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Here's a screenshot:

screenshot

This is from Evaluating Continuous Training Programs Using Generalized Propensity Scores by Jochen Cluve, Hilmar Schneider, Arne Ulendorff and Zhong Zhao. I understand from the text that whether the treatment is at a value t is independent of the covariates X when the generalized propensity score at (t, X) is fixed. But the notation is unfamiliar. It's also in the original GPS paper.

What does it mean? What would 1{T = t} mean? Was it invented specifically for describing GPS?

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This is sometimes used for functions which are 1 when the argument is true and 0 otherwise.

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    $\begingroup$ I'd add that it is called an indicator function: en.wikipedia.org/wiki/Indicator_function $\endgroup$ – Tim Mar 31 '16 at 9:36
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    $\begingroup$ $I()$ is often used with the same meaning. Sometimes the argument is in brackets, sometimes in parentheses, sometimes as a subscript (which looks ugly and awkward in my view). Donald Knuth and others have urged for decades that we should use notation in mathematics like $[T = t]$ with this meaning. Hence $1()$ or $I()$ is dispensable. See e.g. arxiv.org/pdf/math/9205211.pdf In many programming languages something like T == t will be evaluated as 1 or 0: surrounding parentheses are often helpful and indeed prudent given precedence rules. Much here depends on your education! $\endgroup$ – Nick Cox Mar 31 '16 at 9:36

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