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.

Goodness-of-fit tests for multi-dimensional copulas:Expanding application to historical drought data

The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for multi-dimensional copulas. A goodness-of-fit test based on Rosenblatt＇s transformation was mathematically expanded from two dimensions to three dimensions and procedures of a bootstrap version of the test were provided. Through stochastic copula simulation, an empirical application of historical drought data at the Lintong Gauge Station shows that the goodness-of-fit tests perform well, revealing that both trivariate Gaussian and Student t copulas are acceptable for modeling the dependence structures of the observed drought duration, severity, and peak. The goodness-of-fit tests for multi-dimensional copulas can provide further support and help a lot in the potential applications of a wider range of copulas to describe the associations of correlated hydrological variables. However, for the application of copulas with the number of dimensions larger than three, more complicated computational efforts as well as exploration and parameterization of corresponding copulas are required.

Deriving CDF of Kolmogorov-Smirnov Test Statistic

In this review article, we revisit derivation of the cumulative density function (CDF) of the test statistic of the one-sample Kolmogorov-Smirnov test. Even though several such proofs already exist, they often leave out essential details necessary for proper understanding of the individual steps. Our goal is filling in these gaps, to make our presentation accessible to advanced undergraduates. We also propose a simple formula capable of approximating the exact distribution to a sufficient accuracy for any practical sample size.

Asymptomatic Distribution of Goodness-of-Fit Tests in Logistic Regression Model

The logistic regression model has been become commonly used to study the association between a binary response variable;it is widespread application rests on its easy application and interpretation. The subject of assessment of goodness-of-fit in logistic regression model has attracted the attention of many scientists and researchers. Goodness-of-fit tests are methods to determine the suitability of the fitted model. Many of methods proposed and discussed for assessing goodness-of fit in logistic regression model, however, the asymptotic distribution of goodness-of-fit statistics are less examine, it is need more investigated. This work, will focus on assessing the behavior of asymptotic distribution of goodness-of-fit tests, also make comparison between global goodness-of-fit tests, and evaluate it by simulation.

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.

A Test of U-type for Goodness-of-fit in Regression Models Through Martingale Difference Divergence

Based on the martingale difference divergence,a recently proposed metric for quantifying conditional mean dependence,we introduce a consistent test of U-type for the goodness-of-fit of linear models under conditional mean restriction.Methodologically,our test allows heteroscedastic regression models without imposing any condition on the distribution of the error,utilizes effectively important information contained in the distance of the vector of covariates,has a simple form,is easy to implement,and is free of the subjective choice of parameters.Theoretically,our mathematical analysis is of own interest since it does not take advantage of the empirical process theory and provides some insights on the asymptotic behavior of U-statistic in the framework of model diagnostics.The asymptotic null distribution of the proposed test statistic is derived and its asymptotic power behavior against fixed alternatives and local alternatives converging to the null at the parametric rate is also presented.In particular,we show that its asymptotic null distribution is very different from that obtained for the true error and their differences are interestingly related to the form expression for the estimated parameter vector embodied in regression function and a martingale difference divergence matrix.Since the asymptotic null distribution of the test statistic depends on data generating process,we propose a wild bootstrap scheme to approximate its null distribution.The consistency of the bootstrap scheme is justified.Numerical studies are undertaken to show the good performance of the new test.

我国上市公司资本结构与行业关系的实证分析

本文利用One-SampleKolmogorov-SmirnovTest对我国上市公司资本结构与行业关系进行了实证分析,结果发现:我国上市公司资产负债率近似服从正态分布,不同行业各总体的方差无显著性差异,不同行业上市公司资本结构存在显著性差异且在1999-2002年中都保持了相当的稳定性,波动较小。

Spatial and temporal variability of daily precipitation in Haihe River basin, 1958-2007

The seasonal variability and spatial distribution of precipitation are the main cause of flood and drought events. The study of spatial distribution and temporal trend of precipitation in river basins has been paid more and more attention. However, in China, the precipitation data are measured by weather stations （WS） of China Meteorological Administration and hydrological rain gauges （RG） of national and local hydrology bureau. The WS data usually have long record with fewer stations, while the RG data usually have short record with more stations. The consistency and correlation of these two data sets have not been well understood. In this paper, the precipitation data from 30 weather stations for 1958-2007 and 248 rain gauges for 1995-2004 in the Haihe River basin are examined and compared using linear regression, 5-year moving average, Mann-Kendall trend analysis, Kolmogorov-Smirnov test, Z test and F test methods. The results show that the annual precipitation from both WS and RG records are normally distributed with minor difference in the mean value and variance. It is statistically feasible to extend the precipitation of RG by WS data sets. Using the extended precipitation data, the detailed spatial distribution of the annual and seasonal precipitation amounts as well as their temporal trends are calculated and mapped. The various distribution maps produced in the study show that for the whole basin the precipitation of 1958-2007 has been decreasing except for spring season. The decline trend is significant in summer, and this trend is stronger after the 1980s. The annual and seasonal precipitation amounts and changing trends are different in different regions and seasons. The precipitation is decreasing from south to north, from coastal zone to inland area.

Gearbox Deterioration Detection under Steady State,Variable Load, and Variable Speed Conditions

Multiple dominant gear meshing frequencies are present in the vibration signals collected from gearboxes and the conventional spiky features that represent initial gear fault conditions are usually difficult to detect. In order to solve this problem, we propose a new gearbox deterioration detection technique based on autoregressive modeling and hypothesis testing in this paper. A stationary autoregressive model was built by using a normal vibration signal from each shaft. The established autoregressive model was then applied to process fault signals from each shaft of a two-stage gearbox. What this paper investigated is a combined technique which unites a time-varying autoregressive model and a two sample Kolmogorov-Smimov goodness-of-fit test, to detect the deterioration of gearing system with simultaneously variable shaft speed and variable load. The time-varying autoregressive model residuals representing both healthy and faulty gear conditions were compared with the original healthy time-synchronous average signals. Compared with the traditional kurtosis statistic, this technique for gearbox deterioration detection has shown significant advantages in highlighting the presence of incipient gear fault in all different speed shafts involved in the meshing motion under variable conditions.

A Review of the Logistic Regression Model with Emphasis on Medical Research

This study explored and reviewed the logistic regression (LR) model, a multivariable method for modeling the relationship between multiple independent variables and a categorical dependent variable, with emphasis on medical research. Thirty seven research articles published between 2000 and 2018 which employed logistic regression as the main statistical tool as well as six text books on logistic regression were reviewed. Logistic regression concepts such as odds, odds ratio, logit transformation, logistic curve, assumption, selecting dependent and independent variables, model fitting, reporting and interpreting were presented. Upon perusing the literature, considerable deficiencies were found in both the use and reporting of LR. For many studies, the ratio of the number of outcome events to predictor variables (events per variable) was sufficiently small to call into question the accuracy of the regression model. Also, most studies did not report on validation analysis, regression diagnostics or goodness-of-fit measures;measures which authenticate the robustness of the LR model. Here, we demonstrate a good example of the application of the LR model using data obtained on a cohort of pregnant women and the factors that influence their decision to opt for caesarean delivery or vaginal birth. It is recommended that researchers should be more rigorous and pay greater attention to guidelines concerning the use and reporting of LR models.

INTEGRAL-TYPE TESTS FOR GOODNESS-OF-FIT

A family of integral-type goodness-of-fit tests is investigated. This family includes some existing tests, such as the Cramer-von Mises test and Anderson-Darling test, etc. The asymptotic distributions of the tests in the family under the null and local alternative hypotheses are established. The almost sure convergence under a fixed underlying distribution is obtained. Furthermore, simulations are conducted to compare the powers of the tests in the family. Simulation results show that for different alternatives, the more powerful tests are different, and the parameter ）, has great influence on the tests in small sample cases.

GOODNESS-OF-FIT TEST WITH GENETIC BACKGROUND

The chi-square test is a well-known goodness-of-fit test. It is available for arbitrary alternative hypothesis, particularly for a very general alternative. However, when the alternative is a “one-sided” hypothesis, which usually appears in genetic linkage analysis, the chi-square test does not use the information offered by the one-sided hypothesis.Therefore, it is possible that an appropriate one-sided test, which uses the information,will be better than the chi-square test. This paper gives such an efficient one-sided test.Monte Carlo simulation results show that it is more powerful than the chi-square test, and its power has been increased by 30 percent as compared with that of the chi-square test in most situations.

Parameter Estimations for Some Modifications of the Weibull Distribution

Proposed by the Swedish engineer and mathematician Ernst Hjalmar Waloddi Weibull (1887-1979), the Weibull distribution is a probability distribution that is widely used to model lifetime data. Because of its flexibility, some modifications of the Weibull distribution have been made from several researches in order to best adjust the non-monotonic shapes. This paper gives a study on the performance of two specific modifications of the Weibull distribution which are the exponentiated Weibull distribution and the additive Weibull distribution.

A Statistical Analysis of the Modulus of Elasticity and Compressive Strength of Concrete C45/55 for Pre-stressed Precast Beams

Random behavior of concrete C45/55 XF2 used for prefabricated pre-stressed bridge beams is described on the basis of evaluating a vast set of measurements. A detailed statistical analysis is carried out on 133 test results of cylinders 150 ~ 300 mm in size. The tests have been running in laboratories of the Klokner Institute. A single worker took all specimens throughout the period, and the subsequent measurements of the static modulus of elasticity and the compressive strength of the concrete were performed. The measurements were made at the age of 28 days after specimens casting, and only one testing machine with the same capping method was used. Suitable theoretical models of division are determined on the basis of tests in good congruence, with the use of Z2 and the Bernstein criterion. A set of concrete compressive strength （carried out on 133 test results of cylinders 150 ~ 300 mm after test of static modulus of elasticity） shows relatively high skewness in this specific case. This cause that limited beta distribution is better than generally recommended theoretical distribution for strength the normal or lognormal. The modulus of elasticity is not significantly affected due to skewness because the design value is based on mean value.

Analysis of inhibition of concrete steel-rebar corrosion by Na_2Cr_2O_7 concentrations:Implications for conflicting reports on inhibitor effectiveness

Corrosion test data were measured using non-destructive electrochemical techniques and analysed for studying inhibition effectiveness by different concentrations of NazCr207 on the corrosion of concrete steel-rehar in NaC1 and in H2SO4 media. For these, specifications of ASTM G16-95 R04 were combined with the normal and the Gumbel probability density functions as model analytical methods for addressing issues of conflicting reports of inhibitor effectiveness that had generated concerns. Results show that reinforced concrete samples admixed with concentrations having 4 g （0.012 7 tool）, 8 g （0.025 4 mol） and 6 g （0.019 l tool） NaaCr207 exhibited, in that order, high inhibition effectiveness, with respective efficiency, r/, of （90.46±1.30）%, （88.41＋2.24）% and （84.87±4.74）%, in the NaC1 medium. These exhibit good agreements within replicates and statistical methods for the samples. Also, optimal inhibition effectiveness model in the H2SO4 medium was exhibited by 8 g （0.025 4 mol） Na2Cr207 concentration having r/=（78.44±1.10）%. These bear implications for addressing conflicting test data in the study of effective inhibitors for mitigating steel-rebar corrosion in aggressive environments.

Goodness-of-fit tests for vector autoregressive models in time series

The paper proposes and studies some diagnostic tools for checking the goodness-of-fit of general parametric vector autoregressive models in time series. The resulted tests are asymptotically chi-squared under the null hypothesis and can detect the alternatives converging to the null at a parametric rate. The tests involve weight functions,which provides us with the flexibility to choose scores for enhancing power performance,especially under directional alternatives. When the alternatives are not directional,we construct asymptotically distribution-free maximin tests for a large class of alternatives. A possibility to construct score-based omnibus tests is discussed when the alternative is saturated. The power performance is also investigated. In addition,when the sample size is small,a nonparametric Monte Carlo test approach for dependent data is proposed to improve the performance of the tests. The algorithm is easy to implement. Simulation studies and real applications are carried out for illustration.

GOODNESS-OF-FIT TEST ON TWO SAMPLES

In this paper, a new statistics for testing two samples coming from the same population is derived from a simple linear model with an artificial parameter. Its limit distribution is a chi-squared distribution with 2 degrees of freedom under null hypothesis and the limit distribution is a noncentral chi-squared distribution with 2 degrees of freedom under certain sequence of alternative hypothesis. Finally, we make power comparison with other tests on two samples, especially, with Smirnov statistics.

THE GOODNESS-OF-FIT TEST BY USING MULTIVARIATE RANDOM WEIGHTING METHOD

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A viable method for goodness-of-fit test in maximum likelihood fit

A test statistic is proposed to perform the goodness-of-fit test in the unbinned maximum likelihood fit. Without using a detailed expression of the efficiency function, the test statistic is found to be strongly correlated with the maximum likelihood function if the efficiency function varies smoothly. We point out that the correlation coefficient can be estimated by the Monte Carlo technique. With the established method, two examples are given to illustrate the performance of the test statistic.

Discrete Circular Distributions with Applications to Shared Orthologs of Paired Circular Genomes

For structural comparisons of paired prokaryotic genomes,an important topic in synthetic and evolutionary biology,the locations of shared orthologous genes(henceforth orthologs)are observed as binned data.This and other data,e.g.,wind directions recorded at monitoring sites and intensive care unit arrival times on the 24-hour clock,are counted in binned circular arcs,thus modeling them by discrete circular distributions(DCDs)is required.We propose a novel method to construct a DCD from a base continuous circular distribution(CCD).The probability mass function is defined to take the normalized values of the probability density function at some pre-fixed equidistant points on the circle.Five families of constructed DCDs which have normalizing constants in closed form are presented.Simulation studies show that DCDs outperform the corresponding CCDs in modeling grouped(discrete)circular data,and minimum chi-square estimation outperforms maximum likelihood estimation for parameters.We apply the constructed DCDs,invariant wrapped Poisson and wrapped discrete skew Laplace to compare the structures of paired bacterial genomes.Specifically,discrete four-parameter wrapped Cauchy(nonnegative trigonometric sums)distribution models multi-modal shared orthologs in Clostridium(Sulfolobus)better than the others considered,in terms of AIC and Freedman’s goodness-of-fit test.The result that different DCDs fit the shared orthologs is consistent with the fact they belong to two kingdoms.Nevertheless,these prokaryotes have a common favored site around 70°on the unit circle;this finding is important for building synthetic prokaryotic genomes in synthetic biology.These DCDs can also be applied to other binned circular data.