Goodness of fit tests with interval censored data

Cramér–von Mises type goodness of fit tests for interval censored data case 2 are proposed based on a resampling method called the leveraged bootstrap, and their asymptotic consistency is shown. The proposed tests are computationally efficient, and in fact can be applied to other types of censored data, including right censored data, doubly censored data and (mixture of) case k interval censored data. Some simulation results and an example from AIDS research are presented.     

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.     

Goodness of fit tests for Nakagami distribution based on smooth tests

ABSTRACT

Nakagami distribution is one of the most common distributions used to model positive valued and right skewed data. In this study, we interest goodness of fit problem for Nakagami distribution. Thus, we propose smooth tests for Nakagami distribution based on orthonormal functions. We also compare these tests with some classical goodness of fit tests such as Cramer–von Mises, Anderson–Darling, and Kolmogorov–Smirnov tests in respect to type-I error rates and powers of tests. Simulation study indicates that smooth tests give better results than these classical tests give in respect to almost all cases considered.     

Goodness of fit tests for the multiple logistic regression model

Several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model. The test statistics are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions. The null distributions of these statistics are examined by applying the theory for chi-square tests of Moore Spruill (1975) and through computer simulations. All statistics are shown to have a chi-square distribution or a distribution which can be well approximated by a chi-square. The degrees of freedom are shown to depend on the particular statistic and the distributional assumptions.

The power of each of the proposed statistics is examined for the normal, linear, and exponential alternative models using computer simulations.     

One-way analysis of proportions for misclassified binomial data

We consider data with a nominal grouping variable and a binary response variable. The grouping variable is measured without error, but the response variable is measured using a fallible device subject to misclassification. To achieve model identifiability, we use the double-sampling scheme which requires obtaining a subsample of the original data or another independent sample. This sample is then classified by both the fallible device and another infallible device regarding the response variable. We propose two Wald tests for testing the association between the two variables and illustrate the test using traffic data. The Type-I error rate and power of the tests are examined using simulations and a modified Wald test is recommended.     

Goodness of fit for the inverse Gaussian distribution

For testing the fit of the inverse Gaussian distribution with unknown parameters, the empirical distribution-function statistic A² is studied. Two procedures are followed in constructing the test statistic; they yield the same asymptotic distribution. In the first procedure the parameters in the distribution function are directly estimated, and in the second the distribution function is estimated by its Rao-Blackwell distribution estimator. A table is given for the asymptotic critical points of A². These are shown to depend only on the ratio of the unknown parameters. An analysis is provided of the effect of estimating the ratio to enter the table for A². This analysis enables the proposal of the complete operating procedure, which is sustained by a Monte Carlo study.     

Goodness of Fit via Non-parametric Likelihood Ratios

Abstract.  To test if a density f is equal to a specified f ₀, one knows by the Neyman–Pearson lemma the form of the optimal test at a specified alternative f ₁. Any non-parametric density estimation scheme allows an estimate of f . This leads to estimated likelihood ratios. Properties are studied of tests which for the density estimation ingredient use log-linear expansions. Such expansions are either coupled with subset selectors like the Akaike information criterion and the Bayesian information criterion regimes, or use order growing with sample size. Our tests are generalized to testing the adequacy of general parametric models, and to work also in higher dimensions. The tests are related to, but are different from, the 'smooth tests' that go back to Neyman [Skandinavisk Aktuarietidsskrift 20(1937) 149] and that have been studied extensively in recent literature. Our tests are large-sample equivalent to such smooth tests under local alternative conditions, but different from the smooth tests and often better under non-local conditions.     

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.     

Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models

We use Owen's (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z ₁, ..., z _n which is observed directly from a distribution F ( z ). The other one is x ₁, ..., x _m which is observed indirectly from F ( z ), where the x _i s have density ∫ p ( x | z ) dF ( z ) and p ( x | z ) is a conditional density function. We are interested in testing H ₀: p ( x | z ) = p ( x | z ; θ ), for some specified smooth density function. A semiparametric likelihood ratio based statistic is proposed and it is shown that it converges to a chi-squared distribution. This is a simple method for doing goodness of fit tests, especially when x is a discrete variable with finitely many values. In addition, we discuss estimation of θ and F ( z ) when H ₀ is true. The connection between upgraded mixture models and general estimating equations is pointed out.     

Smooth goodness of fit tests for the weibull distribution with singly censored data

The smooth goodness of fit tests are generalized to singly censored data and applied to the problem of testing Weibull (or extreme value) fit. Smooth tests, Pearson-type tests, and the spacings tests proposed by Mann, Schemer, and Fertig (1973) are compared on the basis of local asymptotic relative efficiency with respect to the asymptotic best test against generalized gamma alternatives, The smooth test of order one Is found to be most efficient for the generalized gamma alternatives.     

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.     

Bayesian Goodness of Fit Testing with Mixtures of Triangular Distributions

Abstract.  We consider the consistency of the Bayes factor in goodness of fit testing for a parametric family of densities against a non-parametric alternative. Sufficient conditions for consistency of the Bayes factor are determined and demonstrated with priors using certain mixtures of triangular densities.     

Chi-square tests for multivariate normality with application to common stock prices

The theory of chi-square tests with data-dependent cells is applied to provide tests of fit to the family of p-variate normal distributions. The cells are bounded by hyperellipses (x-[Xbar])'S^-1 (x-[Xbar]) = c_i centered at the sample mean [Xbar] and having shape deter-mined by the sample covariance matrix S. The Pearson statistic with these cells is affine-invariant, has a null distribution not depending on the true mean and covariance, and has asymptotic critical points between those of x² (M-1) and x² (M-2) when M cells are employed. The test is insensitive to lack of symmetry, but peakedness, broad shoulders and heavy tails are easily discerned in the cell counts. Multivariate normality of logarithms of relative prices of common stocks, a common assumption in finan-cial markets theory, is studied using the statistic described here and a large data base.     

One sample tests of location for nonnormal symmetric data

A score test of location is derived for data from a distorted normal distribution. A simulation study compares the performance of this test to the t-test and Wilcoxon test for symmetric data from such a distribution. For this type of data the score test can be considerably more powerful than both the t-test and Wilcoxon test. This suggests that such a score test may be useful in practice when variations from normality can be modeled by such a family of distributions.     

On the influence of misclassified data on results of goodness of fit testing

In this paper we focus on the chi-square test of goodness of fit, which compares an observed discrete distribution to an expected known one. We show that the results of this test, using the common Pearson statistic, are very sensitive to misclassified observations between two or more categories. We also propose a general rule of thumb for analysing data set stability with respect to such classification errors. Practical analysis of a real example illustrates our purpose.     

Improved likelihood ratio tests and pearson chi-square tests for independence in two dimensional contingency tables

When an I×J contingency table has many cells having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio goodness-of-fit statistic, G² and Pearson chi-square statistic, X², for testing independence, are not satisfactory. In this paper we consider the problem of adjusting G² and X². Suitable adjustments are suggested on the basis of analytical investigation of asymptotic bias terms for G² and X². A Monte Carlo simulation is performed for several tables to assess the adjustments of G² and X² in order to obtain a closer approximation to the nominal level of significance.     

Smooth tests of fit for censored gmma samples

In this paper properties of two estimators of Cpm are investigated in terms of changes in the process mean and variance. The bias and mean squared error of these estimators are derived. It can be shown that the estimate of Cpm proposed by Chan, Cheng and Spiring (1988) has smaller bias than the one proposed by Boyles (1991) and also has a smaller mean squared error under certain conditions. Various approximate confidence intervals for Cpm are obtained and are compared in terms of coverage probabilities, missed rate and average interval width.     

On goodness of fit tests based on spacings for type ii censored samples

A class of tests based on spacings is obtained for milticensored samples. Their asymptotic null as well as alternative distributions are obtained.     

Comparison of goodness of fit tests for the cox proportional hazards model

The proportional hazards regression model of Cox(1972) is widely used in analyzing survival data. We examine several goodness of fit tests for checking the proportionality of hazards in the Cox model with two-sample censored data, and compare the performance of these tests by a simulation study. The strengths and weaknesses of the tests are pointed out. The effects of the extent of random censoring on the size and power are also examined. Results of a simulation study demonstrate that Gill and Schumacher's test is most powerful against a broad range of monotone departures from the proportional hazards assumption, but it may not perform as well fail for alternatives of nonmonotone hazard ratio. For the latter kind of alternatives, Andersen's test may detect patterns of irregular changes in hazards.     

分类数据的计量误差模型

二分类总体单元被错误分类情形下,样本比例是总体比例的有偏估计。给出调整比例估计的两种方法：双样本方法和极大似然法,补充了经典抽样理论比例估计的内容。