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In both Tweedie and geometric Tweedie models, the common power parameter p ∉ (0 , 1) works as an automatic distribution selection. It separates two subclasses of semicontinuous (12) ones; and two datasets for illustration purposes are investigated. [ABSTRACT FROM AUTHOR]

Consider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$ , extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions. [ABSTRACT FROM AUTHOR]

This paper studies clustering of data sequences using the k-medoids algorithm. All the data sequences are assumed to be generated from unknown continuous distributions, which form clusters with each cluster containing a composite set of closely located distributions (based on a certain distance metric between distributions). The maximum intracluster distance is assumed to be smaller than the minimum intercluster distance, and both values are assumed to be known. The goal is to group the data sequences together if their underlying generative distributions (which are unknown) belong to one cluster. Distribution distance metrics based k-medoids algorithms are proposed for known and unknown number of distribution clusters. Upper bounds on the error probability and convergence results in the large sample regime are also provided. It is shown that the error probability decays exponentially fast as the number of samples in each data sequence goes to infinity. The error exponent has a simple form regardless of the distance metric applied when certain conditions are satisfied. In particular, the error exponent is characterized when either the Kolmogrov–Smirnov distance or the maximum mean discrepancy are used as the distance metric. Simulation results are provided to validate the analysis. [ABSTRACT FROM AUTHOR]

We investigate the nonparametric, composite hypothesis testing problem for arbitrary unknown distributions in the asymptotic regime where both the sample size and the number of hypothesis grow exponentially large. Such asymptotic analysis is important in many practical problems, where the number of variations that can exist within a family of distributions can be countably infinite. We introduce the notion of discrimination capacity, which captures the largest exponential growth rate of the number of hypothesis relative to the sample size so that there exists a test with asymptotically vanishing probability of error. Our approach is based on various distributional distance metrics in order to incorporate the generative model of the data. We provide analyses of the error exponent using the maximum mean discrepancy and Kolmogorov–Smirnov distance and characterize the corresponding discrimination rates, i.e., lower bounds on the discrimination capacity, for these tests. Finally, an upper bound on the discrimination capacity based on Fano's inequality is developed. Numerical results are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of the model selection/discrimination among three different positively skewed lifetime distributions. All these three distributions, namely; the Weibull, log-normal, and log-logistic, have been used quite effectively to analyze positively skewed lifetime data. In this article, we have used three different methods to discriminate among these three distributions. We have used the maximized likelihood method to choose the correct model and computed the asymptotic probability of correct selection. We have further obtained the Fisher information matrices of these three different distributions and compare them for complete and censored observations. These measures can be used to discriminate among these three distributions. We have also proposed to use the Kolmogorov-Smirnov distance to choose the correct model. Extensive simulations have been performed to compare the performances of the three different methods. It is observed that each method performs better than the other two for some distributions and for certain range of parameters. Further, the loss of information due to censoring are compared for these three distributions. The analysis of a real dataset has been performed for illustrative purposes. [ABSTRACT FROM AUTHOR]

Schumann resonances' statistical parameters vary over time, which gives way to analyze them as a stochastic process. By splitting the Schumann resonance's records into segments and obtaining their empirical distribution function, the differences can be evaluated by calculating the Kolmogorov–Smirnov distance between them. The analysis' results allow for the characterization of the Schumann resonance's variations, along with the typical values it reaches under the chosen metric. It is shown how divergence is mitigated through sample averaging, and also how divergent samples impact on the Fast Fourier Transform algorithm. Divergence quantification adds a layer of information for data processing. Knowing the changes experienced over time in Schumann resonances gives a way to know what kind of mathematical procedures can be applied to the signal. Quantification of signal variations over time can identify error sources in specific procedures, filter out samples unfit under certain analyses, or serve as a stop criteria for cumulative analyses. • Schumann Resonances' temporal segments compared with Kolmogorov–Smirnov distance. • Flexible methodology to compare segments of different duration allowing overlap. • Methodology tested over real data, showing relationship with lightning activity. • Kolmogorov–Smirnov distance as support for further analysis procedures discussed. [ABSTRACT FROM AUTHOR]

This paper considers the problem of feature extraction of pipeline acoustic signals under different working conditions. A novel method is proposed based on multi-scale dispersion entropy partial mean (MDEPM) of multi-modal component to extract features of pipeline signals. First, variational mode decomposition (VMD) algorithm is applied to decompose the acoustic signals to obtain several mode components. Then, Kolmogorov–Smirnov distance (KSD) is introduced as the index to measure the correlation between each mode component and the original signal, and the mode component with a smaller KSD is selected as the feature component. Finally, the MDEPM of the feature component is calculated so as to form the feature vector, which realizes signal feature extraction. In experiment, 13 different condition signals collected are divided into three categories, the experimental results show that the proposed method could extract the characteristics of the different pipeline acoustic signals. Furthermore, the extracted features could be accurately identified and classified by extreme learning machine (ELM) under different working conditions, and then through comparing with other methods, the feasibility and superiority of the proposed method are verified. • To select the feature components by analyzing the Kolmogorov–Smirnov distance between each mode component and the original signal. • The multi-scale dispersion entropy partial mean could characterize the features of different pipeline signals at different scales. • The method of MMC-MDEPM could be used to extract the different signal features. [ABSTRACT FROM AUTHOR]

Clickstream data analysis is considered as the process of collecting, analysing and reporting the aggregate data about the web pages a visitor clicks. Visualizing the clickstream data has gained significant importance in many applications like web marketing, customer prediction, product management, etc. Most existing works employ different tools for visualizing along with techniques like Markov chain modelling. However the accuracy of the methods can be improved when the shortcomings are resolved. Markov chain modelling has problems of occlusion and unable to provide clear display of data visualizing. These issues can be resolved by improving the Markov chain model by introducing a heuristic method of Kolmogorov– Smirnov distance and maximum likelihood estimator for visualizing. These concepts are employed between the underlying distribution states to minimize the Markov distribution. The proposed model named as WebClickviz is performed in Hadoop Apache Flume which is a highly advanced tool. The clickstream data visualization accuracy can be improved when Apache Flume tools are used. The performance evaluation are made on a specific website clickstream data which shows the proposed model of visualization has better performance than existing models like VizClick.

The exponential-geometric distribution has been proposed as a simple and useful reliability model for analyzing lifetime data. For this distribution, some recurrence relations are established for the single moments and product moments of order statistics. Using these recurrence relations, the means, variances and covariances of all order statistics can be computed for all sample sizes in a simple and efficient recursive manner. Next, we discuss the maximum likelihood estimation of the model parameters as well as some simple modified methods of estimation. Then, a Monte Carlo simulation study is carried out to evaluate the performance of all these methods of estimation in terms of their bias and mean square error as well as the percentage of times the estimates converged. Two illustrative examples are finally presented to illustrate all the inferential results developed here. [ABSTRACT FROM PUBLISHER]

Background: Chemical or small interfering (si) RNA screens measure the effects of many independent experimental conditions, each applied to a population of cells (e.g., all of the cells in a well). High-content screens permit a readout (e.g., fluorescence, luminescence, cell morphology) from each cell in the population. Most analysis approaches compare the average effect on each population, precluding identification of outliers that affect the distribution of the reporter in the population but not its average. Other approaches only measure changes to the distribution with a single parameter, precluding accurate distinction and clustering of interesting outlier distributions. Results: We describe a methodology to identify outlier conditions by considering the cell-level measurements from each condition as a sample of an underlying distribution. With appropriate selection of a distance metric, all effects can be embedded in a fixed-dimensionality Euclidean basis, facilitating identification and clustering of biologically interesting outliers. We demonstrate that measurement of distances with the Hellinger distance metric offers substantial computational efficiencies over alternative metrics. We validate this methodology using an RNA interference (RNAi) screen in mouse embryonic stem cells (ESC) with a Nanog reporter. The methodology clusters effects of multiple control siRNAs into their true identities better than conventional approaches describing the median cell fluorescence or the commonly used Kolmogorov-Smirnov distance between the observed fluorescence distribution and the null distribution. It identifies outlier genes with effects on the reporter distribution that would have been missed by other methods. Among them, siRNA targeting Chek1 leads to a wider Nanog reporter fluorescence distribution. Similarly, siRNA targeting Med14 or Med27 leads to a narrower Nanog reporter fluorescence distribution. We confirm the roles of these three genes in regulating pluripotency by mRNA expression and alkaline phosphatase staining using independent short hairpin (sh) RNAs. Conclusions: Using our methodology, we describe each experimental condition by a probability distribution. Measuring distances between probability distributions permits a multivariate rather than univariate readout. Clustering points derived from these distances allows us to obtain greater biological insight than methods based solely on single parameters. We find several outliers from a mouse ESC RNAi screen that we confirm to be pluripotency regulators. Many of these outliers would have been missed by other analysis methods. [ABSTRACT FROM AUTHOR]

Due to an error in the computer program used by Kent County, Michigan, for about 15 months, the African-American proportion of jury pools was about one-half their proportion of the eligible population. Subsequently, a number of defendants appealed their convictions because the jury pool did not fairly represent the community. Although a Federal Court of Appeals found that the statistical evidence helped the defendant establish that African-Americans were under-represented; in a different case, People v. Bryant, the Michigan Supreme Court found the statistics insufficient. The different conclusions arose because the Michigan Court adopted a new criterion that the "disparity of the risk" measure should be at least 0.50. The statistical properties of the measure will be described and it will be seen that it is equivalent to requiring that the Kolmogorov-Smirnov distance between the two relevant distributions is at least 0.50. If one is comparing two normal distributions, with different means and the same variance, the requirement implies that the effect size would need to be at least 1.35 before one could conclude that they differed. Since effect sizes of 0.8 are considered "large," the criteria used by the Michigan Court are far too stringent and if adopted nation-wide would allow individuals to have trials in front of juries where minorities are substantially under-represented. [ABSTRACT FROM AUTHOR]

Abstract: Information criteria based methods are proposed to select the best probability law to model the distribution of samples resulting from the small-scale fading of the propagation channel. The first is based on the estimation of an optimal histogram approximating the probability density function. The second one employs the direct use of an information criterion. Indeed, the modelling of the radio mobile channel small-scale fading is crucial in digital communications. It is the reason why several propagation models have been implemented to take into account the electromagnetic phenomena inherent in radio wave channels. Amongst these models is the family of statistical distributions which is rapid in computation time. In the context of this study our concern is to find, among different probability laws, the one which best coincides with radio channel behaviour. The experimental results show that the proposed methods are better than those methods already employed, such as the classical Kolmogorov–Smirnov test using cumulative distribution functions, or methods using different estimators of probability density functions, like the kernel density estimator and the Gaussian mixture model. Results are provided in supervised and unsupervised contexts. [Copyright &y& Elsevier]

Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes. [ABSTRACT FROM AUTHOR]

Log-normal and log-logistic distributions are often used to analyze lifetime data. For certain ranges of the parameters, the shape of the probability density functions or the hazard functions can be very similar in nature. It might be very difficult to discriminate between the two distribution functions. In this article, we consider the discrimination procedure between the two distribution functions. We use the ratio of maximized likelihood for discrimination purposes. The asymptotic properties of the proposed criterion are investigated. It is observed that the asymptotic distributions are independent of the unknown parameters. The asymptotic distributions are used to determine the minimum sample size needed to discriminate between these two distribution functions for a user specified probability of correct selection. We perform some simulation experiments to see how the asymptotic results work for small sizes. For illustrative purpose, two data sets are analyzed. [ABSTRACT FROM AUTHOR]

We consider model selection and discrimination among three important lifetime distributions. These three distributions have been used quite effectively to analyze lifetime data. We study the probability of correct selection using the maximized likelihood method, as it has been used in the literature. We further compute the asymptotic probability of correct selection, and compare the theoretical, and simulation results for different sample sizes, and for different model parameters. The results have been extended for Type-I censored data also. The theoretical, and simulation results match quite well. Two real data sets have been analyzed for illustrative purposes. We also suggest a method to determine the minimum sample size required to discriminate among the three distributions for a given probability of correct selection, and a user specified protection level. [ABSTRACT FROM AUTHOR]

Abstract: The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov–Smirnov distance. Quantile–quantile plots are also provided to better understand the different distributions. [Copyright &y& Elsevier]

The number of comparisons Xn used by Quicksort to sort an array of n distinct numbers has mean μn of order nlogn and standard deviation of order n. Using different methods, Re´gnier and Ro¨sler each showed that the normalized variate Yn:=(Xn−μn)/n converges in distribution, say to Y; the distribution of Y can be characterized as the unique fixed point with zero mean of a certain distributional transformation.We provide the first rates of convergence for the distribution of Yn to that of Y, using various metrics. In particular, we establish the bound 2n−1/2 in the d2-metric, and the rate O(nϵ−(1/2)) for Kolmogorov–Smirnov distance, for any positive ϵ. [Copyright &y& Elsevier]

In this paper we study stability of optimal solutions of stochastic programming problems with fixed recourse. An upper bound for the rate of convergence is given in terms of the objective functions of the associated deterministic problems. As an example it is shown how it can be applied to derivation of the Law of Iterated Logarithm for the optimal solutions. It is also shown that in the case of simple recourse this upper bound implies upper Lipschitz continuity of the optimal solutions with respect to the Kolmogorov-Smirnov distance between the corresponding cumulative probability distribution functions. [ABSTRACT FROM AUTHOR]