Estimate the unknown component weight (and possibly a location shift parameter in case of a symmetric unknown component density), using different estimation techniques. We recall that the \(i\)-th admixture model has probability density function \(\ell_i\) such that: $$ \ell_i = p_i f_i + (1 - p_i) g_i, $$ where \(g_i\) is the known component density. The unknown quantities \(p_i\) and \(f_i\) then have to be estimated.
Usage
admix_estim(samples, admixMod, est_method = c("PS", "BVdk", "IBM"), ...)Arguments
- samples
A list of the K (K>0) samples to be studied, all following admixture distributions.
- admixMod
A list of objects of class admix_model, with information about known distributions and known parameters.
- est_method
The estimation method to be applied. Can be one of 'BVdk' (Bordes and Vandekerkhove estimator), 'PS' (Patra and Sen estimator), or 'IBM' (Inversion Best-Matching approach) in the continuous case (continuous random variable). Only 'IBM' for discrete random variables. The same estimation method is performed on each sample if several samples are provided.
- ...
Optional arguments to estim_PS, estim_BVdk or estim_IBM depending on the choice made by the user for the estimation method.
Value
An object of class estim_BVdk, estim_PS or estim_IBM (that inherits from class admix_estim),
with two attributes, 'class' and 'names'. The latter contains three elements, among which 'estim_objects' that lists for each
sample under study all the information of the estimation procedure.
Details
For further details on the different estimation techniques, see references below on i) Patra and Sen estimator ; ii) Bordes and Vandekerkhove estimator ; iii) Inversion Best-Matching approach. Important note: estimation by 'IBM' requires at least two samples at hand, and provides unbiased estimators only if the distributions of unknown components are equal (meaning that it requires to perform previously this test between the pairs of samples, see admix_test). When selecting 'BVdk' estimation method, the initialization parameters for the optimization process have been arbitrarily set. The mixing proportion is fixed to 0.5. For the localization parameter, it is based on taking the first moment in the model (\(\ell(x) = p f(x-\mu) + (1 - p) g(x)\)), and isolating \(\mu\) by an inversion formula.
References
Patra RK, Sen B (2016). “Estimation of a two-component mixture model with applications to multiple testing.” Journal of the Royal Statistical Society Series B, 78(4), 869-893. Bordes L, Delmas C, Vandekerkhove P (2006). “Semiparametric Estimation of a Two-Component Mixture Model Where One Component Is Known.” Scandinavian Journal of Statistics, 33(4), 733–752. ISSN 03036898, 14679469, http://www.jstor.org/stable/4616955. Bordes L, Vandekerkhove P (2010). “Semiparametric two-component mixture model with a known component: An asymptotically normal estimator.” Mathematical Methods of Statistics, 19(1), 22–41. doi:10.3103/S1066530710010023 . Milhaud X, Pommeret D, Salhi Y, Vandekerkhove P (2024). “Two-sample contamination model test.” Bernoulli, 30(1), 170–197. doi:10.3150/23-BEJ1593 .
See also
get_mixing_weights() to access the estimated mixing weight(s), get_known_component() to access the known component(s), print.admix_estim() for a brief description of the results,
and summary.admix_estim() for an overview of the estimation process. More precisely, 1) the number of samples under study;
2) the information about the known mixture components (distributions and parameters); 3) the sizes of the samples;
4) the chosen estimation technique (one of 'BVdk', 'PS' or 'IBM'); 5) the estimated mixing proportions (weights of the
unknown component distributions in the mixture model). In case of 'BVdk' estimation, one additional attribute corresponding
to the estimated location shift parameter is included.
Author
Xavier Milhaud xavier.milhaud.research@gmail.com
Examples
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 300, weight = 0.7,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = -2, "sd" = 0.5),
list("mean" = 0, "sd" = 1)))
mixt2 <- twoComp_mixt(n = 250, weight = 0.85,
comp.dist = list("norm", "exp"),
comp.param = list(list("mean" = -2, "sd" = 0.5),
list("rate" = 1)))
mixt3 <- twoComp_mixt(n = 500, weight = 0.5,
comp.dist = list("pois", "pois"),
comp.param = list(list("lambda" = 2),
list("lambda" = 7)))
mixt4 <- twoComp_mixt(n = 1500, weight = 0.2, comp.dist = list("multinom", "multinom"),
comp.param = list(list("size"=1, "prob" = c(0.8,0.1,0.1)),
list("size"=1, "prob" = c(0.1,0.2,0.7))))
data1 <- get_mixture_data(mixt1)
data2 <- get_mixture_data(mixt2)
data3 <- get_mixture_data(mixt3)
data4 <- get_mixture_data(mixt4)
## Define the admixture models:
admixMod1 <- admix_model(knownComp_dist = mixt1$comp.dist[[2]],
knownComp_param = mixt1$comp.param[[2]])
admixMod2 <- admix_model(knownComp_dist = mixt2$comp.dist[[2]],
knownComp_param = mixt2$comp.param[[2]])
admixMod3 <- admix_model(knownComp_dist = mixt3$comp.dist[[2]],
knownComp_param = mixt3$comp.param[[2]])
admixMod4 <- admix_model(knownComp_dist = mixt4$comp.dist[[2]],
knownComp_param = mixt4$comp.param[[2]])
# Estimation by different methods:
admix_estim(samples = list(data1), admixMod = list(admixMod1), est_method = "BVdk")
#> Mixing weight estimation using 'BVdk' assumes the unknown component
#> distribution to have a symmetric probability density function.
#>
#> Call:
#> admix_estim(samples = list(data1), admixMod = list(admixMod1),
#> est_method = "BVdk")
#>
#> Method: BVdk - Number of samples: 1
#>
#> sample size mix_weight location
#> data1 300 0.685 -2.05
admix_estim(samples = list(data1, data2, data3, data4),
admixMod = list(admixMod1, admixMod2, admixMod3, admixMod4), est_method = "PS")
#>
#> Call:
#> admix_estim(samples = list(data1, data2, data3, data4), admixMod = list(admixMod1,
#> admixMod2, admixMod3, admixMod4), est_method = "PS")
#>
#> Method: PS - Number of samples: 4
#>
#> sample size mix_weight
#> data1 300 0.655
#> data2 250 0.841
#> data3 500 0.453
#> data4 1500 0.180
admix_estim(samples = list(data1,data2), admixMod = list(admixMod1,admixMod2), est_method = "IBM")
#> IBM estimators of two unknown proportions are reliable only if the two corresponding
#> unknown component distributions have previously been tested equal (see ?admix_test).
#>
#> Call:
#> admix_estim(samples = list(data1, data2), admixMod = list(admixMod1,
#> admixMod2), est_method = "IBM")
#>
#> Method: IBM - Pairwise estimation
#>
#> pair size_1st size_2nd mix_weight_1st var_1st mix_weight_2nd var_2nd
#> data1 vs data2 300 250 0.749 NA 0.896 NA