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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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