Parametric estimation of the variance of the variance parameter in Bordes & Vandekerkhove (2010) setting, i.e. considering the admixture model with probability density function (pdf) l: l(x) = p*f(x-mu) + (1-p)*g, where g is the known component of the two-component mixture, p is the mixture proportion, f is the unknown component with symmetric density, and mu is the location shift parameter. The estimation of the variance of the variance related to the density f is made by maximum likelihood optimization through the information matrix, with the assumption that the unknown f is gaussian.

BVdk_ML_varCov_estimators(data, hat_w, hat_loc, hat_var, comp.dist, comp.param)

## Arguments

data The observed sample under study. Estimate of the unknown component weight. Estimate of the location shift parameter. Estimate of the variance of the symmetric density f, obtained by plugging-in the previous estimates. See 'Details' below for further information. 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'). 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

The variance of the estimator of the variance of the unknown component density f.

## Details

Plug-in strategy is defined in Pommeret, D. and Vandekerkhove, P. (2019); Semiparametric density testing in the contamination model; Electronic Journal of Statistics, 13, pp. 4743--4793. The variance of the estimator variance of the unknown density f is needed in a testing perspective, since included in the variance of the test statistic. Other details about the information matrix can be found in 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 = 500, unknownComp_weight = 0.8, list.comp, list.param)\$mixt.data
## Estimate mixture weight and location shift parameters in real-life:
list.comp <- list(f = NULL, g = "norm")
list.param <- list(f = NULL, g = c(mean = 7, sd = 0.5))
estim <- BVdk_estimParam(data = sim.data, method = "L-BFGS-B",
comp.dist = list.comp, comp.param = list.param)
## Estimation of the second-order moment of the known component distribution:
m2_knownComp <- mean(rnorm(n = 1000000, mean = 7, sd = 0.5)^2)
hat_s2 <- (1/estim[1]) * (mean(sim.data^2) - ((1-estim[1])*m2_knownComp)) - estim[2]^2
## Estimated variance of variance estimator related to the unknown symmetric component density:
BVdk_ML_varCov_estimators(data = sim.data, hat_w = estim[1], hat_loc = estim[2],
hat_var = hat_s2, comp.dist = list.comp, comp.param = list.param)
#> [1] 0.0329471