Statement: With my signature I confirm that the solutions are the product of my own work. Name: Signature:.

Transcription

1 MATHEMATICAL STATISTICS Take-home final examination February 1 st -February 8 th, 019 Instructions You do not need to edit the solutions Just make sure the handwriting is legible The final solutions should be your work The deadline for completion is February 8th, 019 by 4pm Turn in your solutions to Petra Vranješ or Lidija Urek or send me a scanned version of solutions For any questions contact me by or call me at Statement: With my signature I confirm that the solutions are the product of my own work Name: Signature:

2 1 (5) In a population od size N there are three types of units: A, B and C We would like to estimate the proportions a, b and c of these units When a unit is chosen it does not necessarily respond truthfully but chooses one of the three types at random If a unit is of type X {A, B, C} it will respond that it is of type Y {A, B, C} with probability p XY Assume that the probabilities p XY are known We choose a simple random sample of size n Assume that the units choose their responses independently of each other and independently of the sampling procedure a Let N X be the number of units in the sample of type X and M X the number of units in the sample who respond X for X {A, B, C} Compute E(M X N A, N B, N C ) b Suggest unbiased estimates for a, b and c When is it possible to estimate the proportions? c Compute cov(m X, M Y N A, N B, N C ) for X, Y {A, B, C} d Give standard errors for the unbiased estimates of a, b and c

3 (5) Suppose {p(x, θ), θ Θ R k } is a (regular) family of distributions Define the vector valued score function s as the column vector with components s(x, θ) = θ log(p(x, θ)) = grad(log(p(x, θ)) and the Fisher information matrix as I(θ) = var(s) Remark: If p(x, θ) = 0 define log (p(x, θ)) = 0 a Let t(x) be an unbiased estimator of θ based on the likelihood function, ie Prove that Deduce that cov(s, t) = I E θ (t(x)) = θ E(s) = 0 and E(st T ) = I Remark: Make liberal assumptions about interchanging integration and differentiation b Let a, c be two arbitrary k dimensional vectors Prove that corr ( a T t, c T s ) = (a T c) a T var(t)a c T I(θ)c The correlation coefficient squared is always less or equal 1 Maximize the expression for the correlation coefficient over c and deduce the Rao-Cramér inequality 3

4 3 (5) Assume the data pairs (x 1, y 1 ),, (x n, y n ) are an iid sample from the bivariate normal distribution with parameters µ1 σ11 σ µ = and Σ = 1 µ σ 1 σ Assume the matrix Σ is invertible We would like to test the hypothesis H 0 : Σ has eigenvalues λ and λ for some λ > 0 versus H 1 : for the eigenvalues λ and µ of Σ we have λ/µ / {, 1/} a Find the maximum likelihood estimators of the parameters in the unrestricted case b Show that every symmetric matrix with eigenvalues λ and λ for λ > 0 is of the form λ(1 + a ) λab a b λ 0 a b λab λ(1 + b = ) b a 0 λ b a for a, b such that a + b = 1 c Find explicitly the likelihood ratio test for the above testing problem Hint: under the assumption α, β > 0 and αγ β > 0 the minimum of the function f(x, y) = αx + βxy + γy subject to the side condition x + y = 1 is α + γ (α γ) + 4β d What can you say about the approximate distribution of the test statistic? 4

5 4 (5) Assume the correct regression model is Y = Xβ + ɛ for E(ɛ) = 0 and var(ɛ) = σ I Assume the matrix X of dimensions n m with m < n has full rank Denote by ˆβ the ordinary least squares estimator of β Assume as known that the upper left corner of the inverse of Σ11 Σ 1, Σ 1 Σ is and the lower right corner is Σ 11 = Σ Σ 1 11 Σ 1 ( Σ Σ 1 Σ 1 11 Σ 1 Σ1 Σ 1 11, Σ = (Σ Σ 1 Σ 1 11 Σ 1 a Assume that we forget some independent variables and fit the regression model Y = X 1 β 1 + ɛ, where X = [X 1 ; X ] and E(ɛ ) = 0 and var(ɛ ) = σ I Write β = ( β1 β ) Assuming the wrong model we estimate β 1 by ˆβ 1 = ( X T 1 X 1 X T 1 Y Let ˆβ 1 be the best unbiased linear estimator of β 1 in the correct model Show that var( ˆβ 1 ) var( ˆβ 1) = σ AB 1 A T, where A = ( X T 1 X 1 X T 1 X and B = X T X X T X 1 A b Show that the matrix AB 1 A T is positive semi-definite This means that ˆβ 1 has smaller variance than ˆβ 1 Why is this not in contradiction with the Gauss- Markov theorem? Explain your answer Hint: all the minors of a positive semi-definite matrix are positive semi-definite 5