# Parameter estimation from a measure

I'm doing an exercise where, given a data sample taken from a physical measure, I have to estimate the parameters of the measure where samples $x_i$ are described by the equation: $$x_i = A \sin(2\pi f t_i) + Y_i + Z_i$$ where:

• $A \sin(2\pi f t)$ is a bias introduced by the measure
• $Y ∼ \operatorname{Logis}(\mu, s)$ is the R.V. describing the population
• $Z ∼ N(0, \sigma_Z)$ is Gaussian noise

I have to estimate the five parameters $A,f,\mu, s, \sigma_Z$.

I've found $\mu$ which simply corresponds to the sample mean $\bar{X}$. To estimate $f$ I think I can analyze the autocorrelation but I'm not sure on how to do it.

My main problem is on finding the variances of $Y$ and $Z$. I'm trying to find them using a system with two equations, however I only have: $$s^2 = \sigma_Y^2 + \sigma_Z^2$$ which indicates that the estimated variance $s^2$, since $Y$ and $Z$ are independent R.V, is just the sum of the two variances.

Is there any other relation between those two variances I don't spot?

• Is this an assignment? – Brethlosze Nov 12 '16 at 8:09