mable (version 4.1.1)

mable.mvar: Maximum Approximate Bernstein Likelihood Estimate of Multivariate Density Function

Description

Maximum Approximate Bernstein Likelihood Estimate of Multivariate Density Function

Usage

mable.mvar(
  x,
  M0 = 1L,
  M,
  search = TRUE,
  interval = NULL,
  mar.deg = TRUE,
  method = c("cd", "em", "lmem"),
  controls = mable.ctrl(),
  progress = TRUE
)

Value

A list with components

• m a vector of the selected optimal degrees by the method of change-point

• p a vector of the mixture proportions \(p(j_1, \ldots, j_d)\), arranged in the column-major order of \(j = (j_1, \ldots, j_d)\), \(0 \le j_i \le m_i, i = 1, \ldots, d\).

• mloglik the maximum log-likelihood at an optimal degree m

• pval the p-values of change-points for choosing the optimal degrees for the marginal densities

• M the vector (m1, m2, ... , md), where mi is the largest candidate degree when the search stoped for the i-th marginal density

• interval support hyperrectangle \([a, b]=[a_1, b_1] \times \cdots \times [a_d, b_d]\)

• 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;

Arguments

x

an n x d matrix or data.frame of multivariate sample of size n

M0

a positive integer or a vector of d positive integers specify starting candidate degrees for searching optimal degrees.

M

a positive integer or a vector of d positive integers specify the maximum candidate or the given model degrees for the joint density.

search

logical, whether to search optimal degrees between M0 and M or not but use M as the given model degrees for the joint density.

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])).

mar.deg

logical, if TRUE, the optimal degrees are selected based on marginal data, otherwise, the optimal degrees are chosen the joint data. See details.

method

method for finding maximum likelihood estimate. "cd": coordinate-descent; less memory for data that are high dimensional/large sample.

controls

Object of class mable.ctrl() specifying iteration limit and the convergence criterion eps. Default is mable.ctrl. See Details.

progress

if TRUE a text progressbar is displayed

Author

Zhong Guan <[e-mail]>

Details

A \(d\)-variate density \(f\) 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 densities on \([a, b]\), \(\beta_{mj}(x) = \prod_{i=1}^d\beta_{m_i,j_i}[(x_i-a_i)/(b_i-a_i)]/(b_i-a_i)\), with proportion \(p(j_1, \ldots, j_d)\), \(0 \le j_i \le m_i, i = 1, \ldots, d\). 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 \(M\). For large data and multimodal density, the search for the model degrees is very time-consuming. In this case, it is suggested that use method="cd" and select the degrees based on marginal data using mable or optimable.

References

Guan, Z. (2016) Efficient and robust density estimation using Bernstein type polynomials. Journal of Nonparametric Statistics, 28(2):250-271. Wang, T. and Guan, Z.,(2019) Bernstein Polynomial Model for Nonparametric Multivariate Density, Statistics, Vol. 53, no. 2, 321-338

See Also

mable, optimable

Examples

## Old Faithful Data
# \donttest{
 a<-c(0, 40); b<-c(7, 110)
 ans<- mable.mvar(faithful, M = c(46,19), search =FALSE, method="em",
         interval = rbind(a,b), progress=FALSE)
 plot(ans, which="density") 
 plot(ans, which="cumulative")
# }