madem.density {mable} | R Documentation |
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 |
m |
a positive integer or a vector of |
p |
initial guess of |
interval |
a vector of two endpoints or a |
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 degreem