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.

hat_w

Estimate of the unknown component weight.

hat_loc

Estimate of the location shift parameter.

hat_var

Estimate of the variance of the symmetric density f, obtained by plugging-in the previous estimates. See 'Details' below for further information.

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

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