I come from physics and would like to derive the chi-square function given by the Particle Data Group:
\begin{equation} \chi^2 (\boldsymbol\theta) = (\boldsymbol y-\boldsymbol\mu(\boldsymbol \theta))^T V^{-1}(\boldsymbol y - \boldsymbol\mu(\boldsymbol \theta)), \end{equation} with $V_{ij} = cov[y_i, y_j]$ being the covariance matrice, $\boldsymbol y = (y_1, \cdots, y_N)$ being the vector of measurements and $\boldsymbol \mu(\boldsymbol \theta)$ is the vector of predicted value.
My second question would be if the nomenclature chisquare and least square have the same meaning or are these two different methods?
So far I started with \begin{equation} \boldsymbol{\hat y} = X \boldsymbol{\hat \beta}, \quad \text{with} \quad \boldsymbol{\hat y} = \pmatrix{y_1 \\ y_2 \\ \vdots \\y_N}, \quad X=\pmatrix{1 & x_{11} & x_{12} & \cdots & x_{1K} \\ 1 & x_{21} & x_{22} & \cdots & x_{2K} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 1 & x_{N1} & x_{N2} & \cdots & x_{NK} }, \quad \boldsymbol{\hat \beta}=\pmatrix{\beta_0 \\ \beta_1 \\ \vdots \\ \beta_K} \end{equation} and the hat (^) marking the estimators.
From here I defined my chisquared function to minimize: \begin{equation} \chi^2 = (\boldsymbol y - \boldsymbol{\hat y})^T (\boldsymbol y- \boldsymbol{\hat y}), \end{equation} where $\boldsymbol y$ represents the given data points and $\boldsymbol{\hat y}$ the theoretically predicted estimators. The above formular looks yet similar to the one I am looking for, apart from the fact that I do not know how to introduce the inverse of the covariance matrix $V^{-1}$.
I also previously derived the relations \begin{equation} \begin{split} \boldsymbol{\hat \beta} = (X^T X)^{-1} X^T \boldsymbol{\hat y} \quad \text{and} \quad Var[\hat \beta] = \sigma(X^T X)^{-1}, \end{split} \end{equation} where I am also insecure that the approximations made to obtain the ladder identity hold true in my case.