madem.density function

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

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

Returns

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

Details

A $d$-variate cdf $F$ on a hyperrectangle c("$[a, b]\n$", "$=[a_1, b_1] \\times \\cdots \\times [a_d, b_d]$") 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.