Empirical Distribution Functions and Strong Approximation Theorems for Dependent Random Variables. A Problem of Baker in Probabilistic Number Theory

<strong class="journal-contentHeaderColor">Abstract.</strong> Distributional data such as detrital age populations or grain size distributions are common in the geological sciences. As analytical techniques become more sophisticated, increasingly large amounts of distributional data are being gathered. These advances require quantitative and objective methods, such as multidimensional scaling (MDS), to analyse large numbers of samples. Crucial to such methods is choosing a sensible measure of dissimilarity between samples. At present, the Kolmogorov-Smirnov (KS) statistic is the most widely used of these dissimilarity measures. However, the KS statistic has some limitations. It is very sensitive to differences between the modes of two distributions, and relatively insensitive to differences between their tails. Here we introduce the Wasserstein-2 distance (<em>W</em>₂) as an alternative to address this issue. Whereas the KS-distance is defined as the maximum vertical distance between two empirical cumulative distribution functions, the <em>W</em>₂-distance is a function of the horizontal distances (i.e., age differences) between individual observations. Using a combination of synthetic examples and a published zircon U-Pb dataset, we show that the <em>W</em>₂ distance produces similar MDS results to the KS-distance in most cases, but significantly different results in some cases. Where the results differ, the <em>W</em>₂ results are geologically more sensible. For the case study, we find that the MDS map that is produced using <em>W</em>₂ can be readily interpreted in terms of the shape and average age of the age spectra. The <em>W</em>₂-distance has been added to the R package IsoplotR, for immediate use in detrital geochronology and other applications. The <em>W</em>₂ distance can be generalised to multiple dimensions, which opens opportunities beyond distributional data.

While often confused, the Kolmogorov–Smirnov test and the Smirnov test are actually distinct. Specifically, the Kolmogorov–Smirnov test is used to test the goodness of fit of a given set of data to a theoretical distribution, making this a one‐sample test. In contrast, the Smirnov test is a two‐sample test, used to determine if two samples appear to follow the same distribution. The intuition behind the two tests is the same, however, in that both compare cumulative distribution functions, either two empirical cumulative distribution functions for the two‐sample Smirnov test, or one empirical cumulative distribution function and one known cumulative distribution function for the one‐sample Kolmogorov–Smirnov test.

While often confused, the Kolmogorov–Smirnov test and the Smirnov test are actually distinct. Specifically, the Kolmogorov–Smirnov test is used to test the goodness of fit of a given set of data to a theoretical distribution, making this a one‐sample test. In contrast, the Smirnov test is a two‐sample test, used to determine if two samples appear to follow the same distribution. The intuition behind the two tests is the same, however, in that both compare cumulative distribution functions, either two empirical cumulative distribution functions for the two‐sample Smirnov test, or one empirical cumulative distribution function and one known cumulative distribution function for the one‐sample Kolmogorov–Smirnov test.

New methods are presented that utilize the optimization of goodness-of-fit statistics in order to estimate Weibull parameters from failure data. It is assumed that the underlying population is characterized by a three-parameter Weibull distribution. Goodness-of-fit tests are based on the empirical distribution function (EDF). The EDF is a step function, calculated using failure data, and represents an approximation of the cumulative distribution function for the underlying population. Statistics (such as the Kolmogorov-Smirnov statistic and the Anderson-Darling statistic) measure the discrepancy between the EDF and the cumulative distribution function (CDF). These statistics are minimized with respect to the three Weibull parameters. Due to nonlinearities encountered in the minimization process, Powell’s numerical optimization procedure is applied to obtain the optimum value of the EDF. Numerical examples show the applicability of these new estimation methods. The results are compared to the estimates obtained with Cooper’s nonlinear regression algorithm.

The Calculation of the Spare Parts in the Auto-service Enterprise on the Base of Real Demand

The human body generates more proteins than it has genes that code for proteins. The diversity of proteins stems from the alternative ways in which RNA can be spliced and reassembled. Each alternative version of RNA produces a different protein, providing a way for our bodies to produce a wide range of proteins with a single gene. Some alternative RNA transcripts, however, have splicing errors and produce faulty proteins involved in genetic diseases. Understanding splicing patterns and profiles has wide implications for our understanding of healthy and diseased tissue states. Currently little is known regarding the splicing profiles of healthy tissue which vary across individuals and within individuals by tissue type. Therefore, this project explored the use of RNA splicing data from the first chromosome to predict the tissue type of non-cancerous samples using distribution analysis and supervised learning methods. The Kolmogorov-Smirnov test was used to classify the samples based on empirical cumulative distribution functions and was not able to reliably distinguish between tissue types. Our SVM model was run using both the quantity of splice junctions observed and their presence, and had a high prediction accuracy for both data sets. The performance between the two SVM model outcomes were not significantly different. Overall, the findings suggest the utility of using splice junction data in biological classification and sets the foundation for future work of mapping splicing patterns with phenotype.

In this study, we quantify vegetation fire activity in India using the MODerate resolution Imaging Spectroradiometer (MODIS) active fire datasets. We assessed different fire regime attributes, i.e., fire frequency, seasonality, intensity and the type of vegetation burnt in diverse geographical regions. MODIS data from 2002–2010 revealed an average of 63696 fire counts per year with the highest during 2009. Fire season in India extends from October to June with the peak during March. The K-means algorithm identified hotspot regions of fire clusters in diverse regions of India. We examined fire radiative power (FRP) data in the hotspot regions to address which fires burn intensively than others based on the vegetation type. We first assessed the best statistical fit distributions for the FRP data using the probability density functions (PDFs) and ranked them based on Kolmogorov-Smirnov statistic. We then described the fire intensities using empirical cumulative distribution functions (CDFs). Results suggest diverse pdfs for the FRP data that included Burr, Dagum, Johnson as well as Pearson distribution and they varied based on the vegetation type burnt. Analysis from empirical CDFs suggested relatively high fire intensity for closed broadleaved evergreen/ semi-deciduous forests than the other vegetation types. Although, annual sum of FRP for agricultural fires was less than the closed broadleaved evergreen forests, the values were higher than the mosaic vegetation category and broadleaved deciduous forests. These results on fire hotspots and FRP will be useful to address the impact of vegetation fires on air pollution and climate in India.

Let X 1,…, X n be i.i.d. random variables from an unknown cumulative distribution function F defined on the real line ℝ. No assumptions are made on the unknown F. The problem is to predict the empirical distribution function of a future sample Y = (Y 1,…,Y m ) from the distribution function F on the basis of the observations X 1,…, X n . To solve the problem, a decision theoretic approach is used. In certain situations, when for example a weighted quadratic loss function is used in the prediction problem, the problem of finding the minimax predictor of the empirical distribution function can be reduced to the problem of minimax estimation of the cumulative distribution function F under a composite weighted quadratic loss function. In such a case, a general method of finding minimax predictors of is presented. The method is also useful when no equalizer rule is available. The problem of prediction considered refers also to prediction of the empirical lifetime distribution function and may have some practical significance. For example, the results enable to predict, under the minimax criterion, the number of devices (from among all the m ones) which will fail up to any fixed time t 0.

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in {\mathbb R}$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $y_1,y_2,\dotsc$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = {\mathbb E}[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi |\chi|$, then $\nu_{\bf y}$ is absolutely continuous with respect to the Lebesgue measure for almost every ${\bf y}$. We also prove that if $h < |\chi|$, then the measure $\nu_{\bf y}$ is singular and has dimension $h/|\chi|$ for almost every ${\bf y}$. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.

ObjectivePanax ginseng and Atractylodes macrocephala Koidz. (AMK) are widely used in treating various diseases; however, research is insufficient on measuring the relationship that exists by combining this drug pair using the copula function. Methods In this study, 279 traditional Chinese medicine prescriptions containing the Panax ginseng and AMK drug pair were extracted from the prescription database for three types of screened indications, namely, diabetes mellitus, diarrhea, and insomnia. Following the principle of dose conversion, each dynasty unit was uniformly converted into a modern unit. Then, the kernel density distribution of Panax ginseng and AMK was fitted with their empirical distribution functions. Finally, the optimal copula function was selected from the copula function family as a t-copula function. Results The empirical distribution and probability density functions of Panax ginseng and AMK were obtained. From the results, their Kendall rank correlation coefficients with indications of diabetes mellitus, insomnia, and diarrhea were 0.8689, 0.7858, and 0.7403, whereas their Spearman rank correlation coefficients were 0.9563, 0.9276, and 0.8958. Results also indicated that the use of the t-copula function can better reflect the correlation between Panax ginseng and AMK doses. Conclusion From the three indications, the dose between Panax ginseng and AMK was positively correlated. This study, therefore, confirms the medicinal principle of Chinese medicine “combining” from the perspective of mathematical statistics. Results from this study are practical to evaluate the correlation between the drug pair doses, Panax ginseng and AMK, using the copula function model, which provides a new model for the scientific explanation of compatibility for Chinese medicines. This study also provides a methodological basis for more drug measurement studies and clinical medications.

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper. A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.

Distribution test, a method to evaluate the goodness-of-fit, has been applied to automatic modulation clas-sification (AMC) in recent years. It compares the empirical cumulative distribution function (ECDF) of the sample signal with the theoretical cumulative distribution function (TCDF) under each candidate modulation format. Most existing works in MIMO systems assume the additive noise is Gaussian. However, additive noise often exhibits non-Gaussian characteristics in practical systems. The TCDF of each modulation developed for Gaussian noise can not perform well in practical system. To solve this issue, this paper proposes a practical MIMO system model, which assumes that the noise is Cauchy-Gaussian bi-parameter mixture and analyzes the TCDF corresponding to different modulation modes under this noise model. The Kolmogorov-Smirnov (KS) distribution test is used to AMC in flat-fading channel for both Quadrature Amplitude Modulation and Phase Shift Keying. Extensive simulation results demonstrate that, compared with two-sample KS test based classifier, one-sample KS test based classifier can achieve good recognition results under the non-Gaussian noise model.

In this paper, we propose a Cumulant Kolmogorov-Smirnov (KS) test for spectrum sensing in Cognitive Radio networks. The KS test is a kind of goodness of fit test that quantifies a distance between the empirical distribution functions of the received signals and the channel noise samples. Due to using 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> order Cumulant statistics, the proposed KS detector has some advantages over previous detectors. We exert KS test to the estimated Cumulant vector and perform detection problem. The proposed method is nonparametric and needs only a short sequence of noise samples. In addition, it is not sensitive to noise uncertainty. Superiority of the proposed KS-based method is shown by comparing it with a popular KS method and the classical Energy Detector.

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.