Maximum Likelihood Estimator Problem

[gajohnson]

Homework Statement

1. Suppose the data consist of a single number X, and the model is that X has the
following probability density:

f(x|θ) =(1+ xθ)/2 for -1≤ x ≤1; =0 otherwise.

Supposing the possible values of θ are 0 ≤ θ ≤ 1; find the maximum likelihood estimate
(MLE) of θ, and find its (exact) probability distribution. Is the MLE unbiased? Find
its bias and MSE. [Hint: First find the MLE for a few sample values of X, such as X = –
.5 and X = .5; that should suggest to you the general solution. Drawing a graph helps!
The distribution of the MLE will of course depend upon θ.]

Homework Equations

The Attempt at a Solution

If we start by finding the partial derivative of f(x|θ) this expression is x/2, or x/(1+xθ) if we take the partial of ln(f(x|θ)) instead. Setting both of these equal to 0 does not seem to be yielding anything in either case.

I'm not sure what to do with this, and the hint is not helping me out too much. Any suggestions about how to implement the hint, or what to do from here?

EDIT: All I can see is that the MLE is simply a piecewise function in which:
θ=0, -1≤x≤0
θ=1, 0<x≤1

 

[Nino MMP]

So the MLE of θ is x+1/2. Is this correct? The probability distribution is 0 if θ<x, or θ>1-x. It is 1/(1-x) if 0≤θ≤1-x and it is 1/(x+1) if x≤θ≤1.The MLE is unbiased since the expectation of the distribution is 0. The bias is 0 and the MSE is 0.