naginterfaces.library.correg.linregm_​estfunc¶

naginterfaces.library.correg.linregm_estfunc(irank, b, cov, p, f, tol=0.0)[source]

linregm_estfunc gives the estimate of an estimable function along with its standard error.

For full information please refer to the NAG Library document for g02dn

https://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/g02/g02dnf.html

Parameters
irankint

, the rank of the independent variables.

bfloat, array-like, shape

The values of the estimates of the parameters of the model, .

covfloat, array-like, shape

The upper triangular part of the variance-covariance matrix of the parameter estimates given in . They are stored packed by column, i.e., the covariance between the parameter estimate given in and the parameter estimate given in , , is stored in .

pfloat, array-like, shape

As returned by linregm_fit() and linregm_update().

ffloat, array-like, shape

, the linear function to be estimated.

tolfloat, optional

, the tolerance value used in the check for estimability.

If then , where is the machine precision, is used instead.

Returns
estbool

Indicates if the function was estimable.

The function is estimable.

The function is not estimable and , and are not set.

statfloat

If , contains the estimate of the function, .

sestatfloat

If , contains the standard error of the estimate of the function, .

tfloat

If , contains the -statistic for the test of the function being equal to zero.

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

Standard error of statistic ; this may be due to rounding errors if the standard error is very small or due to mis-specified inputs and .

Warns
NagAlgorithmicWarning
(errno )

On entry, , i.e., model of full rank. In this case is returned as true and all statistics are calculated.

Notes

linregm_estfunc computes the estimates of an estimable function for a general linear regression model which is not of full rank. It is intended for use after a call to linregm_fit() or linregm_update(). An estimable function is a linear combination of the parameters such that it has a unique estimate. For a full rank model all linear combinations of parameters are estimable.

In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates, , and their variance-covariance matrix. Given the upper triangular matrix obtained from the decomposition of the independent variables the SVD gives

where is a diagonal matrix with nonzero diagonal elements, being the rank of , and and are orthogonal matrices. This gives the solution

being the first columns of , i.e., , being the first columns of , and being the first elements of .

Details of the SVD are made available in the form of the matrix :

as given by linregm_fit() and linregm_update().

A linear function of the parameters, , can be tested to see if it is estimable by computing . If is zero, then the function is estimable; if not, the function is not estimable. In practice is tested against some small quantity .

Given that is estimable it can be estimated by and its standard error calculated from the variance-covariance matrix of , , as

Also a -statistic,

can be computed. The -statistic will have a Student’s -distribution with degrees of freedom as given by the degrees of freedom for the residual sum of squares for the model.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

Hammarling, S, 1985, The singular value decomposition in multivariate statistics, SIGNUM Newsl. (20(3)), 2–25

Searle, S R, 1971, Linear Models, Wiley