I'm new to parameter estimation world and I'm studying this model with two parameters $\mu$ and $\sigma$: $$\tag1 dX_t = \mu X_t dt + \sigma X_t dB_t^H $$ where $B_t^H$ is a fractional Brownian motion (fBm) with Hurst parameter $H\in(0,1)$. Moreover I have real data (annual observations of the population size of whooping cranes) and I would like to see how the model fits them.
I already have formulas for the estimator $\hat\mu$ and $\hat\sigma$, but since they contain $H$, I first have to estimate $H$.
I have read plenty of stuff about simulations for parameters estimation, but they are mostly theoretical, they don't give a sequence of steps to follow and I did not find numerical examples.
I am using matlab and for simplicity, I use the built-in function
wfbmesti(X) which, given a fBm signal $X$, estimates the parameter $H$.
In our case, $X$ has to be generated from equation $(1)$, but since we don't have values for $\mu$ and $\sigma$, I guess we can pick what we want. Please correct me if I'm wrong. So let's say we pick $\mu = 2$ and $\sigma = 0.5$.
In order to generate $X$ we also have to choose $H$, but is there a standard procedure to apply?
From what I have understood, this is what I would do:
- Take $H=0.1$ and generate 1000 (how many exactly?) realizations $(X_1,...,X_n)$ of $X$;
- Obtain 1000 estimates of $H$ by means of
wfbmesti(X), one for each realization;
- Compute the bias and the MSE of the vector containing the 1000 values of $H$.
- Repeat steps 1 to 3 with $H=0.2$, $H=0.3$, ...,$H=0.9$.
I think this procedure is called Monte Carlo experiment (isn't it?), does it work this way?
When all is done, what is the correct estimate of $H$ and how to choose it?