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I am very wondering why we do not use least squares instead of maximum likelihood?

for example we have 3 choices k= 1, 2 ,3

$minimizing: (e^{\beta_{i} X}/(1+\sum e^{\beta_{i} X})- Y)^{2} $ for i=1,2,3

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The short answer is because it is not maximum likelihood estimation, so it is not optimal. Maximum likelihood solves for $\beta$ that makes the observed data most likely to have been observed. The likelihood function for Bernoulli random variables ($Y=0,1$) involves exponents in $Y$, not squares.

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    $\begingroup$ so you mean this method is logical but result is not good enough? $\endgroup$ – sherek_66 Jun 14 '19 at 12:06
  • $\begingroup$ It is not logical given that we have a better estimator, but mainly it's not good enough. $\endgroup$ – Frank Harrell Jun 14 '19 at 16:18

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