The University of British Columbia
MATH 605G/STAT 410/STAT 538/STAT 9924
Practice Test 2
1. Suppose y is a random variable whose distribution is a member of the exponential family
having probability function
f (y; θ) = eω(yθ−b(θ))/φ+c(y,φ)
where
b(θ) = − log(−θ).
Assume that ω = 1 and φ = 0.5.
(a) Find the expected value, µ, of y, as a function of θ.
(b) Find the variance of y as a function of θ as well as a function of µ.
(c) Find the canonical link function g(µ).
(d) Suppose two covariates x1 and x2 were used to predict y using a generalized linear model
and the canonical link function. If the respective coefficient estimates were 0.5 and -0.25
(assume there is no intercept), calculate the expected value of y, when x1 = 3 and x2 = 7.
(e) Calculate the variance of y under the conditions of part (d).
2. Suppose y is a random variable whose probability density function is
(
f (y; λ) =
λ
e−yλ ,
1−e−λ
0,
y ∈ (−1, 1)
otherwise
(a) Show that the distribution of y is a member of the exponential family, identifying the
canonical parameter θ and the function b(θ).
(b) Find the expected value of y in terms of θ.
(c) Find the variance of y in terms of θ.
(d) Show that the canonical link function is implicitly defined by the equation
g(µ) = (g(µ) + 1 + µ)eg(µ) − µ.
(e) Use the above result to show that the variance of y can be expressed as a function of µ.
3. Suppose that y1 , . . . , yn are independent counts which follow a Poisson distribution with probability function
µy e−µ
f (y) =
y!
(a) Show that the Poisson distribution is a member of the exponential family, identifying the
canonical parameter, θ, in terms of µ.
(b) Write out the log likelihood function, `(µ).
(c) Find a formula for the maximum likelihood estimator for θ in terms of the y’s.
(d) Use the weak law of large numbers to show that
∂`
(µ)
∂µ
converges in probability to 0, as n → ∞.
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(e) Show that
∂ 2`
(µ)
∂µ2
converges in probability to −1/µ.
(f) Identify the approximate large sample distribution of the maximum likelihood estimator
for θ.
4. Suppose that y1 , . . . , yn are independent counts which follow Poisson distributions with probability function
µj yj e−µj
f (yj ) =
yj !
and suppose x1 , x2 , . . . , xn are values of a covariate that has been measured in correspondence
with the y’s.
(a) Supposing the canonical link function is used, write out an equation which relates the
expected value of yj to a linear function of xj . Do not include an intercept in the model.
(b) Show that the maximum likelihood estimator for the coefficient (β) of x in the above model
is the solution of
n
X
(yj − eβxj )xj = 0.
j=1
(c) On the other hand, show that one step of iteratively re-weighted least-squares, starting
from a guess βbk is accomplished by the formula
P
βb
k+1
=P
xj yj?
x2j eβbk xj
where y ? = yj − eβbk xj + xj βbk eβbk xj . Hint: start from minimizing
n
X
(yj − eβxj )2
j=1
eβbk xj
with respect to β, remembering to Taylor expand eβxj about βbk before carrying out the
minimization.
5. Consider the following output from the fitting of a generalized linear model to 9 independent
observations on a response variable y and a predictor x.
clotting.glm <- glm(y ~ x, family = Gamma(link = ’log’), data = clotting)
summary(clotting.glm)
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Call:
glm(formula = y ~ x, family = Gamma(link = "log"), data = clotting)
Deviance Residuals:
Min
1Q
Median
-0.39803 -0.31078 -0.18855
3Q
0.02734
2
Max
0.72388
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Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.205057
0.232677 18.072 3.93e-07 ***
x
-0.016098
0.004578 -3.516 0.00977 **
--Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
(Dispersion parameter for Gamma family taken to be 0.185469)
Null deviance: 3.5128
Residual deviance: 1.0642
AIC: 75.539
on 8
on 7
degrees of freedom
degrees of freedom
Number of Fisher Scoring iterations: 7
(a) On the basis of the output, write out the fitted model.
(b) Is there strong evidence that the coefficient of x is nonzero? Explain.
(c) Provide an estimate of the expected value of y when x = 50.
(d) In order to answer the above questions, have you made any assumptions that would need
to be checked? Explain.
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