naginterfaces.library.correg.linregm_​estfunc

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

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://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02dnf.html

Parameters

irankint

$k$, the rank of the independent variables.

bfloat, array-like, shape $\left(\text{ip}\right)$

The $\text{ip}$ values of the estimates of the parameters of the model, $\hat{\beta }$.

covfloat, array-like, shape $(\text{ip}\times (\text{ip}+1)/2)$

The upper triangular part of the variance-covariance matrix of the $\text{ip}$ parameter estimates given in $b$. They are stored packed by column, i.e., the covariance between the parameter estimate given in $b[i-1]$ and the parameter estimate given in $b[j-1]$, $j\ge i$, is stored in $cov[(j\times (j-1)/2+i)-1]$.

pfloat, array-like, shape $(\text{ip}\times \text{ip}+2\times \text{ip})$

As returned by linregm_fit() and linregm_update().

ffloat, array-like, shape $\left(\text{ip}\right)$

$f$, the linear function to be estimated.

tolfloat, optional

$\eta$, the tolerance value used in the check for estimability.

If $tol\le 0.0$ then $\sqrt{ϵ}$, where $ϵ$ is the machine precision, is used instead.

Returns

estbool

Indicates if the function was estimable.

$est=True$

The function is estimable.

$est=False$

The function is not estimable and $stat$, $sestat$ and $t$ are not set.

statfloat

If $est=True$, $stat$ contains the estimate of the function, $f^T\hat{\beta }$.

sestatfloat

If $est=True$, $sestat$ contains the standard error of the estimate of the function, $se\left(F\right)$.

tfloat

If $est=True$, $t$ contains the $t$-statistic for the test of the function being equal to zero.

Raises

NagValueError

(errno $1$)

On entry, $irank=⟨value⟩$ and $\text{ip}=⟨value⟩$.

Constraint: $irank\le \text{ip}$.

(errno $1$)

On entry, $irank=⟨value⟩$.

Constraint: $irank\ge 1$.

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

Constraint: $\text{ip}\ge 1$.

(errno $3$)

Standard error of statistic $=0.0$; this may be due to rounding errors if the standard error is very small or due to mis-specified inputs $cov$ and $f$.

Warns

NagAlgorithmicWarning

(errno $2$)

On entry, $irank=\text{ip}$, i.e., model of full rank. In this case $est$ is returned as true and all statistics are calculated.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

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, $\hat{\beta }$, and their variance-covariance matrix. Given the upper triangular matrix $R$ obtained from the $QR$ decomposition of the independent variables the SVD gives

$$\begin{array}{c}R=Q_{\ast }\left(\begin{array}{cc}D& 0\\ 0& 0\end{array}\right)P^T\text{,}\end{array}$$

where $D$ is a $k\times k$ diagonal matrix with nonzero diagonal elements, $k$ being the rank of $R$, and $Q_{\ast }$ and $P$ are $p\times p$ orthogonal matrices. This gives the solution

$$\hat{\beta }=P_1{D}^{-1}{Q}_{{\ast }_1}^T{c}_1\text{,}$$

$P_1$ being the first $k$ columns of $P$, i.e., $P=\left(P_1{P}_0\right)$, $Q_{{\ast }_1}$ being the first $k$ columns of $Q_{\ast }$, and $c_1$ being the first $p$ elements of $c$.

Details of the SVD are made available in the form of the matrix $P^{\ast }$:

$$\begin{array}{c}{P}^{\ast }=\left(\begin{array}{c}{D}^{-1}{P}_1^T\\ P_0^T\end{array}\right)\text{,}\end{array}$$

as given by linregm_fit() and linregm_update().

A linear function of the parameters, $F=f^T\beta$, can be tested to see if it is estimable by computing $\zeta =P_0^{T}f$. If $\zeta$ is zero, then the function is estimable; if not, the function is not estimable. In practice $\left|\zeta \right|$ is tested against some small quantity $\eta$.

Given that $F$ is estimable it can be estimated by $f^T\hat{\beta }$ and its standard error calculated from the variance-covariance matrix of $\hat{\beta }$, $C_{\beta }$, as

$$se\left(F\right)=\sqrt{f^T{C}_{\beta }f}\text{.}$$

Also a $t$-statistic,

$$t=\frac{f^T\hat{\beta }}{se\left(F\right)}\text{,}$$

can be computed. The $t$-statistic will have a Student’s $t$-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