Exact goodness-of-fit tests for censored data

The statistic introduced in Fortiana and Grané (J R Stat Soc B 65(1):115–126, 2003) is modified so that it can be used to test the goodness-of-fit of a censored sample, when the distribution function is fully specified. Exact and asymptotic distributions of three modified versions of this statistic are obtained and exact critical values are given for different sample sizes. Empirical power studies show the good performance of these statistics in detecting symmetrical alternatives.     

On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data

We study the rate of weak convergence of the distributions of the statistics {t _λ (Y), λ ∈ ℝ} from the power divergence family of statistics to the χ ² distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that

Bahadur effectiveness of Watson-Darling goodness-of-fit tests

The rough asymptotic behavior of the probability of large deviations is found for Watson-Darling goodness-of-fit tests, which are centered versions of Kolomogorov-Smirnov statistics. Their Bahadur effectiveness is calculated and conditions for local asymptotic optimality for the alternative of translation are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 138–145, 1987.     

Nonparametric likelihood ratio goodness-of-fit tests for survival data

Berk and Jones (Z. Wahrsch. Verw. Gebiete 47 (1979) 47) described a nonparametric likelihood test of uniformity that is more efficient, in Bahadur's sense, than any weighted Kolmogorov-Smirnov test at any alternative. This article shows how to obtain a nonparametric likelihood test of a general parametric family for incomplete survival data. A nonparametric likelihood ratio test process is employed to measure the discrepancy between a parametric family and the observed data. Large sample properties of the likelihood ratio test process are studied under both the null and alternative hypotheses. A Monte Carlo simulation method is proposed to estimate its null distribution. We show how to produce a likelihood ratio graphical check as well as a formal test of a parametric family based on the developed theory. Our method is developed for the right-censorship model, but can be easily extended to some other survival models. Illustrations are given using both real and simulated data.     

On score-functions and goodness-of-fit tests for stochastic processes

The problems of the construction of asymptotically distribution free goodness-of-fit tests for two diffusion processes are considered. The null hypothesis is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the maximum likelihood estimators. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct asymptotically distribution-free tests for dynamical systems with small noise and ergodic diffusion processes. The proposed tests are in some sense universal. We discuss the possibilities of the construction of asymptotically distribution free tests for inhomogeneous Poisson processes and nonlinear AR time series.     

Some goodness-of-fit tests for the positive stable distributions

Let S(α, π) be a random variable having the positive stable distribution with exponent α and scale parameter π. As is known (see [7]), the characteristic function of such a distribution can be written in the form $$\varphi (t) = \exp ( - |\tau t|^\alpha \exp {\mathbf{ }}( - \tfrac{\pi }{2}i{\mathbf{ }}sign t)),$$ where 0 < α < 1 and π > 0. In what follows, the notation S(α, π) is also used for the family of the positive stable distributions. The purpose of this paper is twofold: (i) to derive a goodness-of-fit test for the family S(α, π) without any assumptions on the parameters α and π, (ii) to derive a goodness-of-fit test for the family S(α, π) with fixed exponent α and nuisance scale parameter π. The powers of the tests for a number of alternatives are estimated by means of statistical simulation.     

Improvements of goodness-of-fit statistics for sparse multinomials based on normalizing transformations

We consider multinomial goodness-of-fit tests for a specified simple hypothesis under the assumption of sparseness. It is shown that the asymptotic normality of the PearsonX ² statistic (X _k ² ) and the log-likelihood ratio statistic (G _k ² ) assuming sparseness. In this paper, we improve the asymptotic normality ofX _k ² andG _k ² statistics based on two kinds of normalizing transformation. The performance of the transformed statistics is numerically investigated.     

On goodness-of-fit tests for multiple recursive random number generators

This paper considers the problem of reducing the computational time in testing uniformity for a full period multiple recursive generator (MRG). If a sequence of random numbers generated by a MRG is divided into even number of segments, say 2s, then the multinomial goodness-of-fit tests and the empirical distribution function goodness-of-fit tests calculated from the ith segment are the same as those of the (s + i)th segment. The equivalence properties of the goodness-of-fit test statistics for a MRG and its associated reverse and additive inverse MRGs are also discussed.     

Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters

We study a test statistic based on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 1. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular, size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.     

Modified cramer-von mises goodness-of-fit tests for spectral distribution functions

Modifications to the Cramer-von Mises goodness-of-fit test statistic for spectral distributions are discussed. The modifications consist of inserting weight functions into the usual sto¬chastic integral for the test statistic. Conditions on the weight function are given under which the integral of the weighted square of the difference between the empirical and theoretical spectral distribution functions converges in distribution to the corresponding integral of a process related to Brownian Motion. The distributions of the test statistic under certain alternatives to the null hypothesis are also discussed. A discussion is given of the large sample distributions for weight function of the form ψ(t) = at ^k ,k < –2.     

Likelihood ratio tests for goodness-of-fit of a nonlinear regression model

We propose likelihood and restricted likelihood ratio tests for goodness-of-fit of nonlinear regression. The first-order Taylor approximation around the MLE of the regression parameters is used to approximate the null hypothesis and the alternative is modeled nonparametrically using penalized splines. The exact finite sample distribution of the test statistics is obtained for the linear model approximation and can be easily simulated. We recommend using the restricted likelihood instead of the likelihood ratio test because restricted maximum-likelihood estimates are not as severely biased as the maximum-likelihood estimates in the penalized splines framework.     

Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Summary The members of the power divergence family of statistics all have an asymptotically equivalent χ² distribution (Cressie and Read [1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX ² statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.     

General chi-square goodness-of-fit tests with data-dependent cells

Summary The goodness-of-fit of a parametric model for non-categorical data can be tested using the x ² statistic calculated after grouping the data into a finite number of disjoint cells. Work of Watson, ebyev, Moore and others shows that the classical limit distributions still hold even for certain methods of grouping that depend on the data themselves. These results are generalised to cover a much wider class of methods of grouping; the parameters can be estimated from either the grouped or the ungrouped data. The proofs use a Central Limit Theorem for Empirical Measures due to Dudley. The grouping cells are allowed to come from any Donsker class for the underlying sampling distribution.Dedicated to Leopold Schmetterer, on his sixtieth birthdayMost of the work for this paper was carried out with the support of a fellowship of the Alexander von Humboldt Foundation at the Ruhr-Universität Bochum     

Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, R^a, as multinomial goodness-of-fit statistics. Each R^a has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of R^a under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of R^a under nonlocal alternatives can be improved. We apply the expansion to the power approximation for R^a. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.     

Neyman smooth goodness-of-fit tests for the marginal distribution of dependent data

We establish a data-driven version of Neyman??s smooth goodness-of-fit test for the marginal distribution of observations generated by an ??-mixing discrete time stochastic process ${(X_t)_{t \in \mathbb {Z}}}$ . This is a simple extension of the test for independent data introduced by Ledwina (J Am Stat Assoc 89:1000?C1005, 1994). Our method only requires additional estimation of the cumulative autocovariance. Consistency of the test will be shown at essentially any alternative. A brief simulation study shows that the test performs reasonable especially for the case of positive dependence. Finally, we illustrate our approach by analyzing the validity of a forecasting method (??historical simulation??) for the implied volatilities of traded options.     

Random projections and goodness-of-fit tests in infinite-dimensional spaces

In this paper we provide conditions under which a distribution is determined by just one randomly chosen projection. Thenweapply our results to construct goodness-of-fit tests for the one and two-sample problems. We include some simulations as well as the application of our results to a real data set. Our results are valid for every separable Hilbert space. *Partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant MTM2005-08519-C02-02. **Partially supported by grants from NSERC and the Canada research chairs program.     

Empirical Hankel transforms and its applications to goodness-of-fit tests

We introduce a special Hankel transform for probability distributions on the nonnegative half-line and discuss some of its properties. Due to the uniqueness of the transform we suggest an integral type test statistic based on the empirical Hankel transform to treat simple and composite hypotheses goodness-of-fit problems. The special case of exponential distributions is studied in detail.     

Hodges-Lehmann asymptotic efficiency of the Kolmogorov and Smirnov goodness-of-fit tests

One considers the Hodges-Lehmann asymptotic efficiency of the Kolmogorov and Smirnov goodness-of-fit tests, which measures the rate of the exponential decrease of the errors of the second kind, under a fixed significance level. It is shown that the Kolmogorov test is always asymptotically optimal in this sense, while the one-sided Smirnov test is asymptotically optimal under additional conditions imposed on the parametric family of distributions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 119–123, 1985.     

Exact distributions of certain goodness-of-fit tests and their complete asymptotic expansions

Let F_n(x) be an empirical distribution function constructed on the basis of a repeated sample from a population with a continuous distribution function F(x). This paper deals with the classical nonparametric Kolmogorov-Smirnov tests For the exact distribution of the two-dimensional vector (D _n ⁺ , D _n ^— ), a representation is obtained in the form of a contour integral, as well as a complete asymptotic expansion of this distribution in a power series in 1/n¹, and a general algorithm of construction of asymptotic expansions of distributions of the statistics of Kuiper, Pyke, Brunk, and many others that are closely related to the Kolmogorov-Smirnov tests.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 43, pp. 107–132, 1974.