My question concerns using Maximum Likelihood to estimate unknown parameters. I will sincerely appreciate if anyone can help me find out if my approach is flawed.

Assume we have a random vector $V$, and we can observe $M$ samples of it denoted by $V_1,V_2,\ldots,V_M$. Define a scalar random variable $X = f(V\mid\theta)$ where $f(\cdot)$ is function whose closed form is known but parameter $\theta$ is unknown. I cannot observe $X$.

We do not know the distribution of $V$, but we do know the distribution of $X$. In fact, $X$ follows a Normal distribution $N(\mu,\sigma^2)$ where $\mu$ and $\sigma$ are unknown.

My purpose is to JOINTLY estimate $\theta$, $\mu$, and $\sigma$. I take the following approach: let us use $g(.|\mu,\sigma^2)$ to denote the pdf for $N(\mu,\sigma)$. The likelihood function $L$ is then $g(f(V\mid\theta)\mid\mu,\sigma^2)$. Based on MLE, we then maximize $\sum_{m=1}^M \log(g(f(V_m\mid\theta)\mid\mu,\sigma^2))$ of all $M$ samples. Since $f(\cdot)$ and $g(\cdot)$ are both known, this maximization problem can be solved by nonlinear programming (at least in theory). The optimal solution would provide the best estimates for $\theta$, $\mu$, and $\sigma$ simultaneously.

I have multiple questions regarding the approach described above:

Deviation from traditional MLE definition: in textbook examples, we should formulate the joint distribution of $V$ (joint because $V$ is a vector) and maximize it. In my problem, however, the joint distribution of $V$ is unknown. The only thing we know is that it has to satisfy $X = f(V)$. For example, let us assume $V$ has two dimensions, and $ X = V_{(1)} + \theta V_{(2)}$. There are many joint distributions that satisfy this equation above.

Identification: as pointed by the example above, There is no one-to-one mapping between $V$ and $X$. Do I have to run into identification issues when I do the estimation?

Do consistency, asymptotic normality and other decent properties for MLE still apply?

Is there any alternative besides MLE I can take?

Many thanks! Any comments are more than welcome.