Kolmogorov-Smirnov Type Research Articles

A Rehabilitation and Monitoring System for Upper Limb Stroke Based on The Combination of Mirror and Stationary Bicycle

The weakening of motor functions in individuals due to a stroke can impact daily life productivity, necessitating therapy or rehabilitation. One method for restoring the function of body parts, especially the upper limb affected by a stroke, is through mirror therapy on a bicycle. Mirror therapy provides visual stimulation to the brain, influencing the improvement of motor functions. To monitor the effectiveness and safety during therapy, a monitoring system is required to track progress and evaluate vital signs to prevent health issues arising from the exercise process. This research aims to design an integrated upper limb motor exercise system using a mirror therapy bicycle with a website to monitor the therapy process and evaluate the results. Observed parameters include heart rate, oxygen saturation, pedal rotations, and therapy duration. This is because the longer the duration and the more rotations of the static bicycle pedaled by post-stroke patients, the greater the likelihood of improving motor skills. Software reliability was tested for the website and data communication. The study involved 20 healthy subjects, analyzed statistically through descriptive statistics and normality tests using the Kolmogorov-Smirnov type. Then, data similarity was tested against a comparator using the Independent Sample T-Test. Heart rate (bpm) and oxygen saturation (SpO2%) data from the sensor were normally distributed with a p-value ≥ 0.05. MQTT demonstrated optimal performance at QoS 0 with a throughput value of 604.8 bits/s, delay of 3.4 ms, jitter of 2.152 ms, and no packet loss.

A new Kolmogorov-Smirnov test based on representative points in the exponential distribution family

Hypothesis testing for the exponential distribution family consistently gains significant attentions from the statistical community, especially in situations characterized by small sample sizes. In this paper, we propose a novel approach for constructing a Kolmogorov-Smirnov type test based on representative points. This testing statistic is derived through a comparison between two empirical functions: one from the observed data and another from chosen representative points. A distribution-free quantile estimator is used for sample standardization. We also discuss how to select an optimal number of representative points based on the loss function, which is recommended for constructing the empirical function from the exponential distribution family. To evaluate the efficiency of this new test, a comprehensive Monte Carlo simulation is conducted to compare this new test with tailor-made tests. Simulation results show that our test statistic will be more powerful especially for small sample sizes, and it is competitive in the comparison with tailor-made tests. Some datasets are analyzed to further illustrate the efficiency of the proposed method.

Kolmogorov–Smirnov type testing for structural breaks: A new adjusted-range based self-normalization approach

A popular self-normalization (SN) approach in time series analysis uses the variance of a partial sum as a self-normalizer. This is known to be sensitive to irregularities such as persistent autocorrelation, heteroskedasticity, unit roots and outliers. We propose a novel SN approach based on the adjusted-range of a partial sum, which is robust to these aforementioned irregularities. We develop an adjusted-range based Kolmogorov–Smirnov type test for structural breaks for both univariate and multivariate time series, and consider testing parameter constancy in a time series regression setting. Our approach can rectify the well-known power decrease issue associated with existing self-normalized KS tests without having to use backward and forward summations as in Shao and Zhang (2010), and can alleviate the “better size but less power” phenomenon when the existing SN approaches (Shao, 2010; Zhang et al., 2011; Wang and Shao, 2022) are used. Moreover, our proposed tests can cater for more general alternatives. Monte Carlo simulations and empirical studies demonstrate the merits of our approach.

A nonparametric test for paired data

The paper proposes a weighted Kolmogorov–Smirnov type test for the two-sample problem for paired data. The asymptotic distribution of the test statistic under the null model is derived. The dependence of both the finite sample and the asymptotic distribution of the test statistic on the dependence structure of the data requires the use of the wild bootstrap technique for inference. The related wild bootstrap test turns out to be a consistent asymptotically level α test. With the finite sample correction applied, the test keeps the level well. An extensive simulation study demonstrates good finite sample behaviour of the test in comparison to the existing procedures. The main role in the proofs is played by tools from empirical processes. Additional simulation results, as well as an R code, which allow one to easily repeat the conducted numerical experiments, are attached as Supplementary Material.

Fast calculation of p-values for one-sided Kolmogorov-Smirnov type statistics

A novel method for computing exact p-values of one-sided statistics from the Kolmogorov-Smirnov family is presented. It covers the Higher Criticism statistic, one-sided weighted Kolmogorov-Smirnov statistics, and the one-sided Berk-Jones statistics. In addition to p-values, the method can also be used for power analysis, finding alpha-level thresholds, and the construction of confidence bands for the empirical distribution function. With its quadratic runtime and numerical stability, the method easily scales to sample sizes in the hundreds of thousands and takes less than a second to run on a sample size of 25,000. This allows practitioners working on large data sets to use exact finite-sample computations instead of approximation schemes. The method is based on a reduction to the boundary-crossing probability of a pure jump stochastic process. FFT convolutions of two different sizes are then used to efficiently propagate the probabilities of the non-crossing paths. This approach has applications beyond statistics, for example in financial risk modeling.

Goodness-of-Fit Test for Non-Stationary and Strongly Dependent Samples

In this article we improve a goodness-of-fit test, of the Kolmogorov-Smirnov type, for equally distributed- but not stationary-strongly dependent data. The test is based on the asymptotic behavior of the empirical process, which is much more complex than in the classical case. Applications to simulated data and discussion of the obtained results are provided. This is, to the best of our knowledge, the first result providing a general goodness of fit test for non-weakly dependent data.

Testing for the presence of treatment effect under selection on observables

The evaluation of the possible effects of a treatment on an outcome plays a central role in both theoretical and applied statistical and econometrical literature. This paper focuses on nonparametric tests for possible difference in the distribution of potential outcomes due to receiving or not receiving a treatment. The approach is based on weighting observed data on the basis on the estimated propensity score. Kolmogorov–Smirnov type and Wilcoxon–Mann–Whitney type tests are constructed, and their limiting distributions are studied. Rejection regions are obtained by inverting confidence intervals. This involves the study of appropriate estimators of the limiting variance of test statistics. Approximations of quantiles via subsampling are also considered. The merits of the different tests are studied by Monte Carlo simulation. An application to the construction of tests for stochastic dominance is provided.

Supervised Nonparametric Classification in the Context of Replicated Point Patterns

A spatial point pattern is a collection of points in space, representing, e.g. observed locations of trees, bird nests, centers of cells in a histological sample, etc. When several independent realizations of the underlying stochastic process are observed, these realizations are referred to as replicated point patterns. The main objective of this paper is to classify a newly observed pattern into one of the existing classes using a supervised nonparametric classification method, namely the Bayes classifier in combination with the k-nearest neighbors algorithms and the kernel regression method. The dissimilarity between a pair of patterns is defined using the functional summaries extracted from the point patterns via the Cramér-von Mises or Kolmogorov-Smirnov type formula. A set of simulation experiments is presented to investigate the performance of the proposed classifier with a dissimilarity measure based on functional summaries, such as the pair correlation function. The application of such a classifier to a real point pattern dataset is also illustrated.

Semiparametric modelling of two-component mixtures with stochastic dominance.

In this work, we studied a two-component mixture model with stochastic dominance constraint, a model arising naturally from many genetic studies. To model the stochastic dominance, we proposed a semiparametric modelling of the log of density ratio. More specifically, when the log of the ratio of two component densities is in a linear regression form, the stochastic dominance is immediately satisfied. For the resulting semiparametric mixture model, we proposed two estimators, maximum empirical likelihood estimator (MELE) and minimum Hellinger distance estimator (MHDE), and investigated their asymptotic properties such as consistency and normality. In addition, to test the validity of the proposed semiparametric model, we developed Kolmogorov–Smirnov type tests based on the two estimators. The finite-sample performance, in terms of both efficiency and robustness, of the two estimators and the tests were examined and compared via both thorough Monte Carlo simulation studies and real data analysis.Supplementary InformationThe online version contains supplementary material available at 10.1007/s10463-022-00835-5.

Three-Dimensional Control Charts for Regulating Processes Described by a Two-Dimensional Normal Distribution

In the statistical management of processes in the initial phase, the stability of the technological process is determined based on the available samples. If the process is not stable, eliminating possible causes is brought into a statistically controlled position. At the same time, simple Shewhart control charts are used. In practice, some methods bring the process to a stable state (ISO standards, standards of various states). After the process has become stable, the boundaries of control charts are found for further management. Then, with the help of new samples, the process is managed. The article considers a process modeled by a two-dimensional normal distribution. New control charts have been found to check the normality and correlation of two-dimensional random variable components. The process is regulated using these charts, preserving the shape of the density of the individual components of the normal vector and linearity of these components. When constructing control charts, the Kolmogorov-Smirnov type agreement criterion and the Fisher criterion on the strength of the linear coupling of components were used. A concrete example shows the course of the introduction of these charts in production. The work results can be used in the initial phase of regulation and during the control check of the process under study. We used these control charts to assess product quality and quality control coming from the machine that produces the sleeves. It presents statistical methods for analyzing problems in factory practice and solutions for their elimination.

The general goodness-of-fit tests for correlated data

Analyzing correlated data by goodness-of-fit type tests is a critical statistical problem in many applications. A unified framework is provided through a general family of goodness-of-fit tests (GGOF) to address this problem. The GGOF family covers many classic and newly developed tests, such as the minimal p-value test, Simes test, the GATES, one-sided Kolmogorov-Smirnov type tests, one-sided phi-divergence tests, the generalized Higher Criticism, the generalized Berk-Jones, etc. It is shown that the omnibus test that automatically adapts among GGOF statistics for given data, i.e., the GGOF-O, is still contained in the GGOF family and is computationally efficient. For analytically controlling the type I error rate of any GGOF tests, exact calculation is deduced under the Gaussian model with positive equal correlations. Based on that, the effective correlation coefficient (ECC) algorithm is proposed to address arbitrary correlations. Simulations are used to explore how signal and correlation patterns jointly influence typical GGOF tests' statistical power. The GGOF-O is shown robustly powerful across various signal and correlation patterns. As demonstrated by a study of bone mineral density, the GGOF framework has good potential for detecting novel disease genes in genetic summary data analysis. Computational tools are available in the R package SetTest on the CRAN.

Morphometric parameters and indices of the lumbar intervertebral disks at girls and women

Annotation. Individual variability of intervertebral discs is one of the factors that makes it difficult to recognize the early signs of their pathological changes before the onset of clinical manifestations. In the development of evidence-based medicine, it is important to use clear criteria for distinguishing between norm and pathology, taking into account individual variability. The “mathematical modeling” and a comprehensive approach to quantitative and qualitative assessment of intervertebral discs will help in the early detection of abnormalities, in the individualization of norm, as well as in modeling the parameters of the artificial intervertebral disc for reconstructive interventions. The analysis by methods of descriptive statistics and the correlation analysis for the purpose of further mathematical processing, selection of optimal parameters for mathematical modeling is carried out. Anthropometric characteristics (body length and body weight, mass-growth factor and mass-growth index) were determined in young women and women of the first period of adulthood (16-26 years), as in separate age groups and in the combined group. MRI of the lumbar spine was performed, followed by morphometry. Statistical analysis was performed in the license program “STATISTICA 6.1”. The mean values of the variation series, standard deviations, coefficients of variation and asymmetry, paired t-test, d-test of Kolmogorov-Smirnov type, Levene index were evaluated. It was determined that the sums of the transverse and sagittal dimensions and the sums of the three dimensions increase proportionally in the caudal direction, have a distribution of indicators as close as possible to normal, and their coefficients of variation are many times smaller than for cross-sectional areas and volumes. Indicators of the ratio of the sum of sagittal and transverse size and the sum of three sizes to the average height of the intervertebral discs have a variability of less than 10% and correspond to the characteristics of the general population. Body weight and body length have significantly higher correlation coefficients with the sums of sagittal and transverse dimensions, with the sums of three dimensions, and cross-sectional areas in comparison with the partial dimensions of the intervertebral discs.

Empirical likelihood inference and goodness-of-fit test for logistic regression model under two-phase case-control sampling

ABSTRACT Due to cost-effectiveness and high efficiency, two-phase case-control sampling has been widely used in epidemiology studies. We develop a semi-parametric empirical likelihood approach to two-phase case-control data under the logistic regression model. We show that the maximum empirical likelihood estimator has an asymptotically normal distribution, and the empirical likelihood ratio follows an asymptotically central chi-square distribution. We find that the maximum empirical likelihood estimator is equal to Breslow and Holubkov (1997)'s maximum likelihood estimator. Even so, the limiting distribution of the likelihood ratio, likelihood-ratio-based interval, and test are all new. Furthermore, we construct new Kolmogorov–Smirnov type goodness-of-fit tests to test the validation of the underlying logistic regression model. Our simulation results and a real application show that the likelihood-ratio-based interval and test have certain merits over the Wald-type counterparts and that the proposed goodness-of-fit test is valid.

Confidence bands for exponential distribution functions under progressive type-II censoring

Based on a progressively type-II censored sample from the exponential distribution with unknown location and scale parameter, confidence bands are proposed for the underlying distribution function by using confidence regions for the parameters and Kolmogorov–Smirnov type statistics. Simple explicit representations for the boundaries and for the coverage probabilities of the confidence bands are analytically derived, and the performance of the bands is compared in terms of band width and area by means of a data example. As a by-product, a novel confidence region for the location-scale parameter is obtained. Extensions of the results to related models for ordered data, such as sequential order statistics, as well as to other underlying location-scale families of distributions are discussed.

One-two dependence and probability inequalities between one- and two-sided union-intersection tests.

In a paper published in 1939 in The Annals of Mathematical Statistics, Wald and Wolfowitz discussed the possible validity of a probability inequality between one- and two-sided coverage probabilities for the empirical distribution function. Twenty-eight years later, Vandewiele and Noé proved this inequality for Kolmogorov-Smirnov type goodness of fit tests. We refer to this type of inequality as one-two inequality. In this paper, we generalize their result for one- and two-sided union-intersection tests based on positively associated random variables and processes. Thereby, we give a brief review of different notions of positive association and corresponding results. Moreover, we introduce the notion of one-two dependence and discuss relationships with other dependence concepts. While positive association implies one-two dependence, the reverse implication fails. Last but not least, the Bonferroni inequality and the one-two inequality yield lower and upper bounds for two-sided acceptance/rejection probabilities which differ only slightly for significance levels not too large. We discuss several examples where the one-two inequality applies. Finally, we briefly discuss the possible impact of the validity of a one-two inequality on directional error control in multipletesting.

Testing for Structural Breaks – A New Self-Normalization Approach Based on the Adjusted Sample Range

In this paper, we introduce adjusted-range based Kolmogorov-Smirnov (KS) type statistics to test for structural breaks in the mean of a process and also in a more general setting. We propose a normalization based on the adjusted-range of a partial sum, which is stochastically proportional to the true long-run variance (LRV), and is nuisance parameter and distribution free. We find, through simulation studies, that the new statistic offers better performance than the self-normalized KS statistic (Shao and Zhang, 2010) and the KS statistics utilizing the small-b and fixed-b approaches (Kiefer and Vogelsang, 2005). Finally, empirical studies also illustrate the suitability of KS type tests.

Kolmogorov-Smirnov APF Test for Inhomogeneous Poisson Processes with Shift Parameter

In this article, we study the Kolmogorov-Smirnov type goodness-of-fit test for the inhomogeneous Poisson process with the unknown translation parameter as multidimensional parameter. The basic hypothesis and the alternative are composite and carry to the intensity measure of inhomogeneous Poisson process and the intensity function is regular. For this model of shift parameter, we propose test which is asymptotically partially distribution free and consistent. We show that under null hypothesis the limit distribution of this statistic does not depend on unknown parameter.

Kolmogorov–Smirnov type test for generated variables

Distribution homogeneity testing, particularly based on the Kolmogorov–Smirnov statistic, has been applied in various empirical studies. In empirical economic analysis, it is often the case that economic variables of interest are obtained as estimated values or residuals of preliminary model fits, called generated variables. In this paper, we extend the Kolmogorov–Smirnov type homogeneity test to accommodate such generated variables, and propose an asymptotically valid bootstrap inference procedure. A small simulation study illustrates that it is crucial for reliable inference to account for estimation errors in the generated variables. The proposed method is applied to compare the total factor productivities across different countries.

New tests for exponentiality based on a characterization with random shift

We derive the efficiencies of two new tests for exponentiality which are based on a recent characterization that uses the idea of a random shift. The finite-sample performance of the newly proposed tests is evaluated and compared to other existing tests by means of Monte Carlo simulations. It is found that the new tests perform favourably when compared to the other tests. Overall the best performing tests seem to be our new Kolmogorov-Smirnov type test, the score function based test by Cox and Oakes, and the Kolmogorov-Smirnov type test based on the mean residual life. The tests are also applied to a real-world data set with i.i.d. data as well as to simulated data from a Cox-proportional hazards model, where we test whether the so-called Cox–Snell residuals follow a standard exponential distribution.

Directional differentiability for supremum-type functionals: Statistical applications

We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and the amplitude of a function. The (usually non-linear) derivatives of these maps adopt simple expressions under suitable assumptions on the underlying space. As an application, we improve and extend to the multidimensional case the results in Raghavachari (Ann. Statist. 1 (1973) 67–73) regarding the limiting distributions of Kolmogorov–Smirnov type statistics under the alternative hypothesis. Similar results are obtained for analogous statistics associated with copulas. We additionally solve an open problem about the Berk–Jones statistic proposed by Jager and Wellner (In A Festschrift for Herman Rubin (2004) 319–331 IMS). Finally, the asymptotic distribution of maximum mean discrepancies over Donsker classes of functions is derived.