made.density: Minimum Approximate Distance Estimate of Density Function...
In mable: Maximum Approximate Bernstein/Beta Likelihood Estimation

made.density

R Documentation

Minimum Approximate Distance Estimate of Density Function with an optimal model degree

Description

Minimum Approximate Distance Estimate of Density Function with an optimal model degree

Usage

made.density(
  x,
  M0 = 1L,
  M,
  search = TRUE,
  interval = NULL,
  mar.deg = TRUE,
  method = c("qp", "em"),
  controls = mable.ctrl(),
  progress = TRUE
)

Arguments

`x`	an `n x d` matrix or `data.frame` of multivariate sample of size `n`
`M0`	a positive integer or a vector of `d` positive integers specify starting candidate degrees for searching optimal degrees.
`M`	a positive integer or a vector of `d` positive integers specify the maximum candidate or the given model degrees for the joint density.
`search`	logical, whether to search optimal degrees between `M0` and `M` or not but use `M` as the given model degrees for the joint density.
`interval`	a vector of two endpoints or a `2 x d` matrix, each column containing the endpoints of support/truncation interval for each marginal density. If missing, the i-th column is assigned as `c(min(x[,i]), max(x[,i]))`.
`mar.deg`	logical, if TRUE, the optimal degrees are selected based on marginal data, otherwise, the optimal degrees are chosen the joint data. See details.
`method`	method for finding minimum distance estimate. "em": EM like method;
`controls`	Object of class `mable.ctrl()` specifying iteration limit and the convergence criterion `eps`. Default is `mable.ctrl`. See Details.
`progress`	if TRUE a text progressbar is displayed

Details

A d-variate cdf F on a hyperrectangle [a, b] =[a_1, b_1] \times \cdots \times [a_d, b_d] can be approximated by a mixture of d-variate beta cdfs on [a, b], \beta_{mj}(x) = \prod_{i=1}^dB_{m_i,j_i}[(x_i-a_i)/(b_i-a_i)], with proportion p(j_1, \ldots, j_d), 0 \le j_i \le m_i, i = 1, \ldots, d. With a given model degree m, the parameters p, the mixing proportions of the beta distribution, are calculated as the minimizer of the approximate L_2 distance between the empirical distribution and the Bernstein polynomial model. The quadratic programming with linear constraints is used to solve the problem. If search=TRUE then the model degrees are chosen using a method of change-point based on the marginal data if mar.deg=TRUE or the joint data if mar.deg=FALSE. If search=FALSE, then the model degree is specified by M.

Value

An invisible mable object with components

m the given model degree(s)
p the estimated vector of mixture proportions with the given optimal degree(s) m
interval support/truncation interval [a, b]
D the minimum distance at degree m
convergence An integer code. 0 indicates successful completion(the EM iteration is convergent). 1 indicates that the iteration limit maxit had been reached in the EM iteration;