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?


  • $\begingroup$ Please clarify the question: what is theta? The inverse hyperbolic sine function is $asinh(z)=\log(z+\sqrt{z^2+1})$ - no theta in sight. $\endgroup$
    – Aniko
    Apr 12, 2012 at 18:45
  • 1
    $\begingroup$ A useful followup thread (which includes a more accurate description of the IHS that is valid for negative $z$ as well as positive $z$) is at stats.stackexchange.com/questions/157532. $\endgroup$
    – whuber
    Jun 18, 2015 at 11:44

1 Answer 1


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

  • $\begingroup$ yeah this is what i was looking for thanks. i unfortunately don't have access to jstor. just to double check, that first term, its log absolute value correct? $\endgroup$
    – tomas
    Apr 12, 2012 at 19:58
  • $\begingroup$ No, it is a square bracket [], but the term inside the log is a quadratic form then it is always positive. $\endgroup$
    – user10525
    Apr 12, 2012 at 20:05
  • $\begingroup$ I have corrected the $sinh^{-1}$ instead of $sinh$. $\endgroup$
    – user10525
    Apr 12, 2012 at 20:10

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