Calculates the maximum adjusted profile likelihood estimates of the variance parameters for a nonlinear heteroscedastic model.
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a numerical vector whose elements are named after the variance
parameters appearing in the nonlinear model. These will be fixed
to the values specified in
character value indicating which correction term to use.
Admissible values are
a list of iteration and algorithmic constants. See the Details section below for their definition.
logical flag. If
absorbs any additional argument.
mpl.nlreg routine returns nearly unbiased estimates of the
variance parameters of a nonlinear heteroscedastic regression model
by maximizing the corresponding adjusted profile likelihood
(Barndorff-Nielsen, 1983). More precisely, it implements two
approximations derived from the theories developed respectively by
Skovgaard (1996) and Fraser, Reid and Wu (1999). The
core algorithm alternates
minimization of minus the adjusted profile log likelihood with
respect to the variance parameters, and minimization of minus the
profile log likelihood with respect to the regression coefficients.
The first step is omitted if the
offset argument is used in
mpl.nlreg returns the constrained maximum
likelihood estimates of the regression coefficients. The
optim is used in both
steps. Starting values are retrieved from the
passed through the
The algorithm iterates until convergence or until the maximum number
of iterations is reached. The stopping rule considers the relative
difference between successive estimates of the variance parameters
and the relative increment of the adjusted profile log likelihood.
These are governed by the parameters
offset argument is used, the relative difference
between successive estimates of the regression coefficients and the
relative increment of the profile log likelihood are considered
instead. If convergence has been reached, the results are saved in
an object of class
mpl. The output can be examined by
Components can be extracted using
The theory is outlined in Brazzale (2000, Sections 3.1 and 3.2.3). Details of the implementation are given in Brazzale (2000, Section 6.3.1).
an object of class
mpl which inherits from
mpl.object for details.
trace = TRUE and
offset = NULL, the iteration number
and the corresponding adjusted profile log likelihood are printed.
control which controls the convergence criteria
plays an important role. Fine-tuning of this argument helps
surrounding a well-known problem in nonlinear regression, that is,
convergence failure in cases where the likelihood and/or the adjusted
profile likelihood are very flat.
Barndorff-Nielsen, O. E. (1983) On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.
Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.
Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 249–264.
Skovgaard, I. (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145–165.
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data(metsulfuron) metsulfuron.nl <- nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, hoa = TRUE, start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)) ) ## ## MMPLE of the variance parameters ## metsulfuron.mpl <- mpl( metsulfuron.nl, trace = TRUE ) summary( metsulfuron.mpl, corr = FALSE ) ## ## constrained MLEs of the regression coefficients ## metsulfuron.mpl <- mpl( metsulfuron.nl, offset = metsulfuron.nl$varPar, trace = TRUE ) summary( metsulfuron.mpl, corr = FALSE )
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