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Let $Y_{1},Y_{2},\ldots,Y_{n}$ be independent random variables with expected values $\mu_{1},\mu_{2},\ldots,\mu_{n}$, respectively. Suppose that the $\mu_{i}$'s are functions of the parameter vector that we want to estimate $\beta = [\beta_{1},\beta_{2},\ldots,\beta_{n}]^{T}$, $p < n$. Thus \begin{align*} E(Y_{i}) = \mu_{i}(\beta) \end{align*}

The simplest form of the method of least squares consists of finding the estimator $\hat{\beta}$ that minimizes the sum of squares of the differences between $Y_{i}$'s and their expected values \begin{align*} S = \sum[Y_{i}-\mu_{i}(\beta)]^{2} \end{align*}

My question is quite simple and naive, but I still would like to make it: what motivates the application of this method in such general context where there is no prior distribution related to $Y_{i}$ neither an explicit relation between the $Y_{i}$'s and the $\mu_{i}$'s?

I am doing this question because, in the normal linear model, we have the assumptions \begin{align*} \textbf{Y} = \textbf{X}\beta + \epsilon \end{align*}

where $\epsilon\sim\mathcal{N}(\textbf{0},\sigma^{2}\textbf{I})$ and $\mu = \textbf{X}\beta$. Since $\epsilon_{i} = Y_{i} - \mu_{i}$, it sounds reasonable to apply the LSE since we are interested in minimizing the corresponding error.

Hopefully, I have been clear enough at posing my doubts. Any help is appreciated. Thanks!

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You have it backwards. The least squares came first, then came all these Gauss-Markov conditions and underlying theory with BLUE and other nice things. You want to fit a function into data, the function has a bunch of parameters. It's natural to look at the fit errors (deviations) $Y_i-\mu(\beta)$. Wouldn't you want to minimize the deviations? You'd like to get one number that represents the general deviation of the function from the data, and your $S$ is quite convenient. At least that's how we've been taught in high school. We weren't given any theory on Normal distribution, exogeneity and other cool stuff. In fact, we didn't even study the statistics yet. Yet the least squares were easy to comprehend at the time, very intuitive.

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