝n 1 ⎠

2

4

Problem 2.2.10 Let X 1 ,..., X n denote a sample from a population with one of the following densities or frequency functions. Find the MLE of θ . (a) f ( x, θ ) = θ e −θ x , x ≥ 0;θ > 0. (Exponential density) (b) f ( x, θ ) = θ cθ x − (θ +1) , x ≥ c; c constant>0; θ > 0. (Pareto density) (c) Solution: (a).

−θ x f (x | θ ) = θ ne ∑ i = L(θ )

(θ ) = n log(θ ) − θ ∑ x i

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STAT 543 Homework 4 Solution

Feb. 10th, 2005

d n ˆ ∑ xi = − ∑ xi = 0 ⇒ θ = dθ θ n n ˆ Check that ′′ = − 2 < 0 , which indicates that θ reaches the maximum of . '

=

ˆ Hence, θ =

∑x

n

θ

i

is the MLE of θ .

f (x | θ ) = θ nc nθ ∏ x i

−(θ +1)

(b).

(θ ) = n log θ + nθ log c − (θ + 1)∑ log x i

1

'

n

=

d n = + n log c − ∑...