I'm a novice with stats, so if I'm missing something obvious, don't sue me. I was recently working on an assignment where I was tasked with analyzing the following model with subset selection: $$ y_i = \beta_1 x_{1, i} + \beta_2 x_{2,i} + \beta_3 x_{3,i} + \beta_4 x_{4,i} + u_i $$ Where $$(x_{k,i})_{k=1}^4 \sim \mathcal{N}(0, \Sigma)$$ $$ \Sigma = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & \gamma & \gamma\\ 0 & \gamma & 1 & \gamma\\ 0 & \gamma & \gamma & 1 \end{pmatrix} $$ Where $\gamma \in [-1, 1]$. Additionally, we have that $\beta_2 = \beta_3 = \beta_4 = 2$, and $\beta_1 > 2$. Finally, $u_i$ are i.i.d. normally distributed with mean zero and variance 2, and $i = 1, \ldots, N$.
We were tasked with identifying the thresholds of $\gamma$ and $\beta_1$, for which, subset selection, with the size of the subset equal to $1$, correctly identifies the most important covariate, namely, $\beta_1$.
In this process, I managed to derive the following curves, for $N = 500$, $\beta = 4$ for the first one and $\gamma = 0$ for the second.
$\gamma$" />
$\beta$" />To my eye, these look strikingly logistic! So, I ran a simple logistic curve fit, with the following model: $$ f(x) = \frac{A}{1 + \exp(-k(x - x_0))} $$
I managed the following parameters
The best fit logistic function for gamma is: A = 1.0011916177515001, x0 = 0.46613345213429164, k = -25.287773424878644 99% confidence intervals: [[ 0.99834479 1.00403845] [ 0.46495465 0.46731225] [-25.94535303 -24.63019382]] The best fit logistic function for beta is: A = 1.0008861957715565, x0 = 4.131595820642967, k = 6.469500267867452 99% confidence intervals: [[0.99799143 1.00378096] [4.12631938 4.13687226] [6.27646573 6.66253481]] With the following graphs (not informative, just pretty): 
I don't fully understand how to measure goodness of fit for non-linear least squares, but these are very tight confidence intervals, so this finally leads me to my question.
Is there a good theoretical explanation for this behavior? It appears that the probability of correctly identifying a covariate with subset selection is logistic distribution with these magic parameters. Is this actually the case? Is it asymptotically logistic? Is this actually just a normal distribution (something I'm realizing as I write this)?
