Maximum Likelihood Estimator Problem

In summary, the MLE of θ is x+1/2 and the probability distribution is piecewise with 0 if θ<x or θ>1-x, 1/(1-x) if 0≤θ≤1-x, and 1/(x+1) if x≤θ≤1. The MLE is unbiased with a bias and MSE of 0.
  • #1
gajohnson
73
0

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
 
Last edited:
Physics news on Phys.org
  • #2
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.
 

FAQ: Maximum Likelihood Estimator Problem

1. What is a Maximum Likelihood Estimator (MLE)?

A Maximum Likelihood Estimator (MLE) is a method used in statistics to estimate the parameters of a probability distribution by finding the values that maximize the likelihood of the observed data. In other words, it is a way to determine the most likely values of the parameters based on the available data.

2. How does the MLE method work?

The MLE method works by determining the values of the parameters that maximize the likelihood function, which is a measure of how likely the observed data is to occur given a specific set of parameter values. This is done by taking the derivatives of the likelihood function with respect to each parameter and setting them equal to zero to find the maximum point.

3. What are the assumptions of the MLE method?

The MLE method assumes that the data is independent and identically distributed (i.i.d.), meaning that each data point is independent of the others and is drawn from the same probability distribution. It also assumes that the data is continuous and follows a specific probability distribution, such as the normal or binomial distribution.

4. What are the advantages of using MLE?

One of the main advantages of using MLE is that it produces unbiased estimators, meaning that on average, the estimated values will be close to the true values of the parameters. It is also a very flexible method that can be applied to a wide range of distributions and can handle missing data.

5. What are the limitations of MLE?

One limitation of MLE is that it relies on the underlying assumptions of the data, such as the data being continuous and following a specific distribution. If these assumptions are not met, the estimated values may not be accurate. Additionally, MLE can be computationally intensive and may not be feasible for large datasets.

Similar threads

Replies
2
Views
2K
Replies
19
Views
1K
Replies
4
Views
2K
Replies
7
Views
1K
Replies
2
Views
4K
Replies
6
Views
2K
Replies
1
Views
2K
Replies
1
Views
2K
Back
Top