maple.dr {mable}

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
)

Arguments

x, y

original two sample raw data, codex:"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

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.

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

Author(s)

Zhong Guan <[e-mail]>

References

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