For testing a model fit for univariate data, tests based on the K-sign depth introduced by Malcherczyk et al. (2021) are distribution-free, outlier robust and very powerful for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K > 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Here, we propose two extensions of the K-sign depth to p-dimensional data, namely the L-simplex depth based on L simplices spanned by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p + 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> residuals and component L-depth, which is the componentwise application of the K-sign depth with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K=L+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi>L</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The simplex depth as well as the component depth can be used in a full version and in a simplified version. We derive some properties of the two simplex depth versions for bivariate data which show in particular that tests based on these depth versions are distribution-free. Then, we compare the four depth versions with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> by simulations. We also provide a simple algorithm to calculate all component depths in linear time and efficient algorithms for the simplex depths.

Bivariate and multivariate sign depth and related distribution-free tests for model fit

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2026年7月6日

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Christine H. Müller,
Stanislav Nagy,
Samuel Trippler

摘要

For testing a model fit for univariate data, tests based on the K-sign depth introduced by Malcherczyk et al. (2021) are distribution-free, outlier robust and very powerful for $K > 2$ $K > 2$ . Here, we propose two extensions of the K-sign depth to p-dimensional data, namely the L-simplex depth based on L simplices spanned by $p + 1$ $p + 1$ residuals and component L-depth, which is the componentwise application of the K-sign depth with $K=L+1$ $K = L + 1$ . The simplex depth as well as the component depth can be used in a full version and in a simplified version. We derive some properties of the two simplex depth versions for bivariate data which show in particular that tests based on these depth versions are distribution-free. Then, we compare the four depth versions with $L = 1$ $L = 1$ and $L = 2$ $L = 2$ by simulations. We also provide a simple algorithm to calculate all component depths in linear time and efficient algorithms for the simplex depths.