A goodness of fit test for the Poisson distribution based on the empirical generating function
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Goodness of fit
Empirical distribution function
Univariate distribution
The relative power of goodness-of-fit test statistics has long been debated in the literature. Chi-Square type test statistics to determine 'fit' for categorical data are still dominant in the goodness-of-fit arena. Empirical Distribution Function type goodness-of-fit test statistics are known to be relatively more powerful than Chi-Square type test statistics for restricted types of null and alternative distributions. In many practical applications researchers who use a standard Chi-Square type goodness-of-fit test statistic ignore the rank of ordinal classes. This thesis reviews literature in the goodness-of-fit field, with major emphasis on categorical goodness-of-fit tests. The continued use of an asymptotic distribution to approximate the exact distribution of categorical goodness-of-fit test statistics is discouraged. It is unlikely that an asymptotic distribution will produce a more accurate estimation of the exact distribution of a goodness-of-fit test statistic than a Monte Carlo approximation with a large number of simulations. Due to their relatively higher powers for restricted types of null and alternative distributions, several authors recommend the use of Empirical Distribution Function test statistics over nominal goodness-of-fit test statistics such as Pearson's Chi-Square. In-depth power studies confirm the views of other authors that categorical Empirical Distribution Function type test statistics do not have higher power for some common null and alternative distributions. Because of this, it is not sensible to make a conclusive recommendation to always use an Empirical Distribution Function type test statistic instead of a nominal goodness-of-fit test statistic. Traditionally the recommendation to determine 'fit' for multivariate categorical data is to treat categories as nominal, an approach which precludes any gain in power which may accrue from a ranking, should one or more variables be ordinal. The presence of multiple criteria through multivariate data may result in partially ordered categories, some of which have equal ranking. This thesis proposes a modification to the currently available Kolmogorov-Smirnov test statistics for ordinal and nominal categorical data to account for situations of partially ordered categories. The new test statistic, called the Combined Kolmogorov-Smirnov, is relatively more powerful than Pearson's Chi-Square and the nominal Kolmogorov-Smirnov test statistic for some null and alternative distributions. A recommendation is made to use the new test statistic with higher power in situations where some benefit can be achieved by incorporating an Empirical Distribution Function approach, but the data lack a complete natural ordering of categories. The new and established categorical goodness-of-fit test statistics are demonstrated in the analysis of categorical data with brief applications as diverse as familiarity of defence programs, the number of recruits produced by the Merlin bird, a demographic problem, and DNA profiling of genotypes. The results from these applications confirm the recommendations associated with specific goodness-of-fit test statistics throughout this thesis.
Goodness of fit
Categorical variable
Empirical distribution function
Chi-square test
Anderson–Darling test
Statistic
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A chi-square type test statistic for testing the uniformity hypothesis, based on selected values of the sample quantiles of the empirical distribution function, is presented. Critical values of the test statistics are given for several particular choices of the sample quantiles. A good agreement with their asymptotic counterparts occurs for sample sizes of the magnitude 30 and sometimes even for small sample sizes.
Quantile
Goodness of fit
Empirical distribution function
Statistic
Anderson–Darling test
Sample (material)
Chi-square test
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Kolmogorov–Smirnov (KS) statistic is a non-parametric statistic based on the empirical distribution function. For the one-sample case, it uses the supremum distance between an empirical distribution function (EDF) and a pre-specified cumulative distribution function (CDF). For two-sample case, it measures the maximum of the distance between two EDFs. KS test, as well as other EDF-based tests such as the Anderson-Darling (AD) test and Cramer-von Mises (CvM) test, has been widely used in statistical analysis. To address and compare the performance of these test statistics, we have conducted a simulation study comparing the type I error and power of the KS test, the CvM test, the AD test, and the Chi-squared test. Our study includes both one sample and two sample tests and for both independent and correlated samples. Our study showed that if we do not have prior information about the tested distributions, EDF-based tests are better. However, so long as we have prior information about the tested distribution and the density of two distributions is bell-shaped and we are expecting differences in variance/sparseness, then the Chi-squared test may be more preferable. When correlation exists between tested samples, adjustment on the informative sample size is important and required.
Empirical distribution function
Anderson–Darling test
Goldfeld–Quandt test
Normality test
Sampling distribution
Statistic
Infimum and supremum
Sample (material)
Chi-square test
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Goodness of fit
Empirical distribution function
Univariate distribution
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The classical goodness-of-fit problem, in the case of a null continuous and completely specified distribution, is faced by a new version of the Girone--Cifarelli test (see Girone, 1964; Cifarelli, 1974 \& 1975). This latter test was introduced for the two-sample problem and showed a substantial gain of power over other common tests based on the empirical distribution function, notably over the Kolmogorov--Smirnov test. First, the problem of the re-definition of the Girone--Cifarelli test-statistic is considered, by reviewing the literature on the subject. A classical remark by Anderson (1962) is shown to be useful to choose the integrating function in the newly defined test-statistic. The sample properties of such a test-statistic are then studied. A table of critical values is obtained by simulation; moreover, the asymptotic null distribution is considered and its accuracy as an approximation of the finite distribution is discussed. Finally, a simulation study, considering a wide set of distributions mostly used in applications, is conducted to compare the proposed test with its classical competitors. The study gives some indications to locate such situations where the Girone-Cifarelli test performs at its best, notably over the Kolmogorov--Smirnov test.
Goodness of fit
Anderson–Darling test
Empirical distribution function
Statistic
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Statistical data treatment is an essential part of climate and weather modeling. The Kolmogorov goodness-of-fit test is a widely applicable statistical method to determine the cumulative distribution function of a continuous random variable, e.g., a precipitation level, wind force, etc. In this paper, the authors consider a problem of goodness-of-fit testing involving additional information about S-symmetry of the cumulative distribution function and its influence on the Kolmogorov statistic distributions. A definition of S-symmetry is given; it is a generalized classical definition of distribution symmetry. It is proved that any continuous increasing cumulative distribution function is S-symmetric. A uniform distribution is considered as an example of an S-symmetric distribution. A modification of the Kolmogorov statistic using additional information about the new type of symmetry is proposed. The exact and asymptotic distributions under the null and the alternative hypothesis of the modified statistics are described. The authors also provide an example which proves that the modified test is more powerful than the non-modified one. The new test is used to check the hypothesis of a uniform distribution of the average sum of precipitation.
Goodness of fit
Empirical distribution function
Statistic
Anderson–Darling test
Symmetric probability distribution
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The classical goodness-of-fit problem, in the case of a null continuous and completely specified distribution, is faced by a new version of the Girone-Cifarelli test. This latter test was introduced for the two-sample problem and showed a substantial gain of power over other common tests based on the empirical distribution function, like the Kolmogorov-Smirnov test. After considering the problem of the definition of the test-statistic in the goodness-of-fit framework, this paper deals with the sample properties of the test and compares it with its classical competitors. The superiority of the Girone-Cifarelli test is shown via a simulation study considering symmetric and skewed distributions.
Goodness of fit
Anderson–Darling test
Empirical distribution function
Statistic
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Two statistics for testing goodness of fit for small sample sizes are provided. The first statistic, Sn, can be used to test the fit to any completely specified continuous distribution function and is more powerful than the Kolmogorov-Smirnov statistic in the cases tested. The second statistic, S*n, tests the fit to an exponential distribution with mean unknown. It is more powerful than a Kolmogorov-Smirnov type statistic suggested by Lilliefors (1969) for the cases tested. Critical values for Snand S*n are given for sample size n=1 (1)20(5)30 and α=0.20, 0.15, 0.10, 0.05 and 0.01. The critical values and power analyses were obtained by Monte Carlo techniques. These new statistics are closely related to the Kolmogorov-Smirnov statistic and are computationally equival
Goodness of fit
Statistic
PRESS statistic
Anderson–Darling test
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In this paper we compared the powers of several discrete goodness-of-fit test statistics considered by Steele and Chaseling (10) under the null hypothesis of a 'zig-zag' distribution. The results suggest that the Discrete Kolmogorov-Smirnov test statistic is generally more powerful for the decreasing trend alternative. The Pearson Chi-Square statistic is generally more powerful for the increasing, unimodal, leptokurtic, platykurtic and bath-tub shaped alternatives. Finally, both the Nominal Kolomogorov- Smirnov and the Pearson Chi-Square test statistic are generally more powerful for the bimodal alternative. We also address the issue of the sensitivity of the test statistics to the alternatives under the 'zig-zag' null.
Goodness of fit
Chi-square test
Statistic
Null (SQL)
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