Equality test of two unknown component distributions using polynomial expansions

Tests the null hypothesis (H0: f1=f2) using the decomposition of unknown component densities of two admixture distributions in an adequate orthonormal polynomial basis. Recall that we have two admixture models with respective probability density functions (pdf) l1 = p1*f1 + (1-p1)g1 and l2 = p2f2 + (1-p2)*g2, where g1 and g2 are the only known elements and l1 and l2 are observed. The admixture weights p1 and p2 thus have to be estimated. For further information on this method, see 'Details' below.

Usage

orthobasis_test(
  samples,
  admixMod,
  conf_level = 0.95,
  est_method = c("BVdk", "PS"),
  ask_poly_param = FALSE,
  K = 3,
  s = 0.49,
  nb_echBoot = 100,
  support = c("Real", "Integer", "Positive", "Bounded.continuous", "Bounded.discrete"),
  bounds_supp = NULL,
  ...
)

Arguments

samples

List of the two samples, each one following the mixture distribution given by l = p*f + (1-p)*g, with f and p unknown and g known.

admixMod

An object of class admix_model, containing useful information about distributions and parameters.

conf_level

The confidence level, default to 95 percent. Equals 1-alpha, where alpha is the level of the test (type-I error).

est_method

Estimation method to get the component weights, either 'PS' (Patra and Sen estimation) or 'BVdk' (Bordes and Vendekerkhove estimation). Choosing 'PS' requires to specify the number of bootstrap samples.

ask_poly_param

(default to FALSE) If TRUE, ask the user to choose both the order 'K' of expansion coefficients in the orthonormal polynomial basis, and the penalization rate 's' involved on the penalization rule for the test.

K

(K > 0, default to 3) If not asked (see the previous argument), number of coefficients considered for the polynomial basis expansion.

s

(in ]0,1/2[, default to 0.49) If not asked (see the previous argument), rate at which the normalization factor is set in the penalization rule for model selection (in ]0,1/2[). Low values of 's' favors the detection of alternative hypothesis. See reference below.

[, default to 0.49) If not asked (see the previous argument), rate at which the normalization factor is set in the penalization rule for model selection (in ]: R:,%20default%20to%200.49)%20If%20not%20asked%20(see%20the%20previous%20argument),%20rate%20at%20which%20the%20normalization%20factor%20is%20set%20in%0A%20%20%20%20%20%20%20%20%20%20the%20penalization%20rule%20for%20model%20selection%20(in%20

nb_echBoot

(default to 100) Number of bootstrap samples, useful when choosing 'PS' estimation method.

support

Support of the probability distributions, useful to choose the appropriate polynomial orthonormal basis. One of 'Real', 'Integer', 'Positive', or 'Bounded.continuous'.

bounds_supp

(default to NULL) Useful if support = 'Bounded.continuous', a list of minimum and maximum bounds, specified as following: list( list(min.f1,min.g1,min.f2,min.g2) , list(max.f1,max.g1,max.f2,max.g2) )

...

Optional arguments to estim_BVdk or estim_PS, depending on the chosen argument 'est_method' (see above).

Value

An object of class orthobasis_test, containing ten attributes: 1) the number of populations under study (2 in this case); 2) the sizes of samples; 3) the information about the known component distribution; 4) the reject decision of the test; 5) the confidence level of the test, 6) the p-value of the test; 7) the value of the test statistic; 8) the variance of the test statistic at each order in the polynomial orthobasis expansion; 9) the selected rank (order) for the test statistic; 10) a vector of estimates, related to the estimated mixing proportions in the two samples.

References

Milhaud X, Pommeret D, Salhi Y, Vandekerkhove P (2022). “Semiparametric two-sample admixture components comparison test: The symmetric case.” Journal of Statistical Planning and Inference, 216, 135-150. ISSN 0378-3758, doi:10.1016/j.jspi.2021.05.010 .

Author

Xavier Milhaud xavier.milhaud.research@gmail.com

Examples

# \donttest{
#### Under the null hypothesis H0.
mixt1 <- twoComp_mixt(n = 300, weight = 0.77,
                      comp.dist = list("norm", "exp"),
                      comp.param = list(list("mean" = 1, "sd" = 1),
                                        list("rate" = 0.33)))
data1 <- getmixtData(mixt1)
admixMod1 <- admix_model(knownComp_dist = mixt1$comp.dist[[2]],
                        knownComp_param = mixt1$comp.param[[2]])
mixt2 <- twoComp_mixt(n = 500, weight = 0.62,
                      comp.dist = list("norm", "norm"),
                      comp.param = list(list("mean" = 1, "sd" = 1),
                                        list("mean" = -2, "sd" = 0.5)))
data2 <- getmixtData(mixt2)
admixMod2 <- admix_model(knownComp_dist = mixt2$comp.dist[[2]],
                        knownComp_param = mixt2$comp.param[[2]])
## Test procedure:
orthobasis_test(samples = list(data1,data2), admixMod = list(admixMod1,admixMod2),
                conf_level = 0.95, est_method = 'BVdk', support = 'Real')
#> Call:orthobasis_test(samples = list(data1, data2), admixMod = list(admixMod1, 
#>     admixMod2), conf_level = 0.95, est_method = "BVdk", support = "Real")
#> 
#> Is the null hypothesis (gaussian unknown component distribution) rejected? Yes
#> Test p-value: 0.015
# }