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Statistics 111
Quiz 11
1. Let X1 , . . . , Xn be a random sample with density f (x) = θcθ x−(θ+1) for x > c with c > 0 and θ > 0.
Find the MLE of θ.
The likelihood function is
f (x1 , . . . , xn ; θ) =
n
Y
−(θ+1)
θcθ xi
= θn cnθ
i=1
n
Y
−(θ+1)
xi
i=1
so the log-likelihood is
`(θ) = n ln θ + nθ ln c −
n
X
(θ + 1) ln xi .
i=1
Take the derivative of this wrt θ, set it to 0 and solve:
n
X
d`(θ)
n
ln xi = 0
= + n ln c −
dθ
θ
i=1
implies θ̂ = n/[
P
ln xi − n ln c].
2. Let 5, 6, 7 be an independent sample from a Poisson distribution with parameter λ.
The likelihood function for a Poisson is
f (x1 , . . . , xn ; λ) =
n
Y
λ xi
i=1
xi !
P
exp(−λ) = exp(−nλ)λ
xi
Y
(xi !)−1 .
Take the log, and get
−nλ + (
X
xi ) ln λ + stuff that doesn’t depend on λ
so, on diffentiating wrt λ and setting to 0, we see
0 = −n + λ−1
X
xi .
Solving gives λ̂ = X̄.
6 What is the MLE of λ?
√
6 What is the MLE of the standard deviation of this Poisson?
The standard deviation of a Poisson is the square root of lambda, so the MLE of the square root is
the square root of the MLE.
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3. Let X1 , X2 , X3 be independent and suppose Xi is exponentially distributed with parameter λ = i.
(Recall that the mean and standard deviation of an exponential are 1/λ.) Let Y = 3X1 − 2X2 + X3 .
Use the formulæ for the mean and variance of a linear combination. IE[Y ] = 3∗1/1−2∗1/2+1∗1/3 = 2 31
and Var[Y ] = 32 ∗(1/1)2 +(−2)2 ∗(1/2)2 +12 ∗(1/3)2 = 10.11. The Y is not normal—linear combinations
of normals are normal, but not linear combinations of exponentials.
IE[Y ] = 2.33
Var[Y ] = 10.11
Is Y normal? No
4. You estimate the mean of distribution by summing the n random observations and dividing by n − 2.
What is (a) the bias in this estimator, (b) the variance of this estimator, and (c) the mean squared
error of this estimator?
(a)
2
µ
n−2
(b)
2 σ2
n
n−2
n
(c)
In this problem your estimate of the mean is µ̂ =
n
the bias is IE[ n−2
X̄] − µ =
n
µ
n−2
− mu =
n
Similarly, the variance is Var[ n−2
X̄] =
1
n−2
P
Xi =
2 σ2
n
n
n−2
n
n
X̄.
n−2
+
2
2
n−2
µ2
We know that IE[X̄] = µ, so
2
µ.
n−2
2 σ2
n
.
n−2
n
The MSE is variance + the square of the bias.
0.16 4. Assume that the IQs of UNC students follow a Weibull distribution with mean 120 and standard
deviation 20. What is the approximate probability that a class of 16 has mean IQ greater than 125?
√
Central Limit Theorem. The sample mean is approximately Normal with mean 120 and sd 20/ 16 = 5.
So the z-transformation is z = (125 − 120)/5 = 1. The area above 1 in the standard normal table is
0.1587.
5 List all, and only, the true statements. (6 pts) B, D
A. Maximum likelihood estimates are unbiased.
B. As the sample size gets large, MLEs have minimum variance.
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C. The transformation of an unbiased estimator is an unbiased estimate of the transformation.
D. Sir Ronald Fisher invented MLEs.
E. For a 10% trimmed mean, one removes the largest 10% of the data and the smallest 10%, and
takes the average.
F. The trimmed mean is good when the data are exponentially distributed.
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