chapter 1

Chapter 6: Point Estimation29.a. The joint pdf (likelihood function) is−λΣ( −θ)fn⎧λen= ⎨⎩ 0x ≥ θ ,..., x ≥θ( x x ; λ,θ )1,...,Notice thatx ix ≥ θ ,..., xn ≥ θ1otherwisemin ≥ ,1 niff ( x i) θΣ xi −θ= −λΣxi+ nand that − λ ( ) λθ .⎧λn exp( − λΣx i) exp ( nλθ) min ( x ) i≥θThus likelihood = ⎨0min ( x ) < θ⎩iConsider maximization wrt θ . Because the exponent n λθ is positive, increasing θwill increase the likelihood provided that min ( x i) ≥θ; if we make θ larger thanˆ = minmin ( x i) , the likelihood drops to 0. This implies that the mle of θ is ( )The log likelihood is now ln( λ)− λΣ( x −θˆ)and solving yieldsλˆθ .ni. Equating the derivative wrt λ to 0n n=Σ x −θˆΣx− n ˆ( ) θ= .iix iˆ = =10θiand Σx i= 55. 80 , so λ ˆ =. 20255.80 − 6.4=b. min ( x ) .64,⎛ n⎞y; n,p = ⎜ ⎟p1−p where⎝ y⎠y n−y30. The likelihood is ( ) ( )f24−λx−24λ( X ≥ 24) = 1−λedx =p = P ∫ eprinciplee−24λ0y= ⇒ λˆ= −ny[ ln ( )]n24= .0120. We knowp ˆ =ynfor n = 20, y = 15., so by the invarianceSupplementary Exercises⎛ X − µ ε ⎞ ⎛ X − µ − ε ⎞31. ( ) ( ) ( ) ⎟⎠P X − µ > ε = P X − µ > ε + P X − µ < −ε= P⎜> ⎟ + P⎜PZ