Statistical Sum Research Articles

Computing the exact distributions of some functions of the ordered multinomial counts: maximum, minimum, range and sums of order statistics

Starting from seminal neglected work by Rappeport (Rappeport 1968 Algorithms and computational procedures for the application of order statistics to queuing problems. PhD thesis, New York University), we revisit and expand on the exact algorithms to compute the distribution of the maximum, the minimum, the range and the sum of the J largest order statistics of a multinomial random vector under the hypothesis of equiprobability. Our exact results can be useful in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. We describe the algorithms, motivate their use in statistical testing and illustrate two applications. We also provide the codes and ready-to-use tables of critical values.

A general statistic to test an optimally weighted combination of common and/or rare variants.

Both genome-wide association study and next-generation sequencing data analyses are widely employed to identify disease susceptible common and/or rare genetic variants. Rare variants generally have large effects though they are hard to detect due to their low frequencies. Currently, many existing statistical methods for rare variants association studies employ a weighted combination scheme, which usually puts subjective weights or suboptimal weights based on some adhoc assumptions (e.g., ignoring dependence between rare variants). In this study, we analytically derived optimal weights for both common and rare variants and proposed a general and novel approach to test association between an optimally weighted combination of variants (G-TOW) in a gene or pathway for a continuous or dichotomous trait while easily adjusting for covariates. Results of the simulation studies show that G-TOW has properly controlled type I error rates and it is the most powerful test among the methods we compared when testing effects of either both rare and common variants or rare variants only. We also illustrate the effectiveness of G-TOW using the Genetic Analysis Workshop 17 (GAW17) data. Additionally, we applied G-TOW and other competitive methods to test disease-associated genes in real data of schizophrenia. The G-TOW has successfully verified genes FYN and VPS39 which are associated with schizophrenia reported in existing publications. Both of these genes are missed by the weighted sum statistic and the sequence kernel association test. Simulation study and real data analysis indicate that G-TOW is a powerful test.

Asymptotically Exact Framework to Approximate Sums of Positive Correlated Random Variables and Application to Diversity-Combining Receivers

Approximations to sums of random variables (RVs) have been a longstanding issue in view of the usual intricacy of the exact sum statistics. In this letter, a new technique called asymptotic matching is introduced and a corresponding new framework is proposed that allows for accurate statistical approximations to a broad class of sums of positive correlated RVs. As an application example, the proposed framework is used to analyze the performance of an equal-gain combiner operating over correlated Rayleigh fading channels, rendering an outstanding fit, especially at high signal-to-noise ratio - a most representative operational regime for communication systems.

Lower Bound for RIP Constants and Concentration of Sum of Top Order Statistics

Restricted Isometry Property (RIP) is of fundamental importance in the theory of compressed sensing and forms the base of many exact and robust recovery guarantees in this field. A quantitative description of RIP involves bounding the so-called RIP constants of measurement matrices. In this respect, it is noteworthy that most results in the literature concerning RIP are upper bounds of RIP constants, which can be interpreted as a theoretical guarantee of successful sparse recovery. On the contrary, the land of lower bounds for RIP constants remains uncultivated. Lower bounds of RIP constants, if exist, can be interpreted as the fundamental limit aspect of successful sparse recovery. In this paper, the lower bound of RIP constants Gaussian random matrices are derived, along with a guide for generalization to sub-Gaussian random matrices. This provides a new proof of the fundamental limit that the minimal number of measurements needed to enforce the RIP of order $s$ is $\Omega(s\log({\rm e}N/s))$, which is more straight-forward than the classical Gelfand width argument. Furthermore, in the proof, we propose a useful technical tool featuring the concentration phenomenon for top-$k$ sum of a sequence of i.i.d. random variables, which is closely related to mainstream problems in statistics and is of independent interest.

Wilcoxon-type rank-sum control charts based on progressively censored reference data

In this article, we introduce a new distribution − free Shewhart − type control chart implementing a modified Wilcoxon-type rank sum statistic based on progressive Type-II censoring reference data. The proposed chart is also a tool for monitoring the incomplete data, because the censoring scheme applied allows the protection of experimental units at an early stage of the testing procedure. The setup of the new nonparametric control chart is presented in detail, while its operating characteristic function is studied. Explicit formulae for the evaluation of Alarm Rate and Average Run Length values for both in-control and out-of-control situations are established. A numerical study carried out depicts the performance and robustness of the proposed control chart. For illustration purposes, a practical example is also discussed.

Determining population stratification and subgroup effects in association studies of rare genetic variants for nicotine dependence.

Rare variants (minor allele frequency < 1% or 5 %) can help researchers to deal with the confounding issue of 'missing heritability' and have a proven role in dissecting the etiology for human diseases and complex traits. We extended the combined multivariate and collapsing (CMC) and weighted sum statistic (WSS) methods and accounted for the effects of population stratification and subgroup effects using stratified analyses by the principal component analysis, named here as 'str-CMC' and 'str-WSS'. To evaluate the validity of the extended methods, we analyzed the Genetic Architecture of Smoking and Smoking Cessation database, which includes African Americans and European Americans genotyped on Illumina Human Omni2.5, and we compared the results with those obtained with the sequence kernel association test (SKAT) and its modification, SKAT-O that included population stratification and subgroup effect as covariates. We utilized the Cochran-Mantel-Haenszel test to check for possible differences in single nucleotide polymorphism allele frequency between subgroups within a gene. We aimed to detect rare variants and considered population stratification and subgroup effects in the genomic region containing 39 acetylcholine receptor-related genes. The Cochran-Mantel-Haenszel test as applied to GABRG2 (P = 0.001) was significant. However, GABRG2 was detected both by str-CMC (P= 8.04E-06) and str-WSS (P= 0.046) in African Americans but not by SKAT or SKAT-O. Our results imply that if associated rare variants are only specific to a subgroup, a stratified analysis might be a better approach than a combined analysis.

A data‐driven approach to detecting change points in linear regression models

AbstractChange points appear in various types of environmental data—from univariate time series to multivariate data structures—and need to be accounted for in proper analysis and inference. The analysis of change points is challenging when no exact information about their number and locations is available, and statistical tests developed under such conditions often have low power identifying the change points. This paper provides a powerful, data‐driven procedure for detecting at‐most‐m change points in linear regression models by adapting a sieve bootstrap approach for a modified cumulative sum statistic. The new procedure does not assume a particular dependence structure nor a particular distribution of regression residuals. It employs a data‐driven selection of the order of an autoregressive model and a robust estimation of the model coefficients. Our simulation studies show a competitive performance of the new bootstrap‐based procedure compared with its asymptotic counterpart. We apply the new testing procedure to address an important environmental problem in Chesapeake Bay—severe oxygen depletion—and detect two change points in the relationship between the volume of low‐oxygen waters and nutrient inputs to the bay during 1985–2017.

Testing the Structure of a Gaussian Graphical Model With Reduced Transmissions in a Distributed Setting

Testing a covariance matrix following a Gaussian graphical model (GGM) is considered in this paper based on observations made at a set of distributed sensors grouped into clusters. Ordered transmissions are proposed to achieve the same Bayes risk as the optimum centralized energy unconstrained approach but with fewer transmissions and a completely distributed approach. In this approach, we represent the Bayes optimum test statistic as a sum of local test statistics which can be calculated by only utilizing the observations available at one cluster. We select one sensor to be the cluster head (CH) to collect and summarize the observed data in each cluster and intercluster communications are assumed to be inexpensive. The CHs with more informative observations transmit their data to the fusion center (FC) first. By halting before all transmissions have taken place, transmissions can be saved without performance loss. It is shown that this ordering approach can guarantee a lower bound on the average number of transmissions saved for any given GGM and the lower bound can approach approximately half the number of clusters when the minimum eigenvalue of the covariance matrix under the alternative hypothesis in each cluster becomes sufficiently large.

A General Asymptotic Framework for Distribution-Free Graph-Based Two-Sample Tests

SummaryTesting equality of two multivariate distributions is a classical problem for which many non-parametric tests have been proposed over the years. Most of the popular two-sample tests, which are asymptotically distribution free, are based either on geometric graphs constructed by using interpoint distances between the observations (multivariate generalizations of the Wald–Wolfowitz runs test) or on multivariate data depth (generalizations of the Mann–Whitney rank test). The paper introduces a general notion of distribution-free graph-based two-sample tests and provides a unified framework for analysing and comparing their asymptotic properties. The asymptotic (Pitman) efficiency of a general graph-based test is derived, which includes tests based on geometric graphs, such as the Friedman–Rafsky test, the test based on the K-nearest-neighbour graph, the cross-match test and the generalized edge count test, as well as tests based on multivariate depth functions (the Liu–Singh rank sum statistic). The results show how the combinatorial properties of the underlying graph affect the performance of the associated two-sample test and can be used to validate and decide which tests to use in practice. Applications of the results are illustrated both on synthetic and on real data sets.

Thermomechanics of Deformations in an Anharmonic Solid Body

The macroscopic laws determining the temperature and deformations of an anharmonic solid body in a given external temperature force field have been stated in the form of the first thermodynamics law supplemented by equations of state of the body. The internal and free energies necessary for it are found from the statistical sum in which some of degrees of freedom determining the body shape are discharged from statistical averaging. These functions of state have been calculated up to the first order of the perturbation theory in the anharmonicity for the microscopic dynamic model of the body with the interatomic interaction potential energy given as a series in powers of atom coordinates. The classical region of high temperatures is considered.

Allowing for Intermolecular Vibrations in the Thermodynamic Functions of a Liquid Inert Gas

Ways of calculating interatomic vibrations in a fluid are considered using the example of an inert gas. Discrete continuum theory in the lattice gas model is used. An approach is proposed for deriving expressions for the statistical sums of the vibrational motion of atoms in a fluid that are always in the bound state. Equations of the bulk equilibrium molecular distribution for concentrations and pair distribution functions in the quasi-chemical approximation are obtained, allowing for effects of direct correlations. Two approaches are used to describe local relative displacements of bound molecules with different amplitudes, provided that the local momentum equilibrium is preliminarily established and compared to the local mass equilibrium. The resulting equations are compared to approaches proposed earlier.

Thermodynamic properties of a multiferroic antiferromagnetic spin system: The influence of Dzyaloshinskii Moriya interaction

In this paper, we address the influence of Dzyaloshinskii Moriya (DM) interaction on the thermodynamic properties of a multiferroic antiferromagnetic spin system using the spin wave theory (SWT) as a diagonalization method which, associated to the statistical physics, helps to evaluate the statistical sum. The basic factors in thermodynamics, such as the Boltzmann entropy and the specific heat capacity at thermal equilibrium, are obtained. Analyzing the numerical results, it follows that the DM interaction is the best candidate that can help to maintain in long time the system in its coherent state. Furthermore, we find that the influence of the DM interaction is more accentuated at low-temperature and that it enables the system to release heat energy. On the other hand, the fact that the specific heat capacity tends to the constant clearly shows that the system obeys the Dulong–Petit law. In addition, our results also show that for high magnetic field (greater than B[Formula: see text]=[Formula: see text]30 Tesla), the magnetic field dependence of both entropy and specific heat shows a peak-like behavior and that the DM interaction raises the corresponding critical magnetic field but lowers the peak amplitude. It follows that the spin-flop transition occurs in the system and that the strong magnetic field withdraw the antiferromagnetic phase. Overall, we find that the DM interaction is a promising candidate for the control of our system.

Noncoherent Detector Threshold Determination in Correlated Pareto Distributed Clutter

Setting thresholds for sliding window noncoherent detection processes is extremely difficult in the presence of a correlated clutter range profile. This is certainly the case when attempting to apply a trimmed sum of order statistics as the measurement of the clutter level. In order to facilitate this, a new expression for the threshold of such a detector is derived, which is a function of probabilities of two associated random variables. This then permits a Monte Carlo scheme to be applied to determine thresholds. In the case of real data, if a clutter map is assumed to be an accurate representation of the radar operating environment, then one can avoid estimating correlations in the process of determining thresholds. Examples of this will be provided to demonstrate the utility of the proposed scheme.

CUMULATED SUM OF SQUARES STATISTICS FOR NONLINEAR AND NONSTATIONARY REGRESSIONS

We show that the cumulated sum of squares statistic has a standard Brownian bridge–type asymptotic distribution in nonlinear regression models with (possibly) nonstationary regressors. This contrasts with cumulated sum statistics which have been previously studied and whose asymptotic distribution has been shown to depend on the functional form and the stochastic properties, such as persistence and stationarity, of the regressors. A recursive version of the test is also considered. A local power analysis is provided, and through simulations, we show that the test has good size and power properties across a variety of situations.

Measurements of water molecule isotopomer concentrations in a discharge of inert gas with addition of H2O and D2 vapours by the method of external-cavity diode laser spectroscopy

Discharge plasma in inert gases with added water vapours and deuterium is studied by using the optimised diode laser spectroscopy with an external cavity. Formation of a spectrally dense grid of cavity transverse mode eigenfrequencies allows one to determine the parameters of Doppler broadening for absorption lines, a kinetic temperature of molecules, and their statistical sums. By using these parameters and the value of an absorption coefficient, correlations of concentrations are found for H2O and HDO isotopomers at various ratios of plasma-forming gas components. An important contribution of initial gas molecules adsorbed on walls into the formation of plasma composition is stressed. It is found that the ratio of H2O and HDO concentrations is the same at various ratios of H2O and D2 concentrations in plasma-forming gases and is constant during the discharge burning process.

Термомеханика деформаций в ангармоническом твердом теле

AbstractThe macroscopic laws determining the temperature and deformations of an anharmonic solid body in a given external temperature force field have been stated in the form of the first thermodynamics law supplemented by equations of state of the body. The internal and free energies necessary for it are found from the statistical sum in which some of degrees of freedom determining the body shape are discharged from statistical averaging. These functions of state have been calculated up to the first order of the perturbation theory in the anharmonicity for the microscopic dynamic model of the body with the interatomic interaction potential energy given as a series in powers of atom coordinates. The classical region of high temperatures is considered.

Расчет статистической суммы и аппроксимация многочастичных функций распределения для магнетиков в модели Гейзенберга

Расчет статистической суммы и аппроксимация многочастичных функций распределения для магнетиков в модели Гейзенберга

Designing a safety management system for higher education centers.

INTRODUCTION:The system of safety management in higher education centers can prevent the complications caused by harmful issues to students and bring their potential talents closer to perfection. Therefore, this study aimed to design a safety management system (SMS) in higher education centers of Iran in 2016.MATERIALS AND METHODS:This study was a descriptive study of qualitative type. This study was conducted in three independent phases, including (1) evaluating theoretical concepts, (2) developing an initial system by determining the points of sharing and differentiation of the evaluated systems, and (3) validating the SMS using the Delphi technique. Consensus on opinions and identifying similarities and differences of reviewed studies have been used for qualitative data analysis, and the descriptive statistics (sum of scores and mean) by means of SPSS version 21 has been used for quantitative data analysis.RESULTS:In the first stage, 108 indexes were identified by reviewing the studies and evaluating the SMS in the world's educational and noneducational organizations for the 12 main categories of SMS. In the Delphi phase, 83 components were identified as a key index of the SMS in higher education centers of Iran. Furthermore, the average mean of participants’ views on the dimensions of the SMS for higher education centers has been 4.32, with the highest average mean of 4.59 related to the dimension of the facility and the firefighting department and the lowest mean of 4.10 for the student dimension.CONCLUSION:The indexes presented in this study will provide a fairly complete tool for designing SMS in higher education centers, which by applying it will provide a good opportunity to improve the performance of these systems over time.

Effect of a 5% Neem Solution on the Presence of Ticks in Cattle in Two Locations in Jalisco, Mexico

The objective of this work was to evaluate the effect of a 5% neem extract on the presence of ticks. Forty creole cattle were used, divided into four groups: 2 experimental and 2 controls, in the towns of Cocula and El Chante, Jalisco. The tick count was by direct palpation, and with an acarometer (25 cm2). Eight body regions were evaluated: neck, thorax, flank and thigh, left and right. At zero time, after the initial tick count, it was applied by spraying and once, the 5% neem extract, at a dose of 10 ml/L. Ticks were quantified at 7, 14, 21, 28, 35 and 42 days post-treatment. The data were subjected to descriptive statistical analysis and Sum of Kruskal-Wallis Ranges (α = 0.05) (SigmaStat 3.1). In Cocula, at zero time, the average of ticks was in neck 32, in thorax 64, in flanks 96 and in thighs 129 (average per animal 323 ticks). In subsequent sampling, the average decreased in the experimental group to 1 or 4 parasites. In “El Chante”, an average of 60 ticks was found at the time of the neck, in the thorax 121, in flanks 181 and in thighs 242 (average per animal 600 ticks). In subsequent sampling, the average decreased in the experimental group, with averages of 1 to 4. Statistical differences (p ≤ 0.001) were found between the groups. It can be concluded that 5% neem extract has a repellent and tick effect in cattle for at least 42 days.

Phase transitions in the extended particle systems, Hagedorn temperature and fractal dimension of space, as a confinement phase transition order parameter

We consider a formal definition of New physics, fundamental constants of physics, formulation of the quantum statistical theory including extra integrals, monopole mechanism for hadronization in QCD, finite approximation of the zeta-function and fermion factorization of the bosonic statistical sum.