Let a realization of a random sample of a normal distribution be $\{x_i\}_{i=1}^N$ such that $x_i$ is a real number. Assume that it is unknown that the distribution is normal and it is desired to identify a probability density function from the realization $\{x_i\}$. Let the approximation of the pdf of the normal distribution be a pdf $f$ that is a mix of a pdf of a normal distribution and a cauchy distribution, such that $$ f(\theta,x) = \theta_3\frac{1}{\sqrt{2\pi\theta_2^2}}\exp\left(-\frac{(x-\theta_1)^2}{\theta_2^2}\right) + \theta_4\frac{1}{\pi\theta_6\left(1+\left(\frac{x-\theta_5}{\theta_6}\right)^2\right)} $$ where the parameters $\theta$ are in the set of maximizers to the corresponding likelihood: $$ \arg \max_{\theta} \prod_{i=1}^N f(\theta, x_i) $$ subject to $$ \theta_1,\theta_3,\theta_4,\theta_5\in \mathbb{R} \text{ and } \theta_2,\theta_6\in \mathbb{R}^{>0}. $$ and $$ \int_{-\infty}^\infty f(\theta,x) \, \text{d} x = 1 $$ Since $\{x_i\}_{i=1}^N$ is a realization of a random sample of a normal distribution, it should be that if the mean and standard deviation of the normal distribution is $\mu$ and $\sigma$, then asymptotically, as the number of observations $N$ tends to infinity, a maximizer should be $\theta = (\mu,\sigma,1,0,0,0)$. My question is: is this the only maximizer? In general, if the pdf f is a mix of a normal pdf and many other pdfs will the set of maximizers be a singleton?
Edit, a higher level formulation of my question:
Consider another identification problem. Let a dataset $\{(x_i,y_i)\}_{i=1}^N$ be such that $y_i$ is a real number response to the real number input $x_i$. For simplicity let the dataset be such that every $x_i$ is related to a unique $y_i$. Then, the set $$ \arg \min_f \sum_{i=1}^N \|y_i-f(x_i)\|^2 $$ subject to $f$ is a function on $\mathbb{R}$ into $\mathbb{R}$, will contain infinitely many functions, every function which passes through every datapoint. Returning to an identification problem similar to the one in my original question. Let the dataset be a collection of real numbers $\{x_i\}_{i=1}^N$. Is it that the set $$ \arg \max_f \prod_{i=1}^N f(x_i) $$ subject to $f(x)$ is a non-zero real number and $\int_{-\infty}^\infty f(x) \text{d}x= 1$, contains infinitely many functions $f$? Does a "likelihood" optimization criterion have the same property as the mean-squared-error criterion? That a flexible parameterization of models implies that there are many different models with the same performance according to the "likelihood" criterion? Could this be illustrated?