We remind that a random variable \(X\) following an admixture distribution has cumulative distribution function (cdf) \(L\) given by \[L(x) = pF(x) + (1-p)G(x), \qquad x \in \mathbb{R},\] where \(G\) is a mixture component whose distribution is perfectly known, whereas \(p\) and \(F\) are unknown. In this setting, if no parametric assumption is made on the unknown component distribution \(F\), the mixture is considered as a semiparametric mixture. For an overview on semiparametric extensions of finite mixture models, see (Xiang and Yang 2018).

The mixture weight \(p\) of the unknown component distribution can be estimated using diverse techniques depending on the assumptions made on the unknown cdf \(F\), among which the ones discussed in the sequel:

- the estimator provided by Bordes and Vandekerkhove, see (L. Bordes and Vandekerkhove 2010);
- the estimator provided by Patra and Sen, see (Patra and Sen 2016);
- the estimator provided by the Inversion - Best Matching method, see (Milhaud et al. 2024).

All these estimation methods can be performed using one single generic function for estimation with appropriate arguments, the so-called \(admix\_estim\) function.

Many works studied the estimation of the unknown proportion in two-component admixture models. Among them, seminal papers are (Laurent Bordes, Delmas, and Vandekerkhove 2006) and (S. Bordes L. Mottelet and Vandekerkhove 2006). These papers are closely connected to the paper by (L. Bordes and Vandekerkhove 2010), where an asymptotic normal estimator is provided for the unknown component weight.

In this case, we use the Bordes and Vandekerkhove estimator, see (L. Bordes and Vandekerkhove 2010).

```
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 400, weight = 0.7,
comp.dist = list("norm", "norm"),
comp.param = list(c("mean" = 3, "sd" = 0.5),
c("mean" = 0, "sd" = 1)))
data1 <- getmixtData(mixt1)
## Define the admixture model:
admixMod <- admix_model(knownComp_dist = mixt1$comp.dist[[2]],
knownComp_param = mixt1$comp.param[[2]])
admix_estim(samples = list(data1), admixMod = list(admixMod),
est.method = 'BVdk', sym.f = TRUE)
#> Call:
#> admix_estim(samples = list(data1), admixMod = list(admixMod),
#> est.method = "BVdk", sym.f = TRUE)
#>
#> Estimated mixing weight of the unknown component distribution in Sample 1: 0.72
```

Because this estimation method relies on the symmetry of the unknown component density, the estimator provides both the estimated mixing weight of the unknown component distribution and the estimated location shift parameter.

In full generality (no assumptions made on the unknown component distribution), we use the Patra and Sen estimator, see (Patra and Sen 2016).

```
admix_estim(samples = list(data1), admixMod = list(admixMod),
est.method = 'PS')
#> Call:
#> admix_estim(samples = list(data1), admixMod = list(admixMod),
#> est.method = "PS")
#>
#> Estimated mixing weight of the unknown component distribution in Sample 1: 0.69
```

In this case, the only estimated parameter is the mixing proportion related to the unknown component distribution.

In the two-sample setting, one idea could be to use the Inversion - Best Matching (IBM) approach. The IBM method ensures asymptotically normal estimators of the unknown quantities, which will be very useful in a testing perspective. However, it is important to note that such estimators are mostly biased when \(F_1 \neq F_2\), and general one-sample estimation strategies such as (Patra and Sen 2016) or (L. Bordes and Vandekerkhove 2010) may be preferred to estimate the unknown component proportion in general settings (despite that this is more time-consuming). In the latter case, one performs twice the estimation method, on each of the two samples under study.

When we are under the null, Milhaud et al. (2024) show that the estimators is consistent towards the true parameter values.

```
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 450, weight = 0.4,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = -2, "sd" = 0.5),
list("mean" = 0, "sd" = 1)))
mixt2 <- twoComp_mixt(n = 380, weight = 0.7,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = -2, "sd" = 0.5),
list("mean" = 1, "sd" = 1)))
data1 <- getmixtData(mixt1)
data2 <- getmixtData(mixt2)
## 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]])
admix_estim(samples = list(data1, data2), admixMod = list(admixMod1, admixMod2),
est.method = 'IBM')
#> Call:
#> admix_estim(samples = list(data1, data2), admixMod = list(admixMod1,
#> admixMod2), est.method = "IBM")
#>
#> Estimated mixing weight of the unknown component distribution in Sample 1: 0.49
#> Estimated mixing weight of the unknown component distribution in Sample 2: 0.9
```

Indeed, one can see that the two unknown proportions were consistently estimated.

Estimators are also consistent under \(H_1\), although they can be (strongly) biased as compared to their true values as illustrated in the following example.

```
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 800, weight = 0.5,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = 1, "sd" = 0.5),
list("mean" = 0, "sd" = 1)))
mixt2 <- twoComp_mixt(n = 600, weight = 0.7,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = 3, "sd" = 0.5),
list("mean" = 5, "sd" = 2)))
data1 <- getmixtData(mixt1)
data2 <- getmixtData(mixt2)
## 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]])
## Estimate the mixture weights of the two admixture models (provide only hat(theta)_n):
admix_estim(samples = list(data1, data2), admixMod = list(admixMod1, admixMod2),
est.method = 'IBM')
#> Call:
#> admix_estim(samples = list(data1, data2), admixMod = list(admixMod1,
#> admixMod2), est.method = "IBM")
#>
#> Estimated mixing weight of the unknown component distribution in Sample 1: 0.35
#> Estimated mixing weight of the unknown component distribution in Sample 2: 0.62
```

In such a framework, it is therefore better to use the estimator by (Patra and Sen 2016), which shows better performance:

```
admix_estim(samples = list(data1, data2), admixMod = list(admixMod1, admixMod2),
est.method = 'PS')
#> Call:
#> admix_estim(samples = list(data1, data2), admixMod = list(admixMod1,
#> admixMod2), est.method = "PS")
#>
#> Estimated mixing weight of the unknown component distribution in Sample 1: 0.43
#> Estimated mixing weight of the unknown component distribution in Sample 2: 0.66
```

Concerning the unknown cdf \(F\), one usually estimate it thanks to the inversion formula \[F(x) = \dfrac{L(x) - (1-p)G(x)}{p},\] once \(p\) has been consistenly estimated.

This is what is commonly called the decontaminated density of the unknown component. In the following, we propose to compare the two decontaminated densities obtained once the unknown quantities have been consistently estimated by the IBM approach. Note that we are under the null (\(F_1=F_2\)), and thus that the decontaminated densities should look similar.

```
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 800, weight = 0.4,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = 3, "sd" = 0.5),
list("mean" = 0, "sd" = 1)))
mixt2 <- twoComp_mixt(n = 700, weight = 0.6,
comp.dist = list("norm", "norm"),
comp.param = list(list("mean" = 3, "sd" = 0.5),
list("mean" = 5, "sd" = 2)))
data1 <- getmixtData(mixt1)
data2 <- getmixtData(mixt2)
## 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]])
## Estimation:
est <- admix_estim(samples = list(data1,data2), admixMod = list(admixMod1,admixMod2),
est.method = 'PS')
prop <- getmixingWeight(est)
## Determine the decontaminated version of the unknown density by inversion:
res1 <- decontaminated_density(sample1 = data1, estim.p = prop[1], admixMod = admixMod1)
res2 <- decontaminated_density(sample1 = data2, estim.p = prop[2], admixMod = admixMod2)
## Use appropriate sequence of x values:
plot(x = res1, x_val = seq(from = 0, to = 6, length.out = 100), add_plot = FALSE)
plot(x = res2, x_val = seq(from = 0, to = 6, length.out = 100), add_plot = TRUE, col = "red")
```

Bordes, Laurent, Céline Delmas, and Pierre Vandekerkhove. 2006.
“Semiparametric Estimation of a Two-Component Mixture Model Where
One Component Is Known.” *Scandinavian Journal of
Statistics* 33 (4): 733–52. http://www.jstor.org/stable/4616955.

Bordes, L., and P. Vandekerkhove. 2010. “Semiparametric
Two-Component Mixture Model with a Known Component: An Asymptotically
Normal Estimator.” *Mathematical Methods of Statistics* 19
(1): 22–41. https://doi.org/https://doi.org/10.3103/S1066530710010023.

Bordes, S., L. Mottelet, and P. Vandekerkhove. 2006.
“Semiparametric Estimation of a Two Components Mixture
Model.” *Annals of Statistics* 34: 1204–32.

Milhaud, Xavier, Denys Pommeret, Yahia Salhi, and Pierre Vandekerkhove.
2024. “Two-sample contamination model
test.” *Bernoulli* 30 (1): 170–97. https://doi.org/10.3150/23-BEJ1593.

Patra, Rohit Kumar, and Bodhisattva Sen. 2016. “Estimation of a two-component mixture model with
applications to multiple testing.” *Journal of the
Royal Statistical Society Series B* 78 (4): 869–93.

Xiang, Yao, S., and G. Yang. 2018. “An Overview of Semiparametric
Extensions of Finite Mixture Models.” *Statistica Scinica*
34: 391–404.