Condition for Method of Least Squares to match Maximum Likelihood Estimation

Theorem

Let $X_n$ and $Y_n$ be sets of $n$ paired observations of random variables $X$ and $Y$.

For the paired observations $\tuple {x_i, y_i}$:

let $x_i$ be error-free

let the errors in $y_i$ be identical and normally distributed with expectation $0$.

Then the method of least squares yields a result equivalent to the maximum likelihood estimator.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): least squares: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): least squares: 2.