Can I omit using an indicator function when estimating an MLE?

[cdux]

When looking for a maximum likelihood estimator for the Uniform distribution I noticed that a common method is to use an indicator function. My initial understanding is that the reason for that is for taking into account the region of ℝ that x produces - or not - a non-zero probability.

If I go on finding the maximum likelihood function without involving an indicator function in the math and then in the end only mention the boundaries of effect of the result, is it correct?

I'd be happy if I can avoid it (correctly) because I find the indicator function method very unintuitive for my style.

 

[chiro]

Hey cdux.

The uniform distribution can't use derivatives because the PDF is flat and also because the range of the PDF depends on the actual parameter (unlike the Normal distribution and others where it doesn't).

So for this particular case, you need to use either the indicator function or the order statistic to derive the estimator for the parameter of the uniform distribution.

If you are using the order statistic, then make sure you check the bias (if any iexists). Also note that when I say order statistic I mean the maximum order statistic X(n) where n is the sample size and X(n) is the nth order statistic distribution.

 

[cdux]

I've found both types of solutions, and I have a hard time understanding the difference between the "easy" one (the 'ordered statistic' one) and the one with the indicator function: In both cases they seem to conclude with the same thing: "Since b ['a' was zero in that case but it applies for both], since b >= x by our initial assumptions, then it comes to reason that -n/b^n which is a decreasing function, must be satisfied with b = max{X1,X2,...Xn}".

Because that's what I was already doing, evaluating l'(b), then looking at it being a decreasing function and then concluding that b should be minimized to maximize L(b) which is our goal.

Unless.. the solutions with indicator functions I've been reading had been redundant and they "talked too much" so to speak. Because if you are going to explain the order of the random variables why use an indicator function to begin with?

 

[chiro]

The indicator formulation is something that can be maximized in the context of the MLE (which is all about maximizing some quantity).

The order statistics on the other hand is an intuitive method that uses the nature of the distribution and knowing that the last value is a good estimate of the actual parameter.

So in short, MLE looks at maximizing something in general and order statistic is an intuitive estimator based on the specific nature of the uniform distribution.

Whatever estimator is good is what you use and there are always multiple estimators for a single parameter (MLE, moment estimator, non-parametric etc).