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Practice Problems Stats 426 Here are some problems to try in addition to homework, quiz, and midterm problems. 1) Let Y1 , Y2 , . . . Yn be iid from the density function f (y| θ) =

y3 e−y/θ 6 θ4

y>0

a) Find the MLE θˆ for θ. b) State a result about the asymptotic distribution of the MLE as n → ∞ and find it. 2) Let X1 , X2 (n = 2) be independent and identically distributed from Exp(1/2). That is fX (x) =

1 −x/2 e 2

x>0

Without any calculations, use what you know about the chi-square and F distributions to find the distribution of X1 /X2 . 3) Recall the Hardy-Weinberg problem described in your text (page 273-274). The multinomial distribution for random variables Y1 , Y2 , Y3 (can extend to more than 3) is given by P (Y1 = y1 , Y2 = y2 , Y3 = y3 ) =

n! y y y p 1p 2 p 3 y1 !y2 !y3 ! 1 2 3

where y1 + y2 + y3 = n and the parameters p1 , p2 , p3 are subject to the constraint p1 + p2 + p3 = 1. This distribution is an extension of the binomial distribution. In fact, the distribution of each Yi , i = 1, 2, 3 is binomial. So, for example, E[Y1 ] = np1 and V ar(Y1 ) = np1 (1 − p1 ). If gene frequencies are in equilibrium, the genotypes AA, Aa, and aa occur in a population with proportions p1 = (1 − θ)2 ,

p2 = 2θ(1 − θ) , and p3 = θ 2

according to Hardy-Weinberg law. a) Using the multinomial as the likelihood and a prior distribution of θ ∼ Beta(10, 10), find the Bayes estimator of θ. That is find E[θ|y]. b) Is the Bayes estimator unbiased? 4) Consider the following probability model: Likelihood : Y1 , . . . , Yn independent and identically distributed Poisson(θ ) Prior : θ ∼ Gamma(α, λ) ,

α , λ fixed

a) Find the posterior distribution p(θ | y). b) Find E[θ | y] and V ar(θ | y). c) Find the marginal distribution p(y). (This distribution will be conditional on α and λ). 5) Consider the simple linear regression model yi = α + β xi + ϵi ,

ϵi ∼ N (0, σ 2 )

i = 1, . . . , n

ˆ be the MLEs of α and β, respectively. Let yˆi = α ˆ xi Let y¯ be the mean of the yi , and let α ˆ and β ˆ+β be the fitted values, and let ei = yi − yˆi be the residuals. ˆ a) What is Cov(¯ y, β) ˆ b) What is Cov(ˆ α, β) c) Show that d) Show that e) Show that

∑n

i=1 ei

=0

∑n

i=1 xi ei

=0

∑n

ˆi ei i=1 y

= 0.

ˆ and V ar( β)? ˆ Can you calculate E[ β] 6) The data below shows the average weight of varsity football players at the University of Texas for selected years. Fit a linear model of the form (use R code given in Lecture Notes, no need to do by hand) yi = α + βxi + ϵi ϵi ∼ N (0, σ 2 ) Year(x) Weight (y)

1905 164

1919 163

1932 181

1945 192

1955 194

1965 199

a) Find a 95% confidence interval for β . b) Test the hypothesis H0 : β = 0 versus H1 : β =  0 at α = 0.05. c) Make the ANOVA table using the anova command in R and know what each number in the table means. What is the estimate for σ 2 ? 7) Consider the intercept-only logistic regression model yi ∼ Binomial(ni , p) log

(

p 1−p

)

yi independent

=α

a) Find the MLE for α. b) Find the Fisher Information [

∂2ℓ I(α) = −E ∂α 2 How would you estimate V ar(ˆ α) ?

]

i = 1, . . . , n...