ML estimation - solve for x I'm trying to solve the following maximum likelihood estimation but for multiplicative noise instead of additive noise:
So the goal is to do ML-estimation for a scalar constant $x$, which is multiplied with noise:
$$\tilde{d}[n] = a[n]x$$
The noise $a[n]$ is a realization of the i.i.d. random variable $A \sim \mathcal{N}(\mu_A, \sigma_A)$.
The probability density function is given by
$$ \pi(x,\mathbf{\tilde{d}}) = \frac{1}{x^N} \pi_A\left(\frac{\tilde{\mathbf{d}}}{x}\right)$$
with $\pi_A$ being the pdf of $A$.
Now I took the natural logarithm of this expression and derived with respect to $x$, which let to the following term:
$$\frac{\partial}{\partial x} \ln\left(\pi(x,\mathbf{\tilde{d}})\right) = -\frac{N}{x} + \frac{1}{\sigma_A^2} \sum_{n=0}^{N-1} \left[ \frac{\tilde{d}[n](\tilde{d}[n]-\mu_a x)}{x^3} \right] \overset{!}{=}0$$
But how do I continue from that? My idea was to solve this equation for $x$, but I can't figure out a closed form solution...
It looks okay to me. If you define empirical mean $\hat{\mu}_d = \frac 1N \sum \tilde{d}[n]$ and empirical second moment $\hat{\gamma}_d = \frac 1N \sum \tilde{d}^2[n]$ then you effectively have an equation of the form $$-N/x + A/x^3 + B/x^2 \stackrel{!}{=} 0,$$ where $A$ and $B$ directly depend on $\hat{\mu}_d, \hat{\gamma}_d, \mu_A, \sigma_A$. If you multiply this equation with $x^3$ it becomes a quadratic equation in $x$, which you can solve for $x$...
I get something like $$x^2 - \frac{\hat{\mu}_d \mu_A}{\sigma_A^2} x + \frac{\gamma_d}{\sigma_A^2} = 0$$ but that was scribbled down in a rush so I would have to check it again later. Can you confirm or correct?