GOODNESS OF FIT STATISTICS BASED ON THE SPACINGS OF COMPLETE OR CENSORED SAMPLES

A goodness-of-fit statistic Z is defined in terms of the spacings generated by the order statistics of a complete or a censored sample from a distribution of the type (l/)f((x-μ)/), μ and unknown. The distribution of Z is studied, mostly through Monte Carlo methods. The power properties of Z for testing Exponential, Uniform, Normal, Gamma and Logistic distributions are discussed; Z is shown to be more powerful than the Smith & Bain (1976) correlation statistic, except for testing Uniform, Normal and Logistic (symmetric distributions) against symmetric alternatives. The statistic Z is generalized to test the goodness-of-fit from κ 2 independent complete or censored samples.     

GOODNESS OF FIT FOR THE BINOMIAL DISTRIBUTION

Goodness of fit testing for the binomial distribution can be carried out using Pearson's X²_p statistic and its components. Applications of this technique are considered and compared with recently suggested empirical distribution function tests. Diagnostic use of components is discussed.     

GOODNESS OF FIT TESTS FOR HETEROGENEITY IN A CAPTURE-RECAPTURE EXPERIMENT

A class of goodness of fit tests is proposed to test for heterogeneity in a population in a capture-recapture experiment when individuals' capture intensities may depend on time in an unspecified manner.     

ON A TEST OF FIT IN TIME SERIES MODELS

It is shown that a test for adequacy of a fitted Arma (p, 4) model based on forecast errors, has the same asymptotic expected value whether the null hypothesis is true or false. A sampling study supports the conclusion in small to moderately large samples and indicates that for sample sizes commonly used by practitioners, the power of the test is very low.     

ON THE EVALUATION OF CONTROL CHART LIMITS BASED ON PREDICTIVE DISTRIBUTIONS

ABSTRACT

We propose a Bayesian approach to obtaining control charts when there is parameter uncertainty. Our approach consists of two stages, (i) construction of the control chart where we use a predictive distribution based on a Bayesian approach to derive the rejection region, and (ii) evaluation of the control chart where we use a sampling theory approach to examine the performance of the control chart under various hypothetical specifications for the data generation model.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

TEST OF FIT FOR THE INVERSE GAUSSIAN AND GAMMA DISTRIBUTIONS UNDER CENSORING

The problem of testing the fit of the inverse Gaussian and the gamma distribution when the sample is censored and some of the parameters are unknown, is studied. Empirical Distribution Function (EDF) statistics, namely Cramér-von Mises' W ² and the Anderson-Darling's A ², are used. The limiting covariance functions of the corresponding empirical processes are derived. Asymptotic percentage points are given for some parameter values and censoring proportions. Moreover, a numerical routine is made available upon request, to obtain p-values for both test statistics, thus eliminating the need of tables and interpolation. Finally, a simple Monte Carlo study is presented to evaluate first, the approximation when using the asymptotic distributions in finite samples and second, to support the use of estimated parameter values instead of the unknown parameters needed in the limiting covariance function.     

IDENTIFYING THE OUT OF CONTROL VARIABLE IN A MULTIVARIATE CONTROL CHART

ABSTRACT

The identification of the out of control variable, or variables, after a multivariate control chart signals, is an appealing subject for many researchers in the last years. In this paper we propose a new method for approaching this problem based on principal components analysis. Theoretical control limits are derived and a detailed investigation of the properties and the limitations of the new method is given. A graphical technique which can be applied in some of these limiting situations is also provided.     

A TEST OF THE LOCATION PARAMETER OF THE EXPONENTIAL DISTRIBUTION

In this paper, we propose a procedure for testing the location parameter of the exponential distribution for certain alternative hypotheses, which could result in the early rejection of the null hypothesis. This is a consequence of the monotone property of the test statistic which is based on the extremal quotient. The test being scale-free does not require the scale parameter to be known.     

A BETA-OPTIMAL TEST OF THE EQUICORRELATION COEFFICIENT

This paper considers the problem of testing for nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. Recently, SenGupta (1987) proposed a locally best test. We construct a beta-optimal test and present selected one and five percent critical values. An empirical power comparison of SenGupta's test with two versions of the beta-optimal test and the power envelope shows the relative strengths of the three tests. It also allows us to assess and confirm Efron's (1975) rule of when to question the use of a locally best test, at least for this testing problem. On the basis of these results, we argue that the two beta-optimal tests can be considered as approximately uniformly most powerful tests, at least at the five percent significance level.     

A BOOTSTRAP TEST OF THE GOODNESS-OF-FIT OF THE MULTIVARIATE TOLERANCE MODEL

The multivariate tolerance (MT) model is used to model multi-agent bioassay survival responses. When the MT model is appropriate, it can aid in identifying the biological mechanism of the survival response or motivate tolerance-based error reduction. A bootstrap test for the goodness-of-fit of the MT model is proposed. The proposed test has little level error, useful power, and makes a novel adjustment for the multiple testing character of the question of MT model goodness-of-fit. This adjustment may be more generally applicable in bootstrap testing.     

ON THE DISTRIBUTION OF SPHERICITY TEST CRITERION IN THE MULTIVARIATE GAUSSIAN DISTRIBUTION

In this paper, the exact distribution of the likelihood-ratio criterion for testing the hypothesis of sphericity in a multivariate normal distribution is derived as a mixture of incomplete beta functions. Some advantages of the present derivation are also indicated.     

ON THE LIKELIHOOD RATIO TEST FOR A SINGLE MODEL AGAINST THE MIXTURE OF TWO KNOWN DENSITIES

In this paper, we consider the likelihood ratio test for a single model against the mixture of two known densities. We research the accuracy of the limiting null distribution obtained by Titterington 1-2 Titterington, D. M. 1981. Contribution to the discussion of paper by M. Aitkin, D. Amderson and J. Hinde, J. R.. Statist. Soc. A, 144: 459 Titterington, D. M., Smith, A. F.M. and Makov, U. E. 1985. Statistical Analysis of Finite Mixture Distribution New York: Wiley.  .After that, we improve the limiting null distribution and confirm its usefulness by the simulation study.     

ON A SHARED ALLELE TEST OF RANDOM MATING

A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, n_ijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n₀, n₁ and n₂ where n_p is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?_o where n₀, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X² compares observed (n_s) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X² is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in _e. This paper gives two theorems which show that x is indeed the approximate distribution of X² for large n and K₁“, provided that no allele type over-dominates the others.     

A CONSISTENT MODEL SPECIFICATION TEST FOR A REGRESSION FUNCTION BASED ON NONPARAMETRIC WAVELET ESTIMATION

In this paper we present a consistent specification test of a parametric regression function against a general nonparametric alternative. The proposed test is based on wavelet estimation and it is shown to have similar rates of convergence to the more commonly used kernel based tests. Monte Carlo simulations show that this test statistic has adequate size and high power and that it compares favorably with its kernel based counterparts in small samples.