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