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**Efficiency transfer for regression models with responses missing at random.**
*(English)*
Zbl 1382.62017

This article deals with the response \(Y\) missing at random in independent observations of a random pair \((X,Y)\), while the covariate vector \(X\) is always observed. It proves that functionals of the conditional distribution of \(Y\) given \(X\) can be estimated efficiently using only the pairs that are completely observed, that is, simply omitting incomplete cases and working with an appropriate efficient estimator, which remains efficient. This efficiency transfer holds true for all regression models for which the distribution of \(Y\) given \(X\) and the marginal distribution of \(X\) do not share common parameters. This result is applied to the general homoscedastic semiparametric regression model. This includes models where the conditional expectation is modeled by a complex semiparametric regression function, as well as all basic models such as linear regression and nonparametric regression. The efficiency transfer does not apply to functionals of the joint distribution that also involve the marginal distribution of \(X\). Four types of functional are studied, the finite-dimensional regression parameter, linear functionals of the regression function, functionals of the infinite-dimensional regression parameter, and linear functionals of the error distribution such as the error variance and the error distribution function.

### MSC:

62G08 | Nonparametric regression and quantile regression |

62J02 | General nonlinear regression |

62J05 | Linear regression; mixed models |