the 's are known values (possibly containing outliers), , ; we denote the design matrix by ,
the 's are known positive (heteroscedasticity) constants,
the errors are independent and identically distributed (i.i.d.) random variables (r.v.) with zero expectation and unit variance,
it is assumed that is a non-singular matrix.
Remarks. The i.i.d. assumption on the errors is rather strict. This assumption can be replaced by the assumption that the are identically distributedr.v. such that and if and zero otherwise for all , where denotes expectation w.r.t. model in (). Another generalization obtains by requiring that in place of the conditionalexpectation. If the distribution of the errors is asymmetric with non-zero mean,the regression intercept and the errors are confounded. The slope parameters,however, are identifiable with asymmetric distributions(Carroll and Welsh, 1988). In the context of GREG prediction, however, we deal with prediction underthe model. Thus, identifiability is not an issue.
It is assumed that a sample is drawn from with sampling design such that the independence (orthogonality) structure of the model errors in () is maintained. The sample regression M- and GM-estimator of are defined as the root to the following estimating equations (cf. Hampel et al., 1986, Chapter 6.3)
where
is the sampling weight,
is a generic-function indexed by the robustness tuning constant ,
is the standardized residual, defined as
is a weight function,
is the regression scale which is estimated by the (normalized) weighted median of the absolute deviations from the weighted median of the residuals.
The Huber and Tukey bisquare (biweight) -functions are denoted by, respectively, and . The sample-based estimators of can be written as a weighted least squares problem
where
and denotes the robustness tuning constant.
2 Representation of the robust GREG as a QR-predictor
where and . The sampling weights, , are "embedded" into the -weights in ().
In contrast to the non-robust "standard" GREG predictor, the -weights in () depend on the study variable, , through the choice of the constants . This will be easily recognized once we define the set of constants. The predictors of the population -total that are defined in terms of the constants form the class of QR-predictor due to Wright (1983).
Important. We denote the constants by (qi, bi) instead of (qi, ri) because ri is our notation for the residuals.
In passing we note that can be expressed in a "standard" GREG representation. Let
then in () can be written as
In the next two sections, we define the constants of the QR-predictor.
2.1 Constants of the QR-predictor
The set of constants is defined as
where is given in () and is defined in (). The tuning constant in is the one that is used to estimate .
2.2 Constants of the QR-predictor
The constants are predictor-specific. They depend on the argument type. Moreover, the 's depend on the robustness tuning constant k —which is an argument of svymean_reg() and svytotal_reg()—to control the robustness of the prediction. To distinguish it from the tuning constant , which is used in fitting model in (), it will be denoted by . Seven sets are available.
type = "duchesne": , where is defined in () with replaced by (Duchesne, 1999)
where is a modified Huber -function with tuning constants and (in place of ). Duchesne (1999) suggested the default parametrization and .
2.3 Implementation
Let and , where and are defined in, respectively, () and Section 2.2. Put , where denotes Hadamard multiplication and the square root is applied element by element. The vector-valued -weights, , in () can be written as
Define the QR factorization , where is an orthogonal matrix and is an upper triangular matrix (both of conformable size). Note that the matrix QR-factorization and Wright's QR-estimators have nothing in common besides the name; in particular, and are unrelated. With this we have
and multiplying both sides by , we get which can be solved easily for since is an upper triangular matrix (see base::backsolve()). Thus, the -weights can be computed as
where the matrix need not be explicitly transposed when using base::crossprod(). The terms and are easy to compute. Thus,
3 Variance estimation
Important. Inference of the regression estimator is only implemented under the assumption of representative outliers (in the sense of Chambers, 1986). We do not cover inference in presence of nonrepresentative
outliers.
Our discussion for variance estimation follows the line of reasoning in Särndal et al., (1992, 233-234) on the variance of the non-robust GREG estimator. To this end, denote by , , the census residuals, where is the census parameter. With this, any -weighted predictor can be written as
where we have used the fact that the -weights in () satisfy the calibration property
The first term on the r.h.s. of the last equality in () is a population quantity and does therefore not contribute to the variance of . Thus, we can calculate the variance of the robust GREG predictor by
under the assumptions that (1) the are known quantities and (2) the do not depend on the .
Disregarding the fact that the -weights are sample dependent and substituting the sample residual for in (), Särndal et al. (1992, 233-234 and Result 6.6.1) propose to estimate the variance of the GREG predictor by the -weighted variance of the total . Following the same train of thought and disregarding in addition that the depend on , the variance of can be approximated by
where denotes a variance estimator of a total for the sampling design .
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