Q:

Inverse hyperbolic sine transformation: estimation of theta

I'm trying to use an inverse hyperbolic sine transformation to reduce the effect of outliers in my target variable.  Unfortunately, I don't appear to have access to the basic papers on it.  I've found the formulation but am not sure how to estimate the theta parameter for it.  Does anyone know? 
Thanks

A:

The basic idea is as follows,
You have the IHS transformation
$$z_j = g_j(y_j;\theta)= \operatorname{sinh}^{-1}(\theta y_j)/\theta,\,\,j=1,...,n.$$
Then you have to find the value of $\theta$ that maximises the concentrated log-likelihood
$$L(\theta) = -\dfrac{n}{2}\log[g(\theta)^TMg(\theta)] - \dfrac{1}{2}\sum_j\log(1+\theta^2 y_j^2),$$
where $g(\theta)=(g_1(y_1;\theta),...,g_n(y_n;\theta))$, $M = I - X(X^TX)^{-1}X^T,$ and $X$ is the matrix of explanatory variables. 
I hope this helps. 
Ref: Alternative Transformations to Handle Extreme Values of the Dependent Variable
Author(s): John B. Burbidge, Lonnie Magee, A. Leslie Robb
Source: Journal of the American Statistical Association, Vol.  83, No.  401 (Mar., 1988), pp.  123-127x

