maple.dr: Maximum approximate profile likelihood estimate of the density ratio model

Description

Select optimal degree with a given regression coefficients.

Usage

maple.dr(
  x,
  y,
  M,
  regr,
  ...,
  interval = c(0, 1),
  alpha = NULL,
  vb = 0,
  baseline = NULL,
  controls = mable.ctrl(),
  progress = TRUE,
  message = TRUE
)

Value

A list with components

m the given or a selected degree by method of change-point
p the estimated vector of mixture proportions \(p = (p_0, \ldots, p_m)\) with the given or selected degree m
alpha the given regression coefficients
mloglik the maximum log-likelihood at degree m
interval support/truncation interval (a,b)
baseline ="control" if \(f_0\) is used as baseline, or ="case" if \(f_1\) is used as baseline.
M the vector (m0, m1), where m1, if greater than m0, is the largest candidate when the search stoped
lk log-likelihoods evaluated at \(m \in \{m_0, \ldots, m_1\}\)
lr likelihood ratios for change-points evaluated at \(m \in \{m_0+1, \ldots, m_1\}\)
pval the p-values of the change-point tests for choosing optimal model degree
chpts the change-points chosen with the given candidate model degrees

Arguments

x, y: original two sample raw data, x:"Control", y: "Case".
M: a positive integer or a vector (m0, m1).
regr: regressor vector function \(r(x)=(1,r_1(x),...,r_d(x))\) which returns n x (d+1) matrix, n=length(x)
...: additional arguments to be passed to regr
interval: a vector (a,b) containing the endpoints of supporting/truncation interval of x and y.
alpha: a given regression coefficient, missing value is imputed by logistic regression
vb: code for vanishing boundary constraints, -1: f0(a)=0 only, 1: f0(b)=0 only, 2: both, 0: none (default).
baseline: the working baseline, "Control" or "Case", if NULL it is chosen to the one with smaller estimated lower bound for model degree.
controls: Object of class mable.ctrl() specifying iteration limit and the convergence criterion for EM and Newton iterations. Default is mable.ctrl. See Details.
progress: logical: should a text progressbar be displayed
message: logical: should warning messages be displayed

Author

Zhong Guan <zguan@iu.edu>

Details

Suppose that ("control") and y ("case") are independent samples from f0 and f1 which satisfy f1(x)=f0(x)exp[alpha0+alpha'r(x)] with r(x)=(r1(x),...,r_d(x)). Maximum approximate Bernstein/Beta likelihood estimates of f0 and f1 are calculated with a given regression coefficients which are efficient estimates provided by other semiparametric methods such as logistic regression. If support is (a,b) then replace r(x) by r[a+(b-a)x]. For a fixed m, using the Bernstein polynomial model for baseline \(f_0\), MABLEs of \(f_0\) and parameters alpha can be estimated by EM algorithm and Newton iteration. If estimated lower bound \(m_b\) for m based on y is smaller that that based on x, then switch x and y and \(f_1\) is used as baseline. If M=m or m0=m1=m, then m is a preselected degree. If m0<m1 it specifies the set of consective candidate model degrees m0:m1 for searching an optimal degree by the change-point method, where m1-m0>3.

References

Guan, Z., Maximum Approximate Bernstein Likelihood Estimation of Densities in a Two-sample Semiparametric Model