Tobias Schoch
Model-Based Covariance Estimation for Regression M- and GM-Estimators1 Introduction2 Covariance estimation3 ImplementationLiterature
The population regression model is given by
where the population
It is assumed that a sample
where the function
M-estimator | Mallows GM-estimator | Schweppe GM-Estimator |
---|---|---|
where
The model-based covariance matrix of
where
and
M-estimator | Mallows GM-estimator | Schweppe GM-Estimator |
---|---|---|
where
and
Remarks
MASS
Venables and Ripley (2002) and robeth
Marazzi (2020).robeth
for the option averaged; see Marazzi (1993, Chap. 4) and Marazzi (1987, Chap. 2.6) on the earlier ROBETH-85
implementation.The main function—which is only a wrapper function— is cov_reg_model
. The following display shows pseudo code of the main function.
cov_reg_model()
{
get_psi_function() // get psi function (function pointer)
get_psi_prime_function() // get psi-prime function (function pointer)
switch(type) {
case 0: cov_m_est() // M-estimator
case 1: cov_mallows_gm_est() // Mallows GM-estimator
case 2: cov_schweppe_gm_est() // Schweppe GM-estimator
}
robsurvey_error() // terminate with error in case of failure
}
The functions cov_m_est()
, cov_mallows_gm_est()
, and cov_schweppe_gm_est()
implement the covariance estimators; see below. All functions are based on the subroutines in BLAS
(Anderson et al., 1999) and LAPACK
(Blackford et al., 2002).
To fix notation, denote the Hadamard product of the matrices
M-estimator (cov_m_est
). The covariance matrix is (up to a scalar) equal to
and is computed as follows:
dgeqrf
).dtrtri
).dtrmm
).Mallows GM-estimator (cov_mallows_gm_est
). The covariance matrix is (up to a scalar) equal to
and is computed as follows:
dgeqrf
).dtrtri
).dorgqr
).dtrmm
).dgemm
).Schweppe GM-estimator (cov_schweppe_gm_est
). The covariance matrix is (up to a scalar) equal to
Put
dgeqrf
).dtrtri
).dorgqr
).dtrmm
).dgemm
).Note. Marazzi (1987) uses the Cholesky factorization (see subroutines RTASKV
and RTASKW
) which is computationally a bit cheaper than our QR factorization.
ANDERSON, E., BAI, Z., BISCHOF, C., BLACKFORD, L. S. , DEMMEL, J., DONGARRA, J., CROZ, J. D. , GREENHAUM, A., HAMMARLING, S., MCKENNEY, A. AND SORENSEN, D. (1999). LAPACK Users’ Guide. Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 3rd edition. DOI: 10.1137/1.9780898719604
BLACKFORD, L. S., PETITET, A., POZO, R., REMINGTON, K., WHALEY, R. C., DEMMEL, J., DONGARRA, J., DUFF, I., HAMMARLING, S., HENRY, G., HEROUX, M., KAUFMAN, L. AND LUMSDAINE, A. (2002). An Updated Set of Basic Linear Algebra Subprograms (BLAS). ACM Transactions on Mathematical Software 28, 135–151. DOI: 10.1145/567806.567807
HAMPEL, F. R., RONCHETTI, E. M., ROUSSEEUW, P. J. AND STAHEL, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions, New York: John Wiley and Sons. DOI: 10.1002/9781118186435
HUBER, P. J. (1981). Robust Statistics, New York: John Wiley and Sons. DOI: 10.1002/ 0471725250
MARAZZI, A. (1987). Subroutines for robust and bounded influence regression in ROBETH, Cahiers de Recherches et de Documentation, 3 ROBETH 2, Division de Statistique et Informatique, Institut Universitaire de Médecine Sociale et Préventive, Lausanne, ROBETH-85 Document No. 2, August 1985, revised April 1987.
MARAZZI, A. (1993). Algorithms, Routines, and S Functions for Robust Statistics: The FORTRAN Library ROBETH with an interface to S-PLUS, New York: Chapman and Hall.
MARAZZI, A. (2020). robeth: R Functions for Robust Statistics, R package version 2.7-6.
VENABLES, W. N. AND RIPLEY, B. D. (2002). Modern Applied Statistics with S, New York: Springer, 4th edition. DOI: 10.1007/978-0-387-21706-2