NAG Library Function Document
nag_robust_m_regsn_param_var (g02hfc)
1
Purpose
nag_robust_m_regsn_param_var (g02hfc) calculates an estimate of the asymptotic variance-covariance matrix for the bounded influence regression estimates (M-estimates). It is intended for use with
nag_robust_m_regsn_user_fn (g02hdc).
2
Specification
#include <nag.h> |
#include <nagg02.h> |
void |
nag_robust_m_regsn_param_var (Nag_OrderType order,
double |
(*psi)(double t,
Nag_Comm *comm),
|
|
double |
(*psp)(double t,
Nag_Comm *comm),
|
|
Nag_RegType regtype,
Nag_CovMatrixEst covmat_est,
double sigma,
Integer n,
Integer m,
const double x[],
Integer pdx,
const double rs[],
const double wgt[],
double cov[],
Integer pdc,
double comm_arr[],
Nag_Comm *comm,
NagError *fail) |
|
3
Description
For a description of bounded influence regression see
nag_robust_m_regsn_user_fn (g02hdc). Let
be the regression arguments and let
be the asymptotic variance-covariance matrix of
. Then for Huber type regression
where
see
Huber (1981) and
Marazzi (1987).
For Mallows and Schweppe type regressions,
is of the form
where
and
.
is a diagonal matrix such that the th element approximates in the Schweppe case and in the Mallows case.
is a diagonal matrix such that the th element approximates in the Schweppe case and in the Mallows case.
Two approximations are available in
nag_robust_m_regsn_param_var (g02hfc):
1. |
Average over the
|
2. |
Replace expected value by observed
|
In all cases is a robust estimate of .
nag_robust_m_regsn_param_var (g02hfc) is based on routines in ROBETH; see
Marazzi (1987).
4
References
Hampel F R, Ronchetti E M, Rousseeuw P J and Stahel W A (1986) Robust Statistics. The Approach Based on Influence Functions Wiley
Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Subroutines for robust and bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 2 Institut Universitaire de Médecine Sociale et Préventive, Lausanne
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– function, supplied by the userExternal Function
-
psi must return the value of the
function for a given value of its argument.
The specification of
psi is:
double |
psi (double t,
Nag_Comm *comm)
|
|
- 1:
– doubleInput
-
On entry: the argument for which
psi must be evaluated.
- 2:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
psi.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
nag_robust_m_regsn_param_var (g02hfc) you may allocate memory and initialize these pointers with various quantities for use by
psi when called from
nag_robust_m_regsn_param_var (g02hfc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: psi should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_robust_m_regsn_param_var (g02hfc). If your code inadvertently
does return any NaNs or infinities,
nag_robust_m_regsn_param_var (g02hfc) is likely to produce unexpected results.
- 3:
– function, supplied by the userExternal Function
-
psp must return the value of
for a given value of its argument.
The specification of
psp is:
double |
psp (double t,
Nag_Comm *comm)
|
|
- 1:
– doubleInput
-
On entry: the argument for which
psp must be evaluated.
- 2:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
psp.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
nag_robust_m_regsn_param_var (g02hfc) you may allocate memory and initialize these pointers with various quantities for use by
psp when called from
nag_robust_m_regsn_param_var (g02hfc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: psp should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_robust_m_regsn_param_var (g02hfc). If your code inadvertently
does return any NaNs or infinities,
nag_robust_m_regsn_param_var (g02hfc) is likely to produce unexpected results.
- 4:
– Nag_RegTypeInput
-
On entry: the type of regression for which the asymptotic variance-covariance matrix is to be calculated.
-
- Mallows type regression.
- Huber type regression.
-
- Schweppe type regression.
Constraint:
, or .
- 5:
– Nag_CovMatrixEstInput
-
On entry: if
,
covmat_est must specify the approximation to be used.
If , averaging over residuals.
If , replacing expected by observed.
If
,
covmat_est is not referenced.
Constraint:
or .
- 6:
– doubleInput
-
On entry: the value of
, as given by
nag_robust_m_regsn_user_fn (g02hdc).
Constraint:
.
- 7:
– IntegerInput
-
On entry: , the number of observations.
Constraint:
.
- 8:
– IntegerInput
-
On entry: , the number of independent variables.
Constraint:
.
- 9:
– const doubleInput
-
Note: the dimension,
dim, of the array
x
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the values of the matrix, i.e., the independent variables.
must contain the th element of , for and .
- 10:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 11:
– const doubleInput
-
On entry: the residuals from the bounded influence regression. These are given by
nag_robust_m_regsn_user_fn (g02hdc).
- 12:
– const doubleInput
-
On entry: if
,
wgt must contain the vector of weights used by the bounded influence regression. These should be used with
nag_robust_m_regsn_user_fn (g02hdc).
If
,
wgt is not referenced.
- 13:
– doubleOutput
-
Note: the dimension,
dim, of the array
cov
must be at least
.
The
th element of the matrix is stored in
- when ;
- when .
On exit: the estimate of the variance-covariance matrix.
- 14:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
cov.
Constraint:
.
- 15:
– doubleOutput
-
Note: the dimension,
dim, of the array
comm_arr
must be at least
.
On exit: if ,
, for , will contain the diagonal elements of the matrix and
, for , will contain the diagonal elements of matrix .
- 16:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
- 17:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CORRECTION_FACTOR
-
Either the value of ,
or ,
or .
In this situation nag_robust_m_regsn_param_var (g02hfc) returns as .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_POS_DEF
-
matrix not positive definite.
- NE_REAL
-
On entry, .
Constraint: .
- NE_SINGULAR
-
matrix is singular or almost singular.
7
Accuracy
In general, the accuracy of the variance-covariance matrix will depend primarily on the accuracy of the results from
nag_robust_m_regsn_user_fn (g02hdc).
8
Parallelism and Performance
nag_robust_m_regsn_param_var (g02hfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_robust_m_regsn_param_var (g02hfc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
nag_robust_m_regsn_param_var (g02hfc) is only for situations in which has full column rank.
Care has to be taken in the choice of the function since if for too wide a range then either the value of will not exist or too many values of will be zero and it will not be possible to calculate .
10
Example
The asymptotic variance-covariance matrix is calculated for a Schweppe type regression. The values of , and the residuals and weights are read in. The averaging over residuals approximation is used.
10.1
Program Text
Program Text (g02hfce.c)
10.2
Program Data
Program Data (g02hfce.d)
10.3
Program Results
Program Results (g02hfce.r)