Kolmogorov-Smirnov statistic

Rassokhin, D. N., Agrafiotis, D. K. (2000) Kolmogorov-Smirnov statistic and its applications in library design. J Mol Graph Model 18(4-5), 370-384. 

Kolmogorov-Smirnov Statistics are used in this example for the... 

From percentage points of Kolmogorov-Smirnov statistics table (Chandra et al., 1981), we conclude that there is no evidence to reject the GEVD model. 

Roughly, about 250 data points are required to fit the generalized hyperbolic distributions. However, about 100 data points can offer reasonable results. Although maximum-likelihood estimation method can be used to estimate the parameters, it is very difficult to solve such a complicated nonlinear equation system with five equations and five unknown parameters. Therefore, numerical algorithms are suggested such as modified Powell method (Wang, 2005). Kolmogorov-Smirnov statistics can also be used here for the fitness test. 

Kolmogorov Smirnov statistic is equal Xj = 1,18, whereas the limit value, obtained from the tables is equal = 1,36. This proves the compatibility of empirical data with simulation. 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

The Kolmogorov Smirnov test Closely related to the visual approach employed in Fig. 42, the KS test evaluates the equality of two distributions. Under the assumption that a saturation parameter has no impact on stability, its distribution within the stable subset is identical (in a statistical sense) to its initial distribution. Deviations between the two distributions, such as those shown in Fig. 42, thus indicate a dependency between the parameter and dynamic stability. 

A basic assumption underlying r-tests and ANOVA (which are parametric tests) is that cost data are normally distributed. Given that the distribution of these data often violates this assumption, a number of analysts have begun using nonparametric tests, such as the Wilcoxon rank-sum test (a test of median costs) and the Kolmogorov-Smirnov test (a test for differences in cost distributions), which make no assumptions about the underlying distribution of costs. The principal problem with these nonparametric approaches is that statistical conclusions about the mean need not translate into statistical conclusions about the median (e.g., the means could differ yet the medians could be identical), nor do conclusions about the median necessarily translate into conclusions about the mean. Similar difficulties arise when - to avoid the problems of nonnormal distribution - one analyzes cost data that have been transformed to be more normal in their distribution (e.g., the log transformation of the square root of costs). The sample mean remains the estimator of choice for the analysis of cost data in economic evaluation. If one is concerned about nonnormal distribution, one should use statistical procedures that do not depend on the assumption of normal distribution of costs (e.g., nonparametric tests of means). 

Variations in OC values in soil samples collected along the N-S and W-E transects at the Florida site are shown in Figure 4. Note that for both the transects, OC in soils from the 0-15 cm depth were more variable than in samples collected from the 15-30 cm depth. Variograms ( 14 ) calculated using these data indicated that for the 0-15 cm depth, OC values in soil samples collected within a separation distance (i.e., lag) of 15 m would be spatially correlated. On the other hand, OC values for the 15-30 cm depth are spatially independent. OC data for both depth increments could be fitted to a normal frequency distribution the normality was confirmed by the Kolmogorov- Smirnov D-statistic ( 23 ). The coefficient of variation (CV) in OC data for both depths was less than 20%. 

Table A.6 Kolmogorov-Smirnov test statistic d - a, n) to test for a normal distribution at different significance levels a.

Descriptive statistics—mean, median, trimmed means, standard deviation and standard error, variance, minimum, maximum, range, interquartile range, skewness, kurtosis Frequency statistics—outlier identification boxplots, stem-and-leaf plots, and histograms Frequency statistics—description percentiles, probability plots, robust estimates or M-estimators, Kolmogorov-Smirnov and Shapiro-Wilk normality tests Variance homogeneity—Levene s test for equality of variance... 

The value of this Kolmogorov-Smirnov test statistic, together with its P-value, can be obtained directly from Minitab in conjunction with a normal probability plot. 

K-S Test. The Kolmogorov-Smirnov goodness of fit test assesses the ability of a probability distribution calculated from WEiBULL STATISTICS (q.v.) to fit the experimental data. It and the A-D (Anderson-Darling) test (which is more sensitive to discrepancies at low and high probabilities of failure) are used as part of the CARES (q.v.) computer program for failure prediction. For a discussion of... 

Table 3 shows the results of Kolmogorov-Smirnov distances for different equities and states, and the difference in statistical proprieties (Skewness and Kurtosis) of simulated data from the estimated model with respect to those of the real dataset. Note that the Kolmogorov-Smirnov test always leads to the null hypothesis HO which means that the distribution of the simulated data is... 

There are many statistics that could be used to characterize the difference between two distributions. One could take the area between flie distributions for example, or the sum of the difference squared as characterizing the difference. The Kolmogorov-Smirnov (K-S) technique to be discussed here takes a simple measure which is the maximum value of the difference between the two distributions. This is identified as D in Figure 8.19 and is defined as ... 

Basic Statistical Properties and Functions. 3 Distributions and More Distributions. 4 Analysis of Mean and Variance. 5 Comparing Distribution Functions - The Kolmogorov-Smirnov Test. 6 Monte Carlo Simulations and Confidence Limits. 7 Non-Gaussian Distributions and Reliability Modeling. 8 Summary... 

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