# Estimating a regression equation with a hill function transformed independent variables

I'm currently trying to estimate the following equation:

$$y = \text{const} + \beta \frac{x^\gamma}{x^\gamma+\rho^\gamma}$$

So, I have to estimate 3 parameters for every variable plus a constant. A natural choice would be to estimate the equation with Nonlinear LS. However, as usual my estimates depent greatly on the starting values. I also tried to estimate the equation via Maximum Likelihood, as the y-variable follows a Gamma distribution. Nontheless, as the number of independent variables goes up(in my actual model I have 13 x-variables and for each I have to estimate 3 parameters) the ML-algorithm can't converge.

Has anyone an idea, how to handle such a regression? Up until now I tried to estimate the model like this in R.

logLikFun_Gamma_optim1 <- function(param) {
alpha1 <- param[1]
beta1 <- param[2]
rho1 <- param[3]
gam1 <- param[4]
rate1 <-  param[5]
sum(dgamma(y, shape = alpha1+beta1*((x1^gam1)/((x1^gam1)+(rho1^gam1)))
, rate = rate1, log = TRUE))
}

maxLik(logLik=logLikFun_Gamma_optim1,start=c(alpha1=1,
beta1=0.5,gam1=0.51,rho1=0.5,rate1=0.007),fixed='rate1',
method='CG',control=list(printLevel=3))

• I have added a formula in mathml code for clarity. I hope that is ok. – Martijn Weterings Oct 10 at 12:45
• What is unclear to me in this question is what you mean with "estimate 3 parameters for every datapoint" Does every datapoint get different estimated parameters? How does that work, what kind of estimation are you referring to? – Martijn Weterings Oct 10 at 12:45
• I agree with @MartijnWeterings about the 3 parameters for each point ... I think it's 3 parameters overall. Also, your formula has a single x, but you say there are 13. It would also be helpful to know more about the data set (what is the sample size? What are the variables? etc) – Peter Flom Oct 10 at 12:55
• If $y$ follows a gamma distribution then what does your equation $y= const + \beta \frac{x^\gamma}{x^\gamma+\rho^\gamma}$ mean? – Martijn Weterings Oct 10 at 13:05
• What does ML stand for (maximum likelihood, machine learning, ....) – Martijn Weterings Oct 10 at 13:09