RXtrisk: True MSE Risk of Shrinkage Resulting from Known Regression Parameters

Description

By specifying numerical values for regression parameters (uncorrelated components and error sigma) that usually are unknown, these functions can calculate and display True MSE Risk statistics associated with shrinkage along a given Q-shaped path.

Usage

RXtrisk(form, data, trug, trus, Q = 0, rscale = 1, steps = 8, qmax = 5, qmin = -5)

Arguments

form

A regression formula [y~x1+x2+...] suitable for use with lm().

data

Data frame containing observations on all variables in the formula.

trug

Column vector of numerical values for the true uncorrelated components of the regression coefficient vector.

trus

Numerical value for the true error standard deviation, Sigma.

Numerical value for the shape parameter controlling shrinkage path curvature. Default shape is Q = 0 for Hoerl-Kennard "ordinary" ridge regression.

rscale

One of three possible choices (0, 1 or 2) for rescaling of variables as they are being "centered" to remove non-essential ill-conditioning: 0 implies no rescaling; 1 implies divide each variable by its standard error; 2 implies rescale as in

steps

Number of equally spaced values per unit change along the horizontal M-extent-of-shrinkage axis where estimates are calculated and displayed in TRACES (default = 8.)

qmax

Maximum allowed Q-shape (default = +5.)

qmin

Minimum allowed Q-shape (default = -5.)

Value

An output list object of class RXtrisk:
formThe regression formula specified as the first argument.
dataName of the data.frame object specified as the second argument.
trugVector of numerical values for the true uncorrelated gamma components.
trusNumerical value for the true error standard deviation, Sigma.
qpNumerical value of the Q-shape actually used for shrinkage.
pNumber of regression predictor variables.
nNumber of complete observations after removal of all missing values.
prinstatListing of principal statistics.
coefMatrix of expected shrinkage-ridge regression coefficients.
rmseMatrix of true MSE risk values for shrunken coefficients.
exevMatrix of true excess eigenvalues (ordinary least squares minus ridge.)
infdMatrix of direction cosines for the true inferior direction, if any.
spatMatrix of shrinkage pattern delta factors.
sextListing of summary statistics for all M-extents-of-shrinkage.

Details

The RXridge() functions calculate maximum likelihood estimates (corrected, if necessary, so as to have correct range) for typical statistical inference situations where regression parameters are unknowns. In sharp contrast with this usual situation, the RXtrisk() functions show exactly how expected regression coefficients and true Mean Squared Error Risk actually do change with shrinkage when regression parameters take on specified, KNOWN numerical values.

Examples

Run this code

data(mpg)  
  form <- mpg~cylnds+cubins+hpower+weight
  rxrobj <- RXridge(form, data=mpg)
  # define true parameter values.
  trugam <- matrix(c(-.5,-.1,.1,-.6),4,1)
  trusig <- 0.4
  # create true shrinkage MSE risk scenario.
  trumse <- RXtrisk(form, data=mpg, trugam, trusig, Q=-1, steps=4)
  plot(trumse)