CS代考程序代写 University of California, Los Angeles Department of Statistics

University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Quiz 7
Let Y1, . . . , Yn be random variables with E(Yi) = μ, var(Yi) = σ2, and cov(Yi, Yj ) = ρσ2. The variance covariance matrixisoftheform(a−b)I+bJ,wherea=1,b=ρ,J=11′. InourmodelΣ=σ2[(1−ρ)I+ρJ]. Theinverseof
this special matrix can be obtained as follows: Σ−1 = 1 􏰉I − ρ J􏰊. Suppose the estimator of μ is given σ2 (1−ρ) 1+(n−1)ρ)
Statistics 100B
EXERCISE 1
by μˆ = 1′Σ−1Y . Is μˆ unbiased estimator of μ? Show that var(μˆ) = 1 1′Σ−11 1′Σ−11
variance covariance matrix given above. Explain why ρ > − 1 . n−1
EXERCISE 2
and simplify it using the inverse of the
Let Y1, Y2, . . . , Yn independent random variables, and let Yi ∼ N(iθ, iσ), i.e. E(Yi) = iθ and var(Yi) = i2σ2, for i = 1, 2, . . . , n. Find the maximum likelihood estimator of θ. Is this estimator efficient estimator of θ?
EXERCISE 3
Consider the two samples X1,…,Xn i.i.d. N(μ1,σ) and Y1,…,Ym i.i.d. N(μ2,σ). If μ1 = μ2 = μ find the MLEs for μ and σ2.
EXERCISE 4
Suppose Yi = β1xi +εi. The xi’s are not random and ε1, . . . , εn are independent with E(εi) = 0, var(εi) = σ2. Assume also that εi ∼ N(0,σ). Find βˆ1 and σˆ2, the maximum likelihood estimates of β1 and σ2. Find the expected value and variance of βˆ1 and its distribution.
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