# (log) Likelihood at true parameters

I am considering the linear regression model

\begin{equation} Y_{i} = A t_{i} + B + \varepsilon_{i} \tag{$1$} \end{equation}

where $A$ and $B$ are unknown real parameters and $(\varepsilon_{i})_{1 \leq i \leq n}$ are observed values of independent and identically distributed gaussian random variables with mean $0$ and variance $\sigma^{2}$ ($\sigma^{2}$ is unknown).

In matrix form, $(1)$ writes :

\begin{equation} Y=X\alpha + \varepsilon \tag{2} \end{equation}

where $X = \begin{bmatrix} t_{1} & 1 \\ \vdots & \vdots \\ t_{n} & 1 \end{bmatrix}$, $\alpha = \begin{bmatrix} A \\ B \end{bmatrix}$ and $\varepsilon \sim \mathcal{N}(0,\sigma^{2}\mathrm{I}_{n})$.

Using Matlab, I generated some data artificially : I took $n=100$, $A=3$, $B=-1$ and $\sigma^{2} = 0.64$ and the parameters estimates I obtain are : $\widehat{A}=3.0096$, $\widehat{B}=-0.9260$ and $\widehat{\sigma^{2}}=0.6337$.

The log-likelihood of $(2)$ is the function

$$\ell(A,B,\sigma^{2}) = -n \log(\sigma) - \frac{1}{2\sigma^{2}} \Vert Y-X\alpha \Vert^{2}$$

This may be a silly question but I do not understand why the true values of the parameters do not maximize the log-liklihood in my case. Is it because the estimates I obtained using Matlab are observed values of random variables which asymptotically converge to the true values of the parameters and here, $n$ is not "big enough" ?

So for example, consider MLE for $\mu$ in a $N(\mu,\sigma^2)$ is the sample mean. The sample mean is 'closer to the data' than the population mean (making the likelihood higher), specifically because that value is the one that's as close to the data as possible; any value other than the sample mean must be less likely.