nonparam.Hankel: Estimate Mixture Complexity Based on Hankel Matrix

In anjaweigel/mixComp_package: Estimation of the Order of Finite Mixture Distributions

Description

Estimation of a mixture's complexity based on estimating the determinant of the Hankel matrix of the moments of the mixing distribution. The estimated determinants can be scaled and/or penalized.

Usage

nonparamHankel(obj, j.max = 10, pen.function = NULL, scaled = FALSE, 
               B = 1000, ...)

## S3 method for class 'hankDet'
print(x, ...)

## S3 method for class 'hankDet'
plot(x, type = "b", xlab = "j", ylab = NULL, mar = NULL, 
     ylim = c(min(0, min(obj)), max(obj)), ...)

Arguments

obj	object of class datMix.
j.max	integer specifying the maximal number of components to be considered.
pen.function	a function with arguments j and n specifying the penalty added to the determinant value given sample size n and the currently assumed complexity j. If left empty no penalty will be added. If non-empty and scaled is TRUE, the penalty function will be added after the determinants are scaled.
scaled	logical specifying whether the vector of estimated determinants should be scaled.
B	integer specifying the number of bootstrap replicates used for scaling of the determinants. Ignored if scaled is FALSE.
x	object of class hankDet.
type	character denoting type of plot, see, e.g. lines. Defaults to "b".
xlab,ylab	labels for the x and y axis with defaults (the default for ylab is created within the function, if no value is supplied).
mar	numerical vector of the form c(bottom, left, top, right) which gives the number of lines of margin to be specified on the four sides of the plot, see par.
ylim	range of y values to use.
...	in nonparamHankel(): further arguments passed to the boot function if scaled is TRUE. in plot.hankDet(): further arguments passed to plot. in print.hankDet(): further arguments passed to print.

Details

Define the complexity of a finite mixture F as the smallest integer p, such that its pdf/pmf f can be written as

f(x) = w_1*g(x;θ _1) + … + w_p*g(x;θ _p).

nonparamHankel estimates p by iteratively increasing the assumed complexity j and calculating the determinant of the (j+1)x(j+1) Hankel matrix made up of the first 2j raw moments of the mixing distribution. As shown by Dacunha-Castelle & Gassiat (1997), once the correct complexity is reached (i.e. for all j >= p), this determinant is zero.

This suggests an estimation procedure for p based on initially finding a consistent estimator of the moments of the mixing distribution and then choosing the estimator estim_p as the value of j which yields a sufficiently small value of the determinant. Since the estimated determinant is close to 0 for all j >= p, this could lead to choosing estim_p rather larger than the true value. The function therefore returns all estimated determinant values corresponding to complexities up to j.max, so that the user can pick the lowest j generating a sufficiently small determinant. In addition, the function allows the inclusion of a penalty term as a function of the sample size n and the currently assumed complexity j which will be added to the determinant value (by supplying pen.function), and/or scaling of the determinants (by setting scaled = TRUE). For scaling, a nonparametric bootstrap is used to calculate the covariance of the estimated determinants, with B being the size of the bootstrap sample. The inverse of the square root of this covariance matrix (i.e. the matrix S^(-1) such that $A = SS$, where A is the covariance matrix) is then multiplied with the estimated determinant vector to get the scaled determinant vector.

For a thorough discussion of the methods that can be used for the estimation of the moments see the details section of datMix.

Value

The vector of estimated determinants (optionally scaled and/or penalized), given back as an object of class hankDet with the following attributes:

scaled	logical indicating whether the determinants are scaled.
pen	logical indicating whether a penalty was added to the determinants.
dist	character string stating the (abbreviated) name of the component distribution, such that the function ddist evaluates its density function and rdist generates random numbers.

References

D. Dacunha-Castelle and E. Gassiat, "The estimation of the order of a mixture model", Bernoulli, Volume 3, Number 3, 279-299, 1997.

See Also

paramHankel for a similar approach which estimates the component weights and parameters on top of the complexity, datMix for the creation of the datMix object.

Examples

## create 'Mix' object
geomMix <- Mix("geom", w = c(0.1, 0.6, 0.3), prob = c(0.8, 0.2, 0.4))

## create random data based on 'Mix' object (gives back 'rMix' object)
set.seed(1)
geomRMix <- rMix(1000, obj = geomMix)

## create 'datMix' object for estimation

# explicit function giving the estimate for the j^th moment of the 
# mixing distribution, needed for Hankel.method "explicit"

explicit.fct.geom <- function(dat, j){
  1 - ecdf(dat)(j - 1)
}
        
## generating 'datMix' object
geom.dM <- RtoDat(geomRMix, Hankel.method = "explicit", 
                  Hankel.function = explicit.fct.geom)

## function for penalization
pen <- function(j, n){
  (j*log(n))/(sqrt(n))
}

## estimate determinants
set.seed(1)
geomdets_pen <- nonparamHankel(geom.dM, pen.function = pen, j.max = 5)
plot(geomdets_pen, main = "Three component geometric mixture")

anjaweigel/mixComp_package documentation built on Sept. 2, 2020, 3:55 p.m.