Minimum Approximate Distance Estimate of Density Function with an optimal model degree
made.density(
x,
M0 = 1L,
M,
search = TRUE,
interval = NULL,
mar.deg = TRUE,
method = c("qp", "em"),
controls = mable.ctrl(),
progress = TRUE
)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;
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
starting candidate degrees for searching optimal degrees.
a positive integer or a vector of d positive integers specify
the maximum candidate or the given model degrees for the joint density.
logical, whether to search optimal degrees between M0 and M
or not but use M as the given model degrees for the joint density.
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])).
logical, if TRUE, the optimal degrees are selected based on marginal data, otherwise, the optimal degrees are chosen the joint data. See details.
method for finding minimum distance estimate. "em": EM like method;
Object of class mable.ctrl() specifying iteration limit
and the convergence criterion eps. Default is mable.ctrl. See Details.
if TRUE a text progressbar is displayed
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 \(M\).