I have a likelihood function of three parameters $a,b,c$ where distributions of $x_i\mid x_{i-1}$ are normal with mean $x_{i-1}+a(b-x_{i-1})$ and variance $c$. Differentiating and equating the log-likelihood to zero I finally arrive to the following conditions for the MLE of $a,b$ and $c$ based on a sample data $x_0,x_1,\dots,x_n$,
\begin{align*} b &= \frac{1}{na}\sum_{i=1}^n \left(x_i-x_{i-1}(1-a)\right)\\ a&= \frac{\sum_{i=1}^n (x_i-x_{i-1})(b-x_{i-1})}{\sum_{i=1}^n (b-x_{i-1})^2}\\ c&= \frac{1}{n}\sum_{i=1}^n (x_i-x_{i-1}-a(b-x_{i-1}))^2 \end{align*}
I tried but it seems hopeless to try to isolate one parameter and estimate the others from that one. So I need to use R to, given data $x=(x_0,\dots,x_n)$, find the values for $a,b$ and $c$ but I do not know the command or way of doing this in a smart way, given the sums. Does anybody know how to proceed or eventually a different way to solve the problem?
Thanks a lot!