madem.density: Minimum Approximate Distance Estimate of univariate Density...

In mable: Maximum Approximate Bernstein/Beta Likelihood Estimation

Minimum Approximate Distance Estimate of univariate Density Function with given model degree(s)

Description

Minimum Approximate Distance Estimate of univariate Density Function with given model degree(s)

Usage

madem.density(
  x,
  m,
  p = rep(1, prod(m + 1))/prod(m + 1),
  interval = NULL,
  method = c("qp", "em"),
  maxit = 10000,
  eps = 1e-07
)

Arguments

x	an n x d matrix or data.frame of multivariate sample of size n
m	a positive integer or a vector of d positive integers specify the given model degrees for the joint density.
p	initial guess of p
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])).
method	method for finding minimum distance estimate. "em": EM like method;
maxit	the maximum iterations
eps	the criterion for convergence

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.

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

mable documentation built on Oct. 1, 2024, 9:06 a.m.