Maximum Likelihood estimators in linear models

Question

Consider two simple linear models.

$y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and
$y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,\ldots,n>2$ where $ \epsilon_{ij} \sim N(0,\sigma^2)$
I have two questions.

1 ) Show that the point on the $x$-axis where the two lines intersect is given by $x_0={\alpha_1-\alpha_2 \over \beta_2-\beta_1}$
2)Obtain the maximum likelihood estimators of $\sigma^2$ and $x_0$

For question 1) what I did was,
Supposing at $x_0,y_0$ the two lines intersect I substituted $x_0,y_0$ to the two equations.But then what happens to the error terms.Can I suppose that the errors in estimating $x_0,y_0$ from both equations are the same?

For 2)

$y_{ij}$ has a Normal distribution.
I calculated $E(y_{1,j})$ and $E(y_{2,j})$ separately.Can I say that $E(X_{1j}) = E(X_{2j})$
and $E(y_{1,j}) = E(y_{2,j})$.
Thereby I obtained $E(y_{ij})={\beta_2\alpha_1-\beta_1\alpha_2 \over\beta_2-\beta_1}$

Similarly $V(y_{ij})=\sigma^2$

Then joint probability density function for $y_{ij}$ is $\prod_{i=1}^2\prod_{j=1}^n {1\over\sqrt {2\pi\sigma^2}}$$ e^{{-(y_{ij}-\mu})^2} / 2\sigma^2$ where $\mu= {\beta_2\alpha_1-\beta_1\alpha_2\over\beta_2-\beta_1}$. Is this a correct joint p.d.f for $y$?

Does $x_0$ also has a normal distribution. Is $V(x_0)=0$. I don't understand how to obtain MLE for $x_0$ .

Can someone please help me to do this please

Answer 1

1. suppose the errors are zero to find the "ground truth" intersection point.
2. In a vector-matrix notation this problem can be augmented to reflect all measurements: ${\bf{y}}={\bf{H}} \boldsymbol{\theta}+\boldsymbol{\epsilon}$, where ${\bf{y}}$ is a $2n \times 1$ column vector of all $y$'s concatenated,$\boldsymbol{\theta}=(\alpha_1,\beta_1,\alpha_2,\beta_2)^T$ and ${\bf{H}}$ is a $2n \times 4$ block matrix given by

\begin{equation} {\bf{H}} = \begin{pmatrix} {\bf{H_1}} & {\bf{0_{n \times 2}}}\\ {\bf{0_{n \times 2}}} & {\bf{H_2}} \end{pmatrix} \end{equation}

where ${\bf{H_1}}$ and ${\bf{H_2}}$ are $n \times 2$ Vandermonde matrices and are given by \begin{equation} {\bf{H_1}}=\begin{pmatrix} {\bf{1}}_{n \times 1} & {\bf{x_1}} \end{pmatrix} \end{equation}

\begin{equation} {\bf{H_2}}=\begin{pmatrix} {\bf{1}}_{n \times 1} & {\bf{x_2}} \end{pmatrix} \end{equation}

and the augmented error vector is of size $2n \times 1$ ,$\boldsymbol{\epsilon} \sim {\mathcal{N}}({\bf{0}}, \sigma^2 {\bf{I}}_{2n})$.

Thus, the Maximum Likelihood (ML) estimator for $\theta$ for this problem is given by the least squares (LS) estimator and is

\begin{equation} \widehat{\boldsymbol{\theta}}_{ML} = ({\bf{H}}^T{\bf{H}})^{-1} {\bf{H}}^T {\bf{y}} \end{equation}

so we get the ML estimation for the four parameters $\widehat{\boldsymbol{\theta}}_{ML}=(\widehat{\alpha}_{1_{ML}},\widehat{\beta}_{1_{ML}},\widehat{\alpha}_{2_{ML}},\widehat{\beta}_{2_{ML}})^T$ and by the invariance property of the ML estimator we get the ML estimator of $x_0$:

\begin{equation} {\widehat{x}}_{0_{ML}} = \frac{\widehat{\alpha}_{1_{ML}} - \widehat{\alpha}_{2_{ML}}}{\widehat{\beta}_{2_{ML}} - \widehat{\beta}_{1_{ML}}} \end{equation}

To complete the question the sample covariance $\sigma^2$ is derived by differentiating the joint log likelihood PDF and is given by

\begin{equation} {\widehat{\sigma}}^2_{ML} = \frac{1}{N} \left \| {\bf{y}} - {\bf{H}} \boldsymbol{\widehat{\theta}}_{ML} \right \|^2_2 \end{equation}