Comparing Least Squares and Maximum Likelihood?

In summary, the conversation discusses a comparison between Maximum Likelihood Estimation (MLE) and Least Squares Estimation (LSE). MLE is accurate for large sample sizes, but can be mathematically complicated. LSE is accurate for smaller sample sizes and is simpler to solve mathematically, but is less accurate for non-linear models. Both methods make assumptions about the underlying distribution and can be used to estimate parameters. However, the statements made about the properties of each method lack precise mathematical interpretation and may be considered subjective.
  • #1
peripatein
880
0
Hi,
Below is my attempt at a comparison between the two above-mentioned methods of estimation. Does anything in the table lack in validity and/or accuracy? Should any properties, advantages/disadvantages be eked out? Any suggestions/comments would be most appreciated!

MLE:

(1) Very accurate for a large N as the pdf of a^ would be unbiased

(2) No loss of information; all data are represented

(3) Quite complicated to solve mathematically

(4) Applicable for varied models, even non-linear

(5) Errors of estimation could be readily found: the 1sigma
error bars are those at which the logarithm falls by 0.5 from
its maximum

(6) Pdf must be known in advance

(7) In case pdf is false, goodness of fit may not be determined

LSE:

(1) Very accurate for a relatively small N as estimators would be biased

(2) -

(3) Finding the suitable linear model is quite simple mathematically

(4) Very convenient to use for linear models; very intricate for
non-linear ones

(5) -

(6) Variance and mean must be known in advance

(7) Method is very sensitive to unusual data values but
goodness of fit may be determined, through chi-squared
test e.g.
 
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  • #2
In my view, one should always ask how the conclusions will be used, i.e. what is the cost function for the decision error? E.g. if you're trying to estimate a mean, and the cost of getting it wrong varies as the square of the error, I suspect least squares is going to be the ideal (but that's an off-the-cuff guess, so don't quote me). But if you need to be within some tight range of accuracy and anything beyond that is simply a miss then MLE may be more appropriate.
 
  • #3
Hey peripatein.

One of the best properties of the MLE is that it can be used with the invariance principle.

This leads you to get estimates of functions of a parameter and it's useful for example when you use the Wald test-statistic with the parameter to get a standard error term as a function of your estimate of the proportion which gives you the Normal approximation.

Other estimators aren't guaranteed to have this property and it is a nice property.

Both methods make assumptions about the underlying distribution in some form (the non-additive linear models) so I don't think you can really use this as a way to differentiate the two approaches.

You can always transform data with the Linear Models approach and link functions provide a way of systematizing this.

Also you can also estimate quantities in the LSE formulation just like you can do with the MLE approach.
 
  • #4
Thank you for your replies! So much appreciated :-).
However, are there any cardinal inaccuracies? Anything of true significance which ought to be added and was left out?
 
  • #5
What do you mean?
 
  • #6
I mean, are the properties for each method, as presented in my initial post, accurate? Are any additional, significant properties missing? What may I write for numbers 2 and 5 under LSE (i.e. what will be the equivalent LSE properties for numbers 2 and 5 under MLE)?
Again, any comments whatsoever would be highly appreciated!
 
  • #7
peripatein said:
I mean, are the properties for each method, as presented in my initial post, accurate?

Most of your statements have no precise mathematical interpretation. You haven't defined what phrases like "very accurate" and "errors of estimation" mean. Can you rephrase your statements using the standard terminology of mathematical statistics? If not, I think you are asking for "rules of thumb" which are empirical or subjective.
 

1. How do least squares and maximum likelihood differ?

Least squares and maximum likelihood are both methods used to estimate the parameters of a statistical model. However, they differ in the way they approach the problem. Least squares minimizes the sum of squared residuals, while maximum likelihood maximizes the likelihood function, which is a measure of how likely the observed data is to occur given the model parameters.

2. Which method is better for parameter estimation: least squares or maximum likelihood?

It depends on the specific situation and the underlying assumptions of the model. In some cases, least squares may be more appropriate, while in others, maximum likelihood may be a better choice. Generally, maximum likelihood is preferred when the data is normally distributed and there are no outliers, while least squares may be more robust to outliers.

3. Can least squares and maximum likelihood be used interchangeably?

No, they cannot be used interchangeably. While both methods can be used for parameter estimation, they have different assumptions and may produce different results. It is important to understand the differences between the two and choose the appropriate method for the specific problem at hand.

4. Is there a relationship between least squares and maximum likelihood?

Yes, there is a relationship between least squares and maximum likelihood. In fact, under certain conditions, the least squares estimate is equivalent to the maximum likelihood estimate. This is when the data is normally distributed and there are no outliers.

5. Which method is more commonly used in practice: least squares or maximum likelihood?

Both least squares and maximum likelihood are commonly used in practice, and the choice depends on the specific problem and the goals of the analysis. In some cases, one method may be more appropriate than the other, while in others, both methods may yield similar results. It is important to understand the underlying assumptions and limitations of each method in order to make an informed decision.

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