# Alternative to plug-in estimation for log-tranformed linear model

I want to estimate a relationship of the form:

$$y=ax^b\times\epsilon$$

If I log this model i get:

$$\log(y)=\log(a)+b\log(x)+ \log(\epsilon)$$

If I then proceed and estimate this model using a linear Regression, my residuals look reasonably normally distributed and homoscedastic, suggesting that this is an acceptable transformation.

However I am not interested in the Parameter $$\log(a)$$ but in the Parameter $$a$$. The obvious way to estimate $$a$$ would be to set $$\hat{a}= \exp(\widehat{\log(a)})$$, which is also the Maximum Likelihood Estimator under the assumption of Normal Distribution as the ML-estimate of the Transformation of a Parameter is the tranformation of the ML-Estimate. However this estimator loses properties of unbiasedness etc. .

I wonder now whether there is a "better" way to estimate $$a$$ under these conditions.

• What properties do you want your estimator to have? – whuber Oct 17 at 11:43
• Ideally I want it to be unbiased and variance-minimal, but I don`t know whether this is possible so I would prefer a reccomendation of someone who knows what is possible. – Sebastian Oct 17 at 12:29