Comparison of Type I Error Rates of Siegel-Tukey and Savage Tests among Non-Parametric Tests

This study aimed to examine the performance of the Siegel-Tukey and Savage tests on data sets with heterogeneous variances. The analysis, considering Normal, Platykurtic, and Skewed distributions and a standard deviation ratio of 1, was conducted for both small and large sample sizes. For small sample sizes, two main categories were established: equal and different sample sizes. Analyses were performed using Monte Carlo simulations with 20,000 repetitions for each scenario, and the simulations were evaluated using SAS software. For small sample sizes, the I. type error rate of the Siegel-Tukey test generally ranged from 0.045 to 0.055, while the I. type error rate of the Savage test was observed to range from 0.016 to 0.041. Similar trends were observed for Platykurtic and Skewed distributions. In scenarios with different sample sizes, the Savage test generally exhibited lower I. type error rates. For large sample sizes, two main categories were established: equal and different sample sizes. For large sample sizes, the I. type error rate of the Siegel-Tukey test ranged from 0.047 to 0.052, while the I. type error rate of the Savage test ranged from 0.043 to 0.051. In cases of equal sample sizes, both tests generally had lower error rates, with the Savage test providing more consistent results for large sample sizes. In conclusion, it was determined that the Savage test provides lower I. type error rates for small sample sizes and that both tests have similar error rates for large sample sizes. These findings suggest that the Savage test could be a more reliable option when analyzing variance differences.

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.

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.

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.

Evaluation of Rainfall Tendency for the Twentieth Century over Indira Sagar Region in Central India

The study investigates long-term changes in annual and seasonal rainfall patterns in the Indira Sagar Region of Madhya Pradesh, India, from 1901 to 2010. Agriculture sustainability, food supply, natural resource development, and hydropower system reliability in the region rely heavily on monsoon rainfall. Monthly rainfall data from three stations (East Nimar, Barwani, and West Nimar) were analyzed. Initially, the pre-whitening method was applied to eliminate serial correlation effects from the rainfall data series. Subsequently, statistical trends in annual and seasonal rainfall were assessed using both parametric (student-t test) and non-parametric tests [Mann-Kendall, Sen’s slope estimator, and Cumulative Sum (CUSUM)]. The magnitude of the rainfall trend was determined using Theil-Sen’s slope estimator. Spatial analysis of the Mann-Kendall test on an annual basis revealed a statistically insignificant decreasing trend for Barwani and East Nimar and an increasing trend for West Nimar. On a seasonal basis, the monsoon season contributes a significant percentage (88.33%) to the total annual rainfall. The CUSUM test results indicated a shift change detection in annual rainfall data for Barwani in 1997, while shifts were observed in West and East Nimar stations in 1929. These findings offer valuable insights into regional rainfall behavior, aiding in the planning and management of water resources and ecological systems.

Parameters Estimation of a Bivariate Generalized Poisson Distribution with Applications to Metabolic Syndrome Data

Background: Bivariate count data are commonly encountered in medicine, biology, engineering, epidemiology and many other applications. The Poisson distribution has been the model of choice to analyze such data. In most cases mutual independence among the variables is assumed, however this fails to take into accounts the correlation between the outcomes of interests. A special bivariate form of the multivariate Lagrange family of distribution, names Generalized Bivariate Poisson Distribution, is considered in this paper. Objectives: We estimate the model parameters using the method of maximum likelihood and show that the model fits the count variables representing components of metabolic syndrome in spousal pairs. We use the likelihood local score to test the significance of the correlation between the counts. We also construct confidence interval on the ratio of the two correlated Poisson means. Methods: Based on a random sample of pairs of count data, we show that the score test of independence is locally most powerful. We also provide a formula for sample size estimation for given level of significance and given power. The confidence intervals on the ratio of correlated Poisson means are constructed using the delta method, the Fieller’s theorem, and the nonparametric bootstrap. We illustrate the methodologies on metabolic syndrome data collected from 4000 spousal pairs. Results: The bivariate Poisson model fitted the metabolic syndrome data quite satisfactorily. Moreover, the three methods of confidence interval estimation were almost identical, meaning that they have the same interval width.

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.

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.

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.

RAYLEIGH-DISTRIBUTION BASED MINIMUM ERROR THRESHOLDING FOR SAR IMAGES

This paper presents a minimum error thresholding (MET) algorithm under the hypothesis that the gray level histogram of SAR image fits to a mixture model of shifted Rayleigh distribution. This algorithm is applied to real SAR images and compared with traditional Otsu algorithm and other MET algorithms based on various models of histogram. The hypothesis of using Rayleigh distribution model is confirmed by Kolmogorov-Smirnov testing and the comparison results obtained show that the proposed new algorithm has good performance in thresholding SAR images.

An analysis of tourist’s perceptions toward tourism development:Study of cold desert destination,India

Tourism impacts on society are complex and mixed.However,they are vital to diverse societies,clusters,and individuals dependent upon their morals,attitudes,and resources existing for tourism development.Increasing tourism also brings many problems.Hence,tourist experience is fundamental for destination image and devel-opment.This research examines tourist perceptions and attitudes toward tourism impacts in Chitkul,Kalpa,and Nako in Kinnaur.Random sampling has been used to measure tourist responses on a range of indicators related to tourism development.Likert scale responses were analyzed using factor analysis,ANOVA,Mann-Whitney U-test,Kolmogorov test,and descriptive statistics.The results confirmed that tourists do not perceive any type of pollu-tion or societal barriers.They observed that natural magnetism and the socio-cultural milieu of the destination is what attracts tourists.However,tourists are not satisfied with‘networking services’,‘organization efforts’,‘sup-plementary conveniences’,and‘carriage concerns’at selected destinations in Kinnaur.Moreover,Chitkul emerged as the top tourist destination in Kinnaur.Since the destination would emerge as a hub of tourist activities shortly considering the congestion and exploitation of nearby tourist destinations at Kulu-Manali-Rohtang in Beas Valley.Hence,the assessment of tourist perceptions can be used as an indicator of tourism destination competitiveness and can assist in developing appropriate tourism policies and infrastructure upgrades.

Observation of Suicide Prevention in Hospitalized Patients with Depression Based on Taihe Emotion-distress Index

[Objectives]To reduce suicidal ideation and suicidal behavior through initial screening of patients with suicidal tendencies and implementing suicide prevention interventions during their hospitalization.[Methods]The Taihe Emotion-distress Index(THEI)was used to conduct pre-admission and post-discharge tests to explore the effects of suicide prevention measures during hospitalization on the alleviation of the disease and the reduction of suicidal behaviors.The study selected patients who were diagnosed with depression in the psychological outpatient department of Taihe Hospital from April 2019 to September 2019 and had to be hospitalized,including patients with moderate depressive episodes,severe depressive episodes with or without psychotic symptoms,and patients with suicidal thoughts and self-harming behaviors.[Results]The pre-admission and post-discharge test data of hospitalized patients were analyzed,and the non-parametric paired sample T test was carried out,and the result was P<0.05,showing that there are significant differences between the pre-admission and post-discharge test data.[Conclusions]The measures of suicide prevention intervention are effective to a certain extent.

Modeling the Browsing Behavior of World Wide Web Users

The World Wide Web is essential to general public nowadays. From a data analysis viewpoint, it provides rich opportunities to gather observational data on a large-scale. This paper focuses on modeling the behavior of visitors to an academic website. Although the conventional probability models, which were used by other literature for fitting in a commercial web site, capture the power law behavior in our data, they fail to capture other important features like the long tail. We propose a new model based on the identities of the users. Qualitative and quantitative tests, which are used for comparing the model fitting to our data, show that the new model outperforms other two conventional probability models.

Trend Analysis of Precipitation in Some Selected Stations in Anambra State

State is in the South East geopolitical zone of Nigeria. The major occupation of the people in this region is trading and farming, which depends on rainfall and other climatic factors. This paper presents statistical and trend analyses of the rainfall in some selected stations in Anambra State, which includes Ifite-Ogwari, Awka, Onitsha and Ihiala. Rainfall data for a period of 1971-2010 were obtained from Climate Research Unit (CRU). The existence of trend and statistical analyses was conducted on monthly total rainfalls using non-parametric techniques. The study revealed that overall averages of yearly and monthly total rainfall were 5798.78 mm and 1739.62 mm in Ifite-Ogwari, 6051.8 mm and 1815 mm in Awka, 6288.87 mm and 1886.88 mm in Onitsha, and 6637.19 mm and 1997.1 mm in Ihiala. Yearly total rainfall has Mann-Whitney of 26 and 41 between 1971 and 1990, 1991 and 2010 respectively in Ifite-Ogwari, 32 and 42 between 1971 and 1990, 1991 and 2010 respectively in Awka, 42 and 39 between 1971 and 1990, 1991 and 2010 respectively in Onitsha, and 33 and 45 between 1971 and 1990, 1991 and 2010 respectively in Ihiala. These parameters show that there are significant trends in the rainfall in term of yearly total for the period. Sen’s estimator revealed that there were significant downward trends for yearly total (-0.775 mm/year) and (-0.094 mm/year) within the period of 1971-1990 and 1991-2010 in Ifite-Ogwari. There was an upward trend of yearly total (1.841 mm/year) between 1971 and1990, whereas there was a downward trend of yearly total (-0.211) between 1991 and 2010 in Awka. It was concluded that there was a significant downward trend in the yearly total and mean rainfalls in Ifite-Ogwari, Awka, Onitsha and Ihiala in the last four decades (40 years), which could be attributed to climate change.

On the Diversity of Long-Term Temperature Responses to Varying Levels of Solar Activity at Ten European Observatories

We analyze ten of the longest (127 to 230 year-long) time series of European daily temperatures available from five different K&#246;ppen-Geiger climate classes. We split these according to the level of solar cycle activity (H for “higher than median” and L for “lower than median”). This reveals coherent patterns in the temperature differences: when TH-TL?are stacked according to their calendar date, the daily averages from January 1 to December 31st disclose characteristic features in addition to the dominant annual seasonal wave, namely variations up to 2&#176;C lasting for about 1.5 to 3 months. The five observatories at intermediate latitudes in a band from Oxford in the West to Prague in the East (same climate class) have very similar signatures. These similarities are most unlikely to be due to pure chance (confirmed by confidence levels in excess of 99% with the Kolmogorov-Smirnov and Kuiper nonparametric tests). The TH-TL patterns carry a regional signature, modulated by a more local response function. On the other hand, northern European observatories (St Petersburg and Arkhangelsk), those south of the Alps (Milan and Bologna), and the easternmost one in Astrakhan, corresponding to different climate classes, have different signatures. Similarly, preliminary study of long air pressure recordings confirms what emerges from the analysis of temperatures. These new observations lead us to conclude that the climate in different regions presents different responses to variations in solar activity. Moreover, the distributions of the lower, middle, and higher quartiles of the temperature and pressure indices in solar cycles with high versus low activity are significantly different, providing further robust statistical confirmation to this conclusion (confidence level higher to much higher than 99% using the Kuiper test).

Power Law for the Rates of Different Numbers of Chronic Diseases among Elderly Chinese People

Chronic disease is an important factor that affect the health of elderly people. We analyzed the 2006 and 2010 data from the Chinese Urban and Rural Elderly Population Surveys, which are nationally representative surveys of elderly people aged 60 years and above. We found that there existed a typical power-law distribution for the rates of different numbers of chronic diseases among elderly Chinese people. A Kolmogorov-Smirnov test indicated that the result was robust, and the power exponents were approximately ?2.5. In addition, a paired t-test was conducted, which demonstrated that the rates of different numbers of chronic diseases did not have significant urban-rural differences, time differences or gender differences.

Comparative Analysis of Climatic Change Trend and Change-Point Analysis for Long-Term Daily Rainfall Annual Maximum Time Series Data in Four Gauging Stations in Niger Delta

The aim of this study is to establish the prevailing conditions of changing climatic trends and change point dates in four selected meteorological stations of Uyo, Benin, Port Harcourt, and Warri in the Niger Delta region of Nigeria. Using daily or 24-hourly annual maximum series (AMS) data with the Indian Meteorological Department (IMD) and the modified Chowdury Indian Meteorological Department (MCIMD) models were adopted to downscale the time series data. Mann-Kendall (MK) trend and Sen’s Slope Estimator (SSE) test showed a statistically significant trend for Uyo and Benin, while Port Harcourt and Warri showed mild trends. The Sen’s Slope magnitude and variation rate were 21.6, 10.8, 6.00 and 4.4 mm/decade, respectively. The trend change-point analysis showed the initial rainfall change-point dates as 2002, 2005, 1988, and 2000 for Uyo, Benin, Port Harcourt, and Warri, respectively. These prove positive changing climatic conditions for rainfall in the study area. Erosion and flood control facilities analysis and design in the Niger Delta will require the application of Non-stationary IDF modelling.

A NEW TEST FOR NORMALITY IN LINEAR AUTOREGRESSIVE MODELS

A nonparametric test for normality of linear autoregressive time series is proposed in this paper.The test is based on the best one-step forecast in mean square with time reverse.Some asymptotic theory is developed for the test,and it is shown that the test is easy to use and has good powers.The empirical percentage points to conduct the test in practice are provided and three examples using real data are included.

PERCENTAGE POINTS AND POWER OF AK-S TYPE TEST FOR LINEARITY IN AUTOREGRESSIVE TIME SERIES

The empirical upper percentage points of the null distribution of a Kolmogorov-Smirnov type test for checking linearity in autoregressive models are tabulated in this paper, and the good power property of the test is demonstrated.