Computes the Probit transformation and its inverse.
Syntax
PROBIT(X, Lo, Hi, Return)
 X
 is the real number for which we compute the transformation. X must be between zero and one (exclusive).
 Lo
 is the xdomain lower bound. If missing, Lo is assumed to be 0.
 Hi
 is the xdomain upper bound. If missing, Hi is assumed to be 1.
 Return
 is a number that determines the type of return value: 0 (or missing) = Probit, 1 = Inverse Probit.
Return Description 0 or omitted Probit Transform 1 The inverse of Probit transform
Remarks
 The X value(s) must be between Lo and Hi (exclusive).
 The Probit function is the quantile function associated with the standard normal distribution.
 The Probit function is commonly used for parameters that lie in the unit interval (i.e., $x\in({0,1})$).
 Numerical values of X close to 0 or 1 or out of range result in #VALUE! or #N/A.
 The Probit function is defined as the inverse cumulative distribution function (CDF):
$$y=\textit{Probit}(x)=\Phi^{1}(x)$$ And,
$$x=\textit{Probit}^{1}(y)=\Phi(y)=\frac{1}{\sqrt{2\pi}}\int_{\infty}^{y}e^{\frac{Z^2}{2}}dz$$ Where: $x_{t}$ is the input value of the input time series at time $t$. X must be between 0 and 1 (exclusive).
 $y_{t}$ is the transformed Logit value at time $t$.
 $\Phi(.)$ is the standard Gaussian cumulative density function.
 $\textit{Probit}^{1}(y)$ is the inverse Probit transformation.
 To support a generic interval (a, b), we perform the following mapping:
$$z = \frac{x  a}{b  a}$$ So, the revised transform function is expressed as follows:
$$y=\Phi^{1}(\frac{x  a}{b  a}) \Rightarrow \Phi(y)=\frac{x  a}{b  a}$$ The inverse transform is expressed as follows:
$$x = a+(b  a)\times\Phi(y)$$  The transform first order derivative is calculated as follows:
$$\frac{dy}{dx}=\frac{1}{(b  a)\phi(y)}=\frac{1}{(b  a)\phi(\Phi^{1}[\frac{x  a}{b  a}]})$$ Where:
$\phi(.)$ is the standard Gaussian probability density function.  The transform first order derivative (i.e., $\frac{dy}{dx}$) is positive for all xvalues and goes to infinity as x approaches the interval endpoints.
 In essence, the Probit function converts a bounded xrange from (Lo, Hi) to $(\infty,\infty)$.
 The Probit function accepts a single value or an array of values for X.
Examples
Example 1:


Files Examples
Related Links
References
 John H. Aldrich, Forrest D. Nelson; Linear Probability, Logit, and Probit Models; SAGE Publications, Inc; 1st Edition(Nov 01, 1984), ISBN: 0803921330.
 Finney, D.J. (1947), Probit Analysis. (1st edition) Cambridge University Press, Cambridge, UK.
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