Cramér-von Mises statistics for discrete distributions

Cramér-von Mises statistics are developed for use in testing for discrete distributions, and tables are given for tests for the discrete uniform distribution.     

Cramér‐von Mises regression

Consider a linear regression model with unknown regression parameters β₀ and independent errors of unknown distribution. Block the observations into q groups whose independent variables have a common value and measure the homogeneity of the blocks of residuals by a Cramér‐von Mises q‐sample statistic T_q(β). This statistic is designed so that its expected value as a function of the chosen regression parameter β has a minimum value of zero precisely at the true value β₀. The minimizer β of T_q(β) over all β is shown to be a consistent estimate of β₀. It is also shown that the bootstrap distribution of T_q(β₀) can be used to do a lack of fit test of the regression model and to construct a confidence region for β₀     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.     

Goodness of fit tests for discrete distributions

Some alternative procedures for testing goodness of fit in discrete distributions are discussed here.. These procedures are based on the probability generating functions.. The methods considered are quite general, being applicable in multidimensional situations., The strength of the tests lies in that no ambiguity as to classification of the data arises.. Hov-ever, some difficulties in the proposed procedures are also pointed out.     

Cramér-von Mises tests of fit for the Poisson distribution

Goodness-of-fit tests based on the Cramér-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A², give good overall tests of fit. The statistic A² will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.     

Estimators Based on Data‐Driven Generalized Weighted Cramér‐von Mises Distances under Censoring – with Applications to Mixture Models

Abstract. Estimators based on data‐driven generalized weighted Cramér‐von Mises distances are defined for data that are subject to a possible right censorship. The function used to measure the distance between the data, summarized by the Kaplan–Meier estimator, and the target model is allowed to depend on the sample size and, for example, on the number of censored items. It is shown that the estimators are consistent and asymptotically multivariate normal for every p dimensional parametric family fulfiling some mild regularity conditions. The results are applied to finite mixtures. Simulation results for finite mixtures indicate that the estimators are useful for moderate sample sizes. Furthermore, the simulation results reveal the usefulness of sample size dependent and censoring sensitive distance functions for moderate sample sizes. Moreover, the estimators for the mixing proportion seem to be fairly robust against a ‘symmetric’ contamination model even when censoring is present.     

Goodness‐of‐Fit Tests for Bivariate and Multivariate Skew‐Normal Distributions

Abstract. Goodness‐of‐fit tests are proposed for the skew‐normal law in arbitrary dimension. In the bivariate case the proposed tests utilize the fact that the moment‐generating function of the skew‐normal variable is quite simple and satisfies a partial differential equation of the first order. This differential equation is estimated from the sample and the test statistic is constructed as an L ₂ ‐type distance measure incorporating this estimate. Extension of the procedure to dimension greater than two is suggested whereas an effective bootstrap procedure is used to study the behaviour of the new method with real and simulated data.     

Modified cramer-von mises and anderson-darling tests for weibull distributions with unknown location and scale parameters

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Goodness of fit for the poisson distribution based on the probability generating function

For testing the fit of a discrete distribution, use of the probability generating function and its empirical counterpart has been suggested in Koeherlakota and Kocherlakota (1986). In the present paper, a particular functional of the corresponding empirical probability generating function process is proposed as a measure to test the discrepancy between the evidence and the hypothesis. The asymptotic behavior of the empirical probability generating function when a parameter is estimated is obtained, The study is exemplified for the Poisson case only but the procedure can be extended to other discrete distributions.     

Exact Goodness‐of‐Fit Testing for the Ising Model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness‐of‐fit test for the finite‐lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness‐of‐fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness‐of‐fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.     

On the asymptotic distributions of high-order spacings statistics

Goodness-of-fit tests for the uniform distribution based on sums of smooth functions of m-spacings are studied. A limiting sum-of-weighted-chi-squareds approximation is shown to be accurate uniformly in m for the special cases of analogues of Greenwoo?s statistic and Moran's statistic. Asymptotic critical points are provided; theory and Monte Carlo studies show they are accurate for all m provided n is moderately large.     

Goodness‐of‐Fit based on Downsampling with Applications to Linear Drift Diffusions

Abstract. A goodness‐of‐fit test for continuous‐time models is developed that examines if the parameter estimates are consistent with another for different sampling frequencies. The test compares parameter estimates obtained from estimating functions for downsamples of the data. We prove asymptotic results for stationary and ergodic processes, and apply the downsampling test to linear drift diffusions. Simulations indicate that the test is quite powerful in detecting non‐Markovian deviations from the linear drift diffusions.     

A family of power‐divergence diagnostics for goodness‐of‐fit

The author extends to the Bayesian nonparametric context the multinomial goodness‐of‐fit tests due to Cressie & Read (1984). Her approach is suitable when the model of interest is a discrete distribution. She provides an explicit form for the tests, which are based on power‐divergence measures between a prior Dirichlet process that is highly concentrated around the model of interest and the corresponding posterior Dirichlet process. In addition to providing interesting special cases and useful approximations, she discusses calibration and the choice of test through examples.     

Goodness‐of‐fit tests for linear regression models with missing response data

The authors show how to test the goodness‐of‐fit of a linear regression model when there are missing data in the response variable. Their statistics are based on the L₂ distance between nonparametric estimators of the regression function and a ‐consistent estimator of the same function under the parametric model. They obtain the limit distribution of the statistics and check the validity of their bootstrap version. Finally, a simulation study allows them to examine the behaviour of their tests, whether the samples are complete or not.     

Self-updating clustering algorithm for estimating the parameters in mixtures of von Mises distributions

The EM algorithm is the standard method for estimating the parameters in finite mixture models. Yang and Pan [25] proposed a generalized classification maximum likelihood procedure, called the fuzzy c-directions (FCD) clustering algorithm, for estimating the parameters in mixtures of von Mises distributions. Two main drawbacks of the EM algorithm are its slow convergence and the dependence of the solution on the initial value used. The choice of initial values is of great importance in the algorithm-based literature as it can heavily influence the speed of convergence of the algorithm and its ability to locate the global maximum. On the other hand, the algorithmic frameworks of EM and FCD are closely related. Therefore, the drawbacks of FCD are the same as those of the EM algorithm. To resolve these problems, this paper proposes another clustering algorithm, which can self-organize local optimal cluster numbers without using cluster validity functions. These numerical results clearly indicate that the proposed algorithm is superior in performance of EM and FCD algorithms. Finally, we apply the proposed algorithm to two real data sets.     

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.     

A cramer - von mises type goodness of fit test with asymmetric weight function

A goodness of fit test of the Cramer - von Mises type, which gives more weight to the upper (or to the lower) tail of the distribution, is proposed and studied. It is found the orthogonal representation of the test for the case of a simple null hypothesis. The characteristic function of the asymptotic null distribution is found and inverted to get percentage points. The asymptotic power of the test is obtained for the normal null hypothesis, against mean and variance shifts and more asymmetric alternatives.

Also the case of the exponential null hypothesis is studied. It is found that the test, which emphasizes the upper tail, has more power than those of Anderson - Darling and Cramer - von Mises, against alternatives which differ from the null hypothesis mainly in the upper tail, and less power when the main difference is in the lower tail of the distribution.     

Comparison of methods of estimation for parameters of generalized Poisson distribution through simulation study

In this article, we take a brief overview of different functional forms of generalized Poisson distribution (GPD) and various methods of its parameter estimation found in the literature. We compare the method of moment estimation (ME) and maximum likelihood estimation (MLE) of parameters of GPD through simulation study in terms of bias, MSE and covariance. To simulate random numbers from GPD, we develop a Matlab function gpoissrnd(). The simulation study leads to the important conclusion that the ME performs better or equally good as compared to MLE when sample size is small.

Further we fit the GPD to various datasets in literature using both estimation methods and observe that the results do not differ significantly even though the sample size is large. Overall, we conclude that for GPD, use of ME in place of MLE will lead to almost similar results. The computational simplicity in calculation of ME as compared to MLE also gives support to the use of ME in case of GPD for practitioners.