mable.group {mable}

Mable fit of one-sample grouped data by an optimal or a preselected model degree

Description

Maximum approximate Bernstein/Beta likelihood estimation based on one-sample grouped data with an optimal selected by the change-point method among m0:m1 or a preselected model degree m.

Usage

mable.group(
  x,
  breaks,
  M,
  interval = c(0, 1),
  IC = c("none", "aic", "hqic", "all"),
  vb = 0,
  controls = mable.ctrl(),
  progress = TRUE
)

Arguments

x

vector of frequencies

breaks

class interval end points

M

a positive integer or a vector (m0, m1). If M = m or m0 = m1 = m, then m is a preselected degree. If m0<m1 it specifies the set of consective candidate model degrees m0:m1 for searching an optimal degree, where m1-m0>3.

interval

a vector containing the endpoints of support/truncation interval

IC

information criterion(s) in addition to Bayesian information criterion (BIC). Current choices are "aic" (Akaike information criterion) and/or "qhic" (Hannan–Quinn information criterion).

vb

code for vanishing boundary constraints, -1: f0(a)=0 only, 1: f0(b)=0 only, 2: both, 0: none (default).

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

Details

Any continuous density function f on a known closed supporting interval [a, b] can be estimated by Bernstein polynomial f_m(x; p) = \sum_{i=0}^m p_i\beta_{mi}[(x-a)/(b-a)]/(b-a), where p = (p_0, \ldots, p_m), p_i\ge 0, \sum_{i=0}^m p_i=1 and \beta_{mi}(u) = (m+1){m\choose i}u^i(1-x)^{m-i}, i = 0, 1, \ldots, m, is the beta density with shapes (i+1, m-i+1). For each m, the MABLE of the coefficients p, the mixture proportions, are obtained using EM algorithm. The EM iteration for each candidate m stops if either the total absolute change of the log likelihood and the coefficients of Bernstein polynomial is smaller than eps or the maximum number of iterations maxit is reached.

If m0<m1, an optimal model degree is selected as the change-point of the increments of log-likelihood, log likelihood ratios, for m \in \{m_0, m_0+1, \ldots, m_1\}. Alternatively, one can choose an optimal degree based on the BIC (Schwarz, 1978) which are evaluated at m \in \{m_0, m_0+1, \ldots, m_1\}. The search for optimal degree m is stoped if either m1 is reached with a warning or the test for change-point results in a p-value pval smaller than sig.level. The BIC for a given degree m is calculated as in Schwarz (1978) where the dimension of the model is d=\#\{i: \hat p_i \ge \epsilon, i = 0, \ldots, m\} - 1 and a default \epsilon is chosen as .Machine$double.eps.

Value

A list with components

• m the given or a selected degree by method of change-point

• p the estimated p with degree m

• mloglik the maximum log-likelihood at degree m

• interval supporting interval (a, b)

• convergence An integer code. 0 indicates successful completion (all the EM iterations are convergent and an optimal degree is successfully selected in M). Possible error codes are

  • 1, indicates that the iteration limit maxit had been reached in at least one EM iteration;

  • 2, the search did not finish before m1.

• delta the convergence criterion delta value

and, if m0<m1,

• M the vector (m0, m1), where m1, if greater than m0, is the largest candidate when the search stoped

• lk log-likelihoods evaluated at m \in \{m_0, \ldots, m_1\}

• lr likelihood ratios for change-points evaluated at m \in \{m_0+1, \ldots, m_1\}

• ic a list containing the selected information criterion(s)

• pval the p-values of the change-point tests for choosing optimal model degree

• chpts the change-points chosen with the given candidate model degrees

Author(s)

Zhong Guan <[e-mail]>

References

Guan, Z. (2017) Bernstein polynomial model for grouped continuous data. Journal of Nonparametric Statistics, 29(4):831-848.

See Also

mable.ic

Examples

## Chicken Embryo Data
 data(chicken.embryo)
 a<-0; b<-21
 day<-chicken.embryo$day
 nT<-chicken.embryo$nT
 Day<-rep(day,nT)
 res<-mable.group(x=nT, breaks=a:b, M=c(2,100), interval=c(a, b), IC="aic",
    controls=mable.ctrl(sig.level=1e-6,  maxit=2000, eps=1.0e-7))
 op<-par(mfrow=c(1,2), lwd=2)
 layout(rbind(c(1, 2), c(3, 3)))
 plot(res, which="likelihood")
 plot(res, which="change-point")
 fk<-density(x=rep((0:20)+.5, nT), bw="sj", n=101, from=a, to=b)
 hist(Day, breaks=seq(a,b,  length=12), freq=FALSE, col="grey",
          border="white", main="Histogram and Density Estimates")
 plot(res, which="density",types=1:2, colors=1:2)
 lines(fk, lty=2, col=2)
 legend("topright", lty=c(1:2), c("MABLE", "Kernel"), bty="n", col=c(1:2))
 par(op)