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NAG[g02hfc] NAG[nag_robust_m_regsn_param_var] - Robust regression, variance-covariance matrix following g02hdc (nag_robust_m_regsn_user_fn)
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Calling Sequence
g02hfc(psi, psp, regtype, covmat_est, sigma, m, x, rs, wgt, cov, comm_arr, 'n'=n, 'comm'=comm, 'fail'=fail)
nag_robust_m_regsn_param_var(. . .)
Parameters
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psi - procedure;
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psi must return the value of the  function for a given value of its argument.
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t - float;
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On entry: the argument for which psi must be evaluated.
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comm - table;
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A Maple table, which should be generated using NAG[Nag_Comm], corresponding to the Nag_Comm structure.
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Before calling nag_robust_m_regsn_param_var (g02hfc) this field may be initialized for use by psi when called from nag_robust_m_regsn_param_var (g02hfc).
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psp - procedure;
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psp must return the value of  for a given value of its argument.
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t - float;
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On entry: the argument for which psp must be evaluated.
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comm - table;
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A Maple table, which should be generated using NAG[Nag_Comm], corresponding to the Nag_Comm structure.
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Before calling nag_robust_m_regsn_param_var (g02hfc) this field may be initialized for use by psp when called from nag_robust_m_regsn_param_var (g02hfc).
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regtype - String;
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On entry: the type of regression for which the asymptotic variance-covariance matrix is to be calculated.
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 Huber type regression.
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 Mallows type regression.
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 Schweppe type regression.
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covmat_est - String;
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On entry: if  , covmat_est must specify the approximation to be used.
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If  , averaging over residuals.
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If  , replacing expected by observed.
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If  , covmat_est is not referenced.
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sigma - float;
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Constraint:  . .
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m - integer;
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On entry:  , the number of independent variables.
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Constraint:  . .
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x - Matrix(1..dim1, 1..m, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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rs - Vector(1..n, datatype=float[8]);
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wgt - Vector(1..n, datatype=float[8]);
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If  , wgt is not referenced.
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cov - Matrix(1..dim1, 1..m, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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On exit: the estimate of the variance-covariance matrix.
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comm_arr - Vector(1.. , datatype=float[8]);
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Note: the dimension, dim, of the array comm_arr must be at least  .
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'n'=n - integer; (optional)
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On entry:  , the number of observations.
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Constraint:  . .
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'comm'=comm - table; (optional)
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A Maple table, which should be generated using NAG[Nag_Comm], corresponding to the Nag_Comm structure.
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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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 g02hdc (nag_robust_m_regsn_user_fn).
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Description
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For a description of bounded influence regression see g02hdc (nag_robust_m_regsn_user_fn). Let  be the regression arguments and let  be the asymptotic variance-covariance matrix of  . Then for Huber type regression
where
![f[H]=1/(n-m)((∑)psi^2(r[i],/,sigma))/(((1/n,∑,psi',((r[i])/(sigma))))^2)kappa^2](/support/helpjp/helpview.aspx?si=10010/file08436/math382.png)
![kappa^2=1+m/n(1/n(∑)((psi',(r[i],/,sigma),-,1/n,(∑),psi',(r[i],/,sigma)))^2)/(((1/n,(∑),psi',((r[i])/(sigma))))^2)](/support/helpjp/helpview.aspx?si=10010/file08436/math384.png)
see Huber (1981) and Marazzi (1987b).
For Mallows and Schweppe type regressions,  is of the form
where ![S[1] = X^T*D*X/n](/support/helpjp/helpview.aspx?si=10010/file08436/math395.png) and ![S[2] = X^T*P*X/n](/support/helpjp/helpview.aspx?si=10010/file08436/math397.png) .
 is a diagonal matrix such that the  th element approximates ![E(psi',(r[i],/,(sigma,w[i])))](/support/helpjp/helpview.aspx?si=10010/file08436/math404.png) in the Schweppe case and ![E(psi',(r[i],/,sigma),w[i])](/support/helpjp/helpview.aspx?si=10010/file08436/math406.png) in the Mallows case.
 is a diagonal matrix such that the  th element approximates ![E(psi^2,(r[i],/,(sigma,w[i])),w[i]^2)](/support/helpjp/helpview.aspx?si=10010/file08436/math413.png) in the Schweppe case and ![E(psi^2,(r[i],/,sigma),w[i]^2)](/support/helpjp/helpview.aspx?si=10010/file08436/math415.png) in the Mallows case.
Two approximations are available in nag_robust_m_regsn_param_var (g02hfc):
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Average over the ![r[i]](/support/helpjp/helpview.aspx?si=10010/file08436/math421.png)
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Replace expected value by observed
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See Hampel et al. (1986) and Marazzi (1987b).
In all cases  is a robust estimate of  .
nag_robust_m_regsn_param_var (g02hfc) is based on routines in ROBETH; see Marazzi (1987b).
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Error Indicators and Warnings
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"NE_ALLOC_FAIL"
Dynamic memory allocation failed.
"NE_BAD_PARAM"
On entry, argument  had an illegal value.
"NE_CORRECTION_FACTOR"
Correction factor = 0 (Huber type regression).
"NE_INT"
On entry,  . Constraint:  .
On entry,  . Constraint:  .
"NE_INT_2"
On entry,  ,  . Constraint:  .
On entry,  :  ,  .
"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 consult NAG for assistance.
"NE_POS_DEF"
 matrix not positive-definite.
"NE_REAL"
On entry,  . Constraint:  .
"NE_REAL_ARRAY_ELEM_CONS"
On entry, an element of  .
"NE_SINGULAR"
 matrix is singular or almost singular.
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Examples
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>
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psi := proc(t, comm)
if (t <= -1.5) then
-1.5:
elif (t >= 1.5) then
1.5:
else
t:
end if:
end proc:
psp := proc(t, comm)
if (abs(t) < 1.5) then
1.0:
else
0.0:
end if:
end proc:
regtype := "Nag_SchweppeReg":
covmat_est := "Nag_CovMatAve":
sigma := 20.7783:
n := 5:
m := 3:
comm := NAG:-Nag_Comm():
x := Matrix([[1, -1, -1], [1, -1, 1], [1, 1, -1], [1, 1, 1], [1, 0, 3]], datatype=float[8]):
rs := Vector([0.5643, -1.1286, 0.5643, -1.1286, 1.1286], datatype=float[8]):
wgt := Vector([0.4039, 0.5012, 0.4039, 0.5012, 0.3862], datatype=float[8]):
cov := Matrix(3, 3, datatype=float[8]):
comm_arr := Vector(37, datatype=float[8]):
NAG:-g02hfc(psi, psp, regtype, covmat_est, sigma, m, x, rs, wgt, cov, comm_arr, 'n' = n, 'comm' = comm):
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See Also
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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 (1987b) 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
g02 Chapter Introduction.
NAG Toolbox Overview.
NAG Web Site.
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