I used a Taylor series to expand log(1 - ax) so I could estimate the value of parameter 'a'.
The expansion becomes -ax - a^2*x^2/2 - a^3*x^3/3 . . .
Now I need to estimate the parameter 'a' using regression and for simplicity I am only using the first 3 terms in the expansion.
The equation to be estimated becomes y ~ ax + a^2*x^2/2 + a^3*x^3/3 [I have absorbed the negative sign on the left hand side of the equation]
I wanted to ask if there is a way to estimate the coefficients a , a^2 and a^3 in the above equation, keeping in mind that all the three coefficients are powers of each other.
Is there a package in R for this?
Please do note that the Taylor series expansion was necessary as there are several other terms in the original equation which I haven't mentioned here.
The original equation I have is:
Y ~ (1 - aX)(B^b)(C^c)(D^d)
In the above equation I have to estimate a,b,c,d, where a is to be estimated as aconstant while b,c and d as smooth splines.
So I have taken log on both side, which makes it:
log(Y) ~ log(1 - aX) + blog(B) + clog(C) + d*log(D)
If there is a better way to approach the entire equation, do mention.