Akin Anarat | Searcharxiv

arXiv2026

A test for normality based on self-similarity

Testing for normality is a widely used procedure in statistics and data analysis, often applied prior to employing methods that rely on the assumption of normally distributed data. While several existing tests target distributional characteristics such as higher-order moments, others focus on functional aspects such as the distribution function. In this article, we propose an alternative idea by exploiting the self-similarity property of the normal distribution and introduce the Self-Similarity Test for Normality (SSTN). This procedure leverages the structural property that the distribution of a suitably centered and scaled sum of independent and identically distributed random variables with finite variance coincides with the original distribution if and only if that distribution is normal. The SSTN evaluates normality by applying a self-similarity transformation to the standardized empirical characteristic function and examining how the transformed functions change across successive applications. For the normal distribution, repeated applications preserve the functional form of the characteristic function, whereas deviations from normality manifest in systematic changes between consecutive transforms. These changes are aggregated into a test statistic, whose null distribution is obtained by Monte Carlo calibration, using a sample-size-specific calibration for small samples and an approximation of the asymptotic null distribution for larger ones. A comprehensive simulation study shows that the SSTN performs at least competitively and frequently superior to several well-established tests for normality.

stat.ME↗

arXiv2025

A nonparametric statistical method for deconvolving densities in the analysis of proteomic data

In medical research, often, genomic or proteomic data are collected, with measurements frequently subject to uncertainties or errors, making it crucial to accurately separate the signals of the genes or proteins, respectively, from the noise. Such a signal separation is also of interest in skin aging research in which intrinsic aging driven by genetic factors and extrinsic, i.e.\ environmentally induced, aging are investigated by considering, e.g., the proteome of skin fibroblasts. Since extrinsic influences on skin aging can only be measured alongside intrinsic ones, it is essential to isolate the pure extrinsic signal from the combined intrinisic and extrinsic signal. In such situations, deconvolution methods can be employed to estimate the signal's density function from the data. However, existing nonparametric deconvolution approaches often fail when the variance of the mixed distribution is substantially greater than the variance of the target distribution, which is a common issue in genomic and proteomic data. We, therefore, propose a new nonparametric deconvolution method called N-Power Fourier Deconvolution (NPFD) that addresses this issue by employing the $N$-th power of the Fourier transform of transformed densities. This procedure utilizes the Fourier transform inversion theorem and exploits properties of Fourier transforms of density functions to mitigate numerical inaccuracies through exponentiation, leading to accurate and smooth density estimation. An extensive simulation study demonstrates that NPFD effectively handles the variance issues and performs comparably or better than existing deconvolution methods in most scenarios. Moreover, applications to real medical data, particularly to proteomic data from fibroblasts affected by intrinsic and extrinsic aging, show how NPFD can be employed to estimate the pure extrinsic density.

stat.ME↗