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How did Owen derive this relation?

Owen't equation

in terms of normal cdf G(x), the normal pdf G'(x) and Owen's T-function. I cannot find the derivation nor the derivative of the T-function (only its definition both in Owen's Table of Normal Integrals and on Wikipedia), thus bluntly talking the derivative does not work...

Jenny Reininger
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  • For the definite integral see https://stats.stackexchange.com/questions/61080/how-can-i-calculate-int-infty-infty-phi-left-fracw-ab-right-phiw. Note that the integral on the left hand side of the equation is implicitly over some interval determined by "$x,$" presumably $(-\infty,x].$ Once you interpret this integral in an ordinary regression context, the definition of $T$ on [Wikipedia](https://en.wikipedia.org/wiki/Owen%27s_T_function) makes short work of this formula. – whuber Dec 16 '20 at 14:01

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