@@ -282,6 +282,86 @@ with :math:`\eta=(g, m, r)` and where the components of the linear score are
282282
283283 \psi _b(W; \eta ) &= g(1 ,X) - g(0 ,X) + \frac {Z (Y - g(1 ,X))}{m(X)} - \frac {(1 - Z)(Y - g(0 ,X))}{1 - m(x)}.
284284
285+
286+
287+ **********************************
288+
289+ For ``DoubleMLPQ `` the only valid option is ``score='PQ' ``. For ``treatment=d `` with :math: `d\in \{ 0 ,1 \}` and
290+ a quantile :math: `\tau\in (0 ,1 )` this implements the nonlinear score function:
291+
292+ .. math ::
293+
294+ \psi (W; \theta , \eta ) := g_{d}(X, \tilde {\theta }) + \frac {1 \{ D=d\} }{m(X)}(1 \{ Y\le \theta \} - g_d(X, \tilde {\theta })) - \tau
295+
296+
297+ :math: `\eta =(g_d,m)` with true values
298+
299+ .. math ::
300+
301+ g_{d,0 }(X, \theta _0 ) &= \mathbb {E}[1 \{ Y\le \theta _0 \}|X, D=d]
302+
303+ m_0 (X) &= P(D=d|X).
304+
305+ :math: `g_{d,0 }(X,\theta _0 )` depends on the target parameter :math: `\theta _0 `, such that
306+ the score is estimated with a preliminary estimate :math: `\tilde {\theta }`. For further details, see Kallus et al., (2019).
307+
308+
309+ Local potential quantiles (LPQs)
310+ **********************************
311+
312+ For ``DoubleMLLPQ `` the only valid option is ``score='LPQ' ``. For ``treatment=d `` with :math: `d\in \{ 0 ,1 \}`, instrument :math: `Z` and
313+ a quantile :math: `\tau\in (0 ,1 )` this implements the nonlinear score function:
314+
315+ .. math ::
316+
317+ \psi (W; \theta , \eta ) :=& \Big (g_{d, Z=1 }(X, \tilde {\theta }) - g_{d, Z=0 }(X, \tilde {\theta }) + \frac {Z}{m(X)}(1 \{ D=d\} \cdot 1 \{ Y\le \theta \} - g_{d, Z=1 }(X, \tilde {\theta }))
318+
319+ &\quad - \frac {1 -Z}{1 -m(X)}(1 \{ D=d\} \cdot 1 \{ Y\le \theta \} - g_{d, Z=0 }(X, \tilde {\theta }))\Big ) \cdot \frac {2 d -1 }{\gamma } - \tau
320+
321+
322+ :math: `\eta =(g_{d,Z=1 }, g_{d,Z=0 }, m, \gamma )` with true values
323+
324+ .. math ::
325+
326+ g_{d,Z=z,0 }(X, \theta _0 ) &= \mathbb {E}[1 \{ D=d\} \cdot 1 \{ Y\le \theta _0 \}|X, Z=z],\quad z\in \{ 0 ,1 \}
327+
328+ m_{Z=z,0 }(X) &= P(D=d|X, Z=z),\quad z\in \{ 0 ,1 \}
329+
330+ m_0 (X) &= P(Z=1 |X)
331+
332+ \gamma _0 &= \mathbb {E}[P(D=d|X, Z=1 ) - P(D=d|X, Z=0 )].
333+
334+ :math: `\gamma _0 ` is estimated with the two additional nuisance components
335+
336+ .. math ::
337+
338+ m_{Z=z,0 }(X) = P(D=d|X, Z=z),\quad z\in \{ 0 ,1 \}.
339+
340+ :math: `g_{d,Z=z,0 }(X, \theta _0 )` depends on the target parameter :math: `\theta _0 `, such that
341+ the score is estimated with a preliminary estimate :math: `\tilde {\theta }`. For further details, see Kallus et al., (2019).
342+
343+
344+ Conditional value at risk (CVaR)
345+ **********************************
346+
347+ For ``DoubleMLCVAR `` the only valid option is ``score='CVAR' ``. For ``treatment=d `` with :math: `d\in \{ 0 ,1 \}` and
348+ a quantile :math: `\tau\in (0 ,1 )` this implements the score function:
349+
350+ .. math ::
351+
352+ \psi (W; \theta , \eta ) := g_{d}(X, \gamma ) + \frac {1 \{ D=d\} }{m(X)}(\max (\gamma , (1 - \tau )^{-1 }(Y - \tau \gamma )) - g_d(X, \gamma )) - \theta
353+
354+ :math: `\eta =(g_d,m,\gamma )` with true values
355+
356+ .. math ::
357+
358+ g_{d,0 }(X, \gamma _0 ) &= \mathbb {E}[\max (\gamma _0 , (1 - \tau )^{-1 }(Y - \tau \gamma _0 ))|X, D=d]
359+
360+ m_0 (X) &= P(D=d|X)
361+
362+ :math: `\gamma _0 ` being the potential quantile of :math: `Y(d)`. As for potential quantiles, the estimate :math: `g_d` is constructed via
363+ a preliminary estimate of :math: `\gamma _0 `. For further details, see Kallus et al., (2019).
364+
285365Specifying alternative score functions via callables
286366++++++++++++++++++++++++++++++++++++++++++++++++++++
287367
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