Introduction

In this vignette, we will briefly describe and motivate how we constructed the test statistics used by the function m_test and how it derives a test decision.

Asymptotic distribution of the M-test statistics

For a more detailed description of the asymptotic behaviour of M-estimators, we refer to Maronna et al. (2019, p. 37ff.), which is the main reference for the following motivation.

We consider two independent samples X₁, …, X_m and Y₁, …, Y_n of i.i.d. random variables which are symmetrically distributed with variances σ_X² and σ_Y².

For M-estimators μ̂_X and μ̂_Y with a ψ-function ψ, it can be shown under these conditions that

where μ_X ∈ ℝ and μ_Y ∈ ℝ are the values for which

and

From this, it follows that

implying

In order to use this statistic as a test statistic for our M-tests, we need to estimate σ_X, σ_Y, ν_X, and ν_Y. We use the τ-scale estimator (Maronna and Zamar, 2002) to estimate σ_X² and σ_Y² by σ̂_X² and σ̂_Y² robustly and estimate ν_X and ν_Y by

Under the previous considerations, the test statistic of the M-tests we implemented in the package is given by

where Δ = μ_X − μ_Y is the location difference between both distributions.

The M-tests are implemented in the function m_test. More details on the usage of the function can be found in the vignette Getting started with robnptests. Inside m_test, we use the function scaleTau2 from the R package robustbase (Maechler et al., 2022) to compute the τ-scale estimates for the samples.

Simulation results

The following figure shows the simulated test sizes from a small simulation study with 1000 replications, where we applied the M-tests with different ψ-functions to samples from the 𝒩(0, 1)-distribution, the t₂-distribution, and the χ₃²-distribution. We chose the significance level α = 0.05. The results are shown in the following figure.

Under the 𝒩(0, 1)- and the t₂-distribution we make similar observations: For equal sample sizes m = n ≥ 30, the simulated test size is quite close to the the specified value of α. When m ≠ n, it seems to be important that both values are rather large and do not deviate too much from each other. Otherwise, the tests may become very anti-conservative. In general, the three test statistics lead to similar results for the considered sample sizes.

Under the χ₃²-distribution, all tests are anti-conservative. While there seems to be some improvement when the sample sizes become larger, the estimated sizes are still rather far away from 0.05. A reason might be that the asymptotic variance we use is only a good approximation for symmetric distributions (Maronna et al., 2019, p. 38).

Based on these results, we discourage using the tests for asymmetric distributions. For symmetric distributions, the asymptotic test should only be used for large samples. In all other cases, the randomization or permutation test might be preferable.

Session Info

library(robnptests)

sessionInfo()
#> R version 4.4.2 (2024-10-31)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.2 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=C              
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> time zone: Etc/UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] robnptests_1.1.0 rmarkdown_2.29  
#> 
#> loaded via a namespace (and not attached):
#>  [1] digest_0.6.37     R6_2.6.1          codetools_0.2-20  fastmap_1.2.0    
#>  [5] xfun_0.51         maketools_1.3.2   cachem_1.1.0      knitr_1.49       
#>  [9] htmltools_0.5.8.1 buildtools_1.0.0  lifecycle_1.0.4   Rdpack_2.6.2     
#> [13] cli_3.6.4         sass_0.4.9        jquerylib_0.1.4   compiler_4.4.2   
#> [17] rbibutils_2.3     sys_3.4.3         tools_4.4.2       evaluate_1.0.3   
#> [21] bslib_0.9.0       yaml_2.3.10       jsonlite_1.9.0    rlang_1.1.5