Estimation of the variance of the estimators in admixture models with symmetric unknown density.

Semiparametric estimation of the variance of the estimators, i.e. the mixture weight p and the location shift parameter mu considering the admixture model with probability density function l: l(x) = p*f(x-mu) + (1-p)*g(x), x in R, where g is the known component of the two-component mixture, p is the unknown proportion, f is the unknown component density and mu is the location shift. See 'Details' below for more information.

Usage

BVdk_varCov_estimators(data, loc, p, comp.dist, comp.param)

Arguments

data: The observed sample under study.
loc: The estimated location shift parameter, related to the unknown symmetric density.
p: The estimated unknown component weight.
comp.dist: A list with two elements corresponding to component distributions (specified with R native names for these distributions) involved in the admixture model. Unknown elements must be specified as 'NULL' objects, e.g. when 'f' is unknown: list(f=NULL, g='norm').
comp.param: A list with two elements corresponding to the parameters of the component distributions, each element being a list itself. The names used in this list must correspond to the native R argument names for these distributions. Unknown elements must be specified as 'NULL' objects, e.g. if 'f' is unknown: list(f=NULL, g=list(mean=0,sd=1)).

Value

A list containing 1) the variance-covariance matrix of the estimators (assessed at the specific time points 'u' and 'v' such that u=v=mean(data)); 2) the variance of the mixture weight estimator; 3) the variance of the location shift estimator; 4) the variance of the unknown component cumulative distribution function at points 'u' and 'v' (useless for most of applications, explaining why 'u' and 'v' are set equal to mean(data) by default, with no corresponding arguments here).

Details

See formulas pp.28--30 in Appendix of Bordes, L. and Vandekerkhove, P. (2010); Semiparametric two-component mixture model when a component is known: an asymptotically normal estimator; Math. Meth. Stat.; 19, pp. 22--41.

Author

Xavier Milhaud xavier.milhaud.research@gmail.com

Examples

## Simulate data:
list.comp <- list(f = 'norm', g = 'norm')
list.param <- list(f = c(mean = 4, sd = 1), g = c(mean = 7, sd = 0.5))
sim.data <- rsimmix(n=140, unknownComp_weight=0.9, comp.dist=list.comp, comp.param=list.param)
## Estimate the location shift and mixture weight parameters in real-life setting:
list.comp <- list(f = NULL, g = 'norm')
list.param <- list(f = NULL, g = c(mean = 7, sd = 0.5))
estimators <- BVdk_estimParam(data = sim.data[['mixt.data']], method = "L-BFGS-B",
                              comp.dist = list.comp, comp.param = list.param)
#> Warning: Still needs to be implemented for cases where the unknown density mean is lower then the known density one!
## Estimate the variance of the two estimators (first mixture weight, then location shift):
BVdk_varCov_estimators(data = sim.data[['mixt.data']], loc = estimators[2], p = estimators[1],
                       comp.dist = list.comp, comp.param = list.param)
#> $varCov.mat
#>               [,1]          [,2]          [,3]
#> [1,]  1.184421e-03  5.911774e-04 -2.904904e-09
#> [2,]  5.911774e-04  3.557560e-03 -1.449919e-09
#> [3,] -2.904904e-09 -1.449919e-09  7.124553e-15
#> 
#> $var_pEstim
#> [1] 0.001184421
#> 
#> $var_muEstim
#> [1] 0.00355756
#> 
#> $var_FEstim
#> [1] 7.124553e-15
#>