g02hd performs bounded influence regression (-estimates) using an iterative weighted least squares algorithm.
public static void g02hd( G02..::..G02HD_CHI chi, G02..::..G02HD_PSI psi, double psip0, double beta, int indw, int isigma, int n, int m, double[,] x, double y, double wgt, double theta, out int k, ref double sigma, double rs, double tol, double eps, int maxit, int nitmon, out int nit, out int ifail )
Public Shared Sub g02hd ( _ chi As G02..::..G02HD_CHI, _ psi As G02..::..G02HD_PSI, _ psip0 As Double, _ beta As Double, _ indw As Integer, _ isigma As Integer, _ n As Integer, _ m As Integer, _ x As Double(,), _ y As Double(), _ wgt As Double(), _ theta As Double(), _ <OutAttribute> ByRef k As Integer, _ ByRef sigma As Double, _ rs As Double(), _ tol As Double, _ eps As Double, _ maxit As Integer, _ nitmon As Integer, _ <OutAttribute> ByRef nit As Integer, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g02hd( G02..::..G02HD_CHI^ chi, G02..::..G02HD_PSI^ psi, double psip0, double beta, int indw, int isigma, int n, int m, array<double,2>^ x, array<double>^ y, array<double>^ wgt, array<double>^ theta, [OutAttribute] int% k, double% sigma, array<double>^ rs, double tol, double eps, int maxit, int nitmon, [OutAttribute] int% nit, [OutAttribute] int% ifail )
static member g02hd : chi : G02..::..G02HD_CHI * psi : G02..::..G02HD_PSI * psip0 : float * beta : float * indw : int * isigma : int * n : int * m : int * x : float[,] * y : float * wgt : float * theta : float * k : int byref * sigma : float byref * rs : float * tol : float * eps : float * maxit : int * nitmon : int * nit : int byref * ifail : int byref -> unit
- Type: NagLibrary..::..G02..::..G02HD_CHIIf , chi must return the value of the weight function for a given value of its argument. The value of must be non-negative.
A delegate of type G02HD_CHI.If , the actual parameter chi may be the dummy method G02HDZ. (G02HDZ is included in the NAG Library.)
- Type: NagLibrary..::..G02..::..G02HD_PSIpsi must return the value of the weight function for a given value of its argument.
A delegate of type G02HD_PSI.
- Type: System..::..DoubleOn entry: the value of .
- Type: System..::..DoubleOn entry: if , beta must specify the value of .For Huber and Schweppe type regressions, is the th percentile of the standard Normal distribution (see g01fa). For Mallows type regression is the solution towhere is the standard Normal cumulative distribution function (see s15ab).If , beta must specify the value of .where is the standard normal density, i.e., .If , beta is not referenced.Constraint: if , .
- Type: System..::..Int32On entry: determines the type of regression to be performed.
Constraint: , or .
- Huber type regression.
- Mallows type regression.
- Schweppe type regression.
- Type: System..::..Int32On entry: determines how is to be estimated.
Constraint: , or .
- is held constant at its initial value.
- is estimated by median absolute deviation of residuals.
- is estimated using the function.
- Type: System..::..Int32On entry: , the number of observations.Constraint: .
- Type: System..::..Int32On entry: , the number of independent variables.Constraint: .
- Type: array<System..::..Double,2>[,](,)[,][,]An array of size [dim1, m]Note: dim1 must satisfy the constraint:On entry: the values of the matrix, i.e., the independent variables. must contain the th element of , for and .On exit: unchanged, except as described above.
- Type: array<System..::..Double>()On entry: the data values of the dependent variable.must contain the value of for the th observation, for .On exit: unchanged, except as described above.
- Type: array<System..::..Double>()On entry: the weight for the th observation, for .If , during calculations elements of wgt will be transformed as described in [Description]. Before exit the inverse transformation will be applied. As a result there may be slight differences between the input wgt and the output wgt.If , the th observation is not included in the analysis.If , wgt is not referenced.On exit: unchanged, except as described above.
- Type: array<System..::..Double>()An array of size [m]On entry: starting values of the parameter vector . These may be obtained from least squares regression. Alternatively if and or if and sigma approximately equals the standard deviation of the dependent variable, , then , for may provide reasonable starting values.On exit: the M-estimate of , for .
- Type: System..::..Int32%On exit: the column rank of the matrix .
- Type: System..::..Double%On entry: a starting value for the estimation of . sigma should be approximately the standard deviation of the residuals from the model evaluated at the value of given by theta on entry.Constraint: .On exit: the final estimate of if or the value assigned on entry if .
- Type: array<System..::..Double>()
- Type: System..::..DoubleOn entry: the relative precision for the final estimates. Convergence is assumed when both the relative change in the value of sigma and the relative change in the value of each element of theta are less than tol.It is advisable for tol to be greater than .Constraint: .
- Type: System..::..Double
- Type: System..::..Int32On entry: the maximum number of iterations that should be used during the estimation.A value of should be adequate for most uses.Constraint: .
- Type: System..::..Int32On entry: determines the amount of information that is printed on each iteration.
When printing occurs the output is directed to the current advisory message unit (see (X04ABF not in this release)).
- No information is printed.
- On the first and every nitmon iterations the values of sigma, theta and the change in theta during the iteration are printed.
- Type: System..::..Int32%On exit: the number of iterations that were used during the estimation.
For the linear regression model
g02hd calculates the M-estimates given by the solution, , to the equation
or as the solution to
|where||is a vector of length of the dependent variable,|
|is a by matrix of independent variables of column rank ,|
|is a vector of length of unknown parameters,|
|and||is a vector of length of unknown errors with var ,|
|where||is the th residual, i.e., the th element of the vector ,|
|is a suitable weight function,|
|are suitable weights such as those that can be calculated by using output from g02hb,|
|and||may be estimated at each iteration by the median absolute deviation of the residuals|
for a suitable weight function , where and are constants, chosen so that the estimator of is asymptotically unbiased if the errors, , have a Normal distribution. Alternatively may be held at a constant value.
The above describes the Schweppe type regression. If the are assumed to equal for all , then Huber type regression is obtained. A third type, due to Mallows, replaces (1) by
This may be obtained by use of the transformations
(see Marazzi (1987)).
The calculation of the estimates of can be formulated as an iterative weighted least squares problem with a diagonal weight matrix given by
The value of at each iteration is given by the weighted least squares regression of on . This is carried out by first transforming the and by
and then using (F04JGF not in this release) . If is of full column rank then an orthogonal-triangular () decomposition is used; if not, a singular value decomposition is used.
Observations with zero or negative weights are not included in the solution.
Note: there is no explicit provision in the method for a constant term in the regression model. However, the addition of a dummy variable whose value is for all observations will produce a value of corresponding to the usual constant term.
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
Note: g02hd may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (LDX) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
On entry, , or , or , On entry, , and , or . On entry, , or .
- A value returned by the chi function is negative.
- During iterations a value of was encountered.
- A failure occurred in (F04JGF not in this release) . This is an extremely unlikely error. If it occurs, please contact NAG.
- The weighted least squares equations are not of full rank. This may be due to the matrix not being of full rank, in which case the results will be valid. It may also occur if some of the values become very small or zero, see [Further Comments]. The rank of the equations is given by k. If the matrix just fails the test for nonsingularity then the result and is possible (see (F04JGF not in this release)).
- The method has failed to converge in maxit iterations.
- Having removed cases with zero weight, the value of , i.e., no degree of freedom for error. This error will only occur if .
The accuracy of the results is controlled by tol. For the accuracy of the weighted least squares see (F04JGF not in this release).
In cases when it is important for the value of sigma to be of a reasonable magnitude. Too small a value may cause too many of the winsorized residuals, i.e., , to be zero, which will lead to convergence problems and may trigger the error.
By suitable choice of the functions chi and psi this method may be used for other applications of iterative weighted least squares.
For the variance-covariance matrix of see g02hf.
Having input , and the weights, a Schweppe type regression is performed using Huber's function. The method BETCAL calculates the appropriate value of .