Comparative analysis of the power of goodness-of-fit tests for near competing hypotheses. I. The verification of simple hypotheses

Using statistical modeling methods, we analyze the power of a series of goodness-of-fit tests for simple and complex hypotheses. The estimates we give for the power of the tests for simple hypotheses versus some near competing hypotheses enable us to rank the goodness-of-fit tests.     

On goodness-of-fit tests for parametric hypotheses in perturbed dynamical systems using a minimum distance estimator

We consider the problem of the construction of the goodness-of-fit test in the case of continuous time observations of a diffusion process with small noise. The null hypothesis is parametric and we use a minimum distance estimator of the unknown parameter. We propose an asymptotically distribution free test for this model.     

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.     

The components of chi-squared statistics for goodness-of-fit tests

One obtains a formula for the transformation of a vector with a given covariance matrix. The sum of the squares of the components of the transformed vector is equal to a chi-square type quadratic form, constructed with the aid of the initial vector. Taking into account this property, one finds the components of the Pearson statistic, Pearson-Fisher statistic, etc.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 337–350, 1986.     

Comparison of tests for non-parametric hypotheses

1. Summary The main object of this paper is to find a criterion for comparison of two tests for non-parametric hypotheses, taking advantage of the qualitative information that may exist. After a detailed analysis of the problem and some earlier suggestins for its solution (sections 2–4), a criterion is suggested in section 5. In order to apply it to a concrete case, a location problem is specified in section 6. The rank tests to be compared are analyzed in section 7, and the comparison by way of the criterion is carried out in section 8. It turns out that sign tests, sometimes slightly modified, are very often optimal according to the criterion used.     

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.     

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.     

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.     

Asymptotically minimax tests for nonparametric hypotheses concerning the distribution density

Asymptotically minimax goodness-of-fit, as well as symmetry, homogeneity, and independence tests are constructed, when it is known that the norm of the difference of the densities of the hypothesis and the alternative are bounded away from zero by a constant _n and the densities satisfy some smoothness conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 44–53, 1988.     

Likelihood ratio tests for a class of non-oblique hypotheses

In this paper, we study the likelihood ratio tests for situations in which the null and alternative hypotheses are determined by two polyhedral cones,C ₁ andC ₂, which are nested so thatC ₁ L C ₂ andL is a linear space. The two cones are proved to be non-oblique. Members which satisfy this nesting condition are easily identified and include the cases in whichC ₁=L orL=C ₂. When testing two non-oblique hypotheses with variances unknown, the least favorable point within the null hypothesis has not been determined in general. However, for the situation considered here, the zero vector is shown to be least favorable within the null hypothesis. Two sets of hypotheses are said to be equivalent if they lead to the same likelihood ratio test. For two non-oblique polyhedral cones,C ₁ andC ₂, four sets of equivalent hypotheses are identified. IfC ₁ L C ₂, then the two cones in each of these four sets of hypotheses are similarly nested with a linear space in between.This research was supported in part by the National Institutes for Health under Grant 1 R01 GM42584-01A1.Part of this work is taken from this author's dissertation submitted in partial fulfillment for the Ph.D. Degree at the University of Missouri-Columbia.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

Asymptotic tests of composite hypotheses for non-ergodic type stochastic processes

Limiting distributions of a score statistic and the likelihood ratio statistic for testing a composite hypothesis involving several parameters in non-ergodic type stochastic processes are obtained. It is shown that, unlike in the usual theory (ergodic type processes), the limiting distributions of these statistics are different both under the null and a contiguous sequence of alternative hypotheses. The results are applied to a regression model with explosive autoregressive Gaussian errors. In the discussion of this example a modified score statistic is suggested where the limiting null and non-null distributions are the same as those of the likelihood ratio statistic.     

Asymptotically minimax tests of composite hypotheses for nonergodic type processes

Asymptotically efficient tests satisfying a minimax type criterion are derived for testing composite hypotheses involving several parameters in nonergodic type stochastic processes. It is shown, in particular, that the analogue of the usual Neyman's C (α) type test (i.e., the score test) is not efficient for the nonergodic case. Moreover, the likelihood-ratio statistic is not fully efficient for the model discussed in the paper. The efficient statistic derived here is a modified version of the score-statistic discussed previously by Basawa and Koul (1979).     

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let and be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices ²(X ^t X)^–1 and ²(X ^t X)^–1 respectively. Let the Wald-type statistics based on and be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (²/²). Thus, the asymptotic efficiency-robustness properties of relative to translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that is efficiency-robust compared to . The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.     

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