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Stat-3503 statistics minors and study groups only

 Stat-3503 (statistics minors and study groups only) Airoldi/Fall-20

Single-learning objective final — due Wednesday 12/16 by 5:00pm
Estimating a constant from Uniform observations.
Similarly to an example we explored during lecture, let’s assume we observe n numbers,
denoted by x1, x2, . . . , xn, between 0 and 5θ. Let’s further assume that these observations
are IID samples from a Uniform distribution, that is,
Xi ∼ Uniform[0, 5θ], independent and identically distributed.
We want to obtain estimates for the unknown quantity θ.
a) Using the 2 × 2 table we learned about in class, and practiced in problem set no. 1,
please categorize x1, x2, . . . , xn and θ.
b) Do we have any latent variables in this estimation problem?
c) What is the expected value of Xi = E[Xi
d) Please provide any method-of-moments estimator (MOME) for θ. That is, ˆθMOME =?
d) Please compute the maximum likelihood estimator (MLE) for θ. That is, ˆθMLE =?
e) Given that the density of the maximum value among n samples from a uniform random
variable, denoted Y = maxi Xi
, is
fY (y | θ) = nθn yn 1, for y ∈ [0, θ], (1)
use the transformation theorem to find the density of ˆθMLE.
(Hint. Define Z = ˆθMLE. Then write Z, i.e., the MLE, as a function of Y , i.e., the
maximum, by identifying that function g that leads to the equation Z = g(Y ). Then apply
the transformation theorem to g, assuming the density of Y is given by Equation 1.)
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