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Nonparametric moment-based goodness-of-fit tests are developed for the family of chi-square copulas of arbitrary dimensions. This class of dependence models allows for tail asymmetries and contains the family of multivariate normal copulas as a special case. The proposed tests are based on two rank correlation coefficients whose population versions are equal, up to a monotone transformation, when the underlying dependence structure is a chi-square copula. The test statistics are computed from natural rank-based estimations of these two correlation coefficients and their large-sample distributions under the null hypothesis of a chi-square copula are derived; the validity of a parametric bootstrap procedure for the computation of p-values is formally established as well. Particular attention is given to tests for the families of normal and centered chi-square copulas. The simulations that are reported indicate that the new tests are reliable alternatives to those based on the empirical copula, both in the bivariate and multivariate cases. The usefulness of the introduced methodology is illustrated on the five-dimensional Nutrient dataset.

This research provides a projection-based test to check parametric single-index regression structure in variable-adjusted models. An adaptive-to-model strategy is employed, which makes the proposed test work better on the significance level maintenance and more powerful than existing tests. With mild conditions, the proposed test asymptotically behaves like a test that is for classical regression setup without distortion errors in observations. Numerical studies with simulated and real data are conducted to examine the performance of the test in finite sample scenarios.

Field studies in ecology often make use of data collected in a hierarchical fashion, and may combine studies that vary in sampling design. For example, studies of tree recruitment after disturbance may use counts of individual seedlings from plots that vary in spatial arrangement and sampling density. To account for the multi-level design and the fact that more than a few plots usually yield no individuals, a mixed effects zero inflated Poisson model is often adopted. Although it is a convenient modeling strategy, various aspects of the model could be misspecified. A comprehensive test procedure, based on the cumulative sum of the residuals, is proposed. The test is proven to be consistent, and its convergence properties are established as well. The application of the proposed test is illustrated by a real data example and simulation studies.

In this paper, we study the entropy test for the goodness of fit test in (nonlinear) autoregressive conditional duration (ACD) models. To implement a test, we first explore the null limiting distribution of the residual empirical process from ACD models and verify that it has an asymptotic expansion form that consists of the true empirical process and extra terms yielded by parameter estimation. Then, we show that under regularity conditions, the proposed entropy test approximately follows a distribution that is free from the parameter estimation. For illustration, a simulation study and real data analysis are conducted. In the implementation of the test, a parametric bootstrap method is employed.

Multiple-testing problems have received much attention. Different strategies have been considered in order to deal with this problem. The false discovery rate (FDR) is, probably, the most studied criterion. On the other hand, the sequential goodness of fit (SGoF), is a recently proposed approach. Most of the developed procedures are based on the independence among the involved tests; however, in spite of being a reasonable proviso in some frameworks, independence is not realistic for a number of practical cases. Therefore, one of the main problems in order to develop appropriate methods is, precisely, the effect of the dependence among the different tests on decisions making. The consequences of the correlation on the z-values distribution in the general multitesting problem are explored. Some different algorithms are provided in order to approximate the distribution of the expected rejection proportions. The performance of the proposed methods is evaluated in a simulation study in which, for comparison purposes, the Benjamini and Hochberg method to control the FDR, the Lehmann and Romano procedure to control the tail probability of the proportion of false positives (TPPFP), and the Beta-Binomial SGoF procedure are considered. Three different dependence structures are considered. As usual, for a better understanding of the problem, several practical cases are also studied.

Limited information statistics have been recommended as the goodness-of-fit measures in sparse 2^k contingency tables, but the p-values of these test statistics are computationally difficult to obtain. A Bayesian model diagnostic tool, Relative Entropy-Posterior Predictive Model Checking (RE-PPMC), is proposed to assess the global fit for latent trait models in this paper. This approach utilizes the relative entropy (RE) to resolve possible problems in the original PPMC procedure based on the posterior predictive p-value (PPP-value). Compared with the typical conservatism of PPP-value, the RE value measures the discrepancy effectively. Simulated and real data sets with different item numbers, degree of sparseness, sample sizes, and factor dimensions are studied to investigate the performance of the proposed method. The estimates of univariate information and difficulty parameters are found to be robust with dual characteristics, which produce practical implications for educational testing. Compared with parametric bootstrapping, RE-PPMC is much more capable of evaluating the model adequacy.

In this paper, using simulations, we compare specification procedures for testing the null hypothesis of a Gaussian distribution for the innovations of GARCH models. More precisely, Cramer-von Mises and Kolmogorov-Smirnov type statistics are computed for empirical processes based on the standardized residuals and their squares. For computing P-values, we use the parametric bootstrap method and the multipliers method. We also use the so-called Khmaladze's transform to obtain an approximated Brownian motion under the null hypothesis, for which we compute also Cramer-von Mises and Kolmogorov-Smirnov type statistics, using both the standardized residuals and their squares.

The dependence structure between WTI and Brent crude oil spot log-returns is analysed using modern copula techniques. In a first step, to account for autocorrelation and volatility clustering in the marginals, several single equation models are applied. Second, to select both copulas and tail copulas characterising the joint dynamics between the time series, newly introduced bootstrap-based goodness-of-fit tests are implemented and evaluated. Based on each approach, a comprehensive backtesting is performed by simulating and comparing the risk measures Value-at-Risk and Expected Shortfall with observed values.

The classical spatial scan test, which derived by maximizing the likelihood ratio statistic over a collection of cluster candidates, is widely used in spatial cluster detection. As the likelihood ratio statistic is only a special case in the family of power divergence (PD) goodness-of-fit statistics, the classical spatial scan test is extended to the family of PD spatial scan tests. Therefore, the family of PD spatial scan tests includes not only the classical case but also many other cases. The test statistics, the asymptotic null distribution, and the methods to incorporate overdispersion in the cluster detection are derived. It is found that in the absence of independent variables, the asymptotic null distribution of the PD spatial scan statistics only depends on the ratio of at risks populations and the collection of cluster candidates. Particularly, the focus is on three special cases in the family. They are the deviance, the Pearson, and the Freeman-Tukey spatial scan tests, where the deviance spatial scan test is equivalent to the classical spatial scan test. In simulation studies, it is found that the three test statistics are almost equally powerful for cluster detection.

We propose a frequency domain generalized likelihood ratio test for testing nonstationarity in time series. The test is constructed in the frequency domain by comparing the goodness of fit in the log-periodogram regression under the varying coefficient fractionally exponential models. Under such a locally stationary specification, the proposed test is capable of detecting dynamic changes of short-range and long-range dependences in a regression framework. The asymptotic distribution of the proposed test statistic is known under the null stationarity hypothesis, and its finite sample distribution can be approximated by bootstrap. Numerical results show that the proposed test has good power against a wide range of locally stationary alternatives.

The Generalised Normal Variance-Mean (GNVM) model in which the mixing random variable is Gamma distributed is considered. This model generalises the popular Variance-Gamma (VG) distribution. This GNVM model can be interpreted as the addition of noise to a (skew) VG base. The discussion is based on goodness of fit criteria and on parameter estimation. The conclusion is that the shape of the VG distribution can be adjusted in a favourable way by adding noise.

This paper proposes a new method for choosing regression models which may produce multiple models with sufficient explanatory power and parsimony unlike the traditional model selection procedures that aim at obtaining a single best model. The method ensures interpretability of the resulting models even under strong multicollinearity. The algorithm proceeds in the forward stepwise manner with two requirements for the selected regression models to be fulfilled: goodness of fit and the magnitude of update in loss functions. For the latter criterion, the standardized update is newly introduced, which is closely related with the model selection criteria including the Mallows' C"p, Akaike information criterion and Bayesian information criterion. Simulation studies demonstrate that the proposed algorithm works well with and without strong multicollinearity and even with many explanatory variables. Application to real data is also provided.

The problem of validity of a model on the informativeness of the right-censoring random variable on the inter-event time with recurrent events is considered. The generalized Koziol-Green model for recurrent events has been used in the literature to account for informativeness in the estimation of the gap time distribution or the cumulative hazard rate function. No formal procedure for validating such assumption has been developed for a recurrent failure time data. In this manuscript, we propose procedures for assessing the validity of the assumed model with recurrent events. Our tests are based on the scaled difference of two competing estimators of the cumulative hazard rate possessing nice asymptotic properties. Large sample properties of the proposed procedures are presented. The asymptotic results are applied for the construction of @g^2 and Kolmogorov-Smirnov type tests. Results of a simulation study on Type-I error probabilities and powers are presented. The procedures are also applied to real recurrent event data.

With numerous statistical packages being easily available to conduct the logistic regression analysis, assessment for the goodness-of-fit in the logistic case-control studies becomes more important in practice. While various methods for model checking in conventional case-control studies have been proposed in the literature, methods for checking model adequacy with matched case-control data get relatively less attention. In this study, we propose an omnibus goodness-of-fit test to assess adequacy of the conditional logistic model for matched case-control data. The proposed test can be either constructed based on the discrepancy between two moment estimations or derived to be a score-type test under a general random-effects model. Computation of the proposed test is quite simple in which it does not need to partition the covariate space or to estimate p -value of the test via simulations. The asymptotic null distribution and power calculation of the test are derived under a sequence of alternatives. Empirical type I error rates and powers of the test are performed by simulation studies. An example has been used to illustrate the proposed method as well.

Plotting points of order statistics are often used in the determination of goodness-of-fit of observed data to theoretical percentiles. Plotting points are usually determined by using nonparametric methods which produce, for example, the mean- and median-ranks. Here, we use a distribution-based approach which selects plotting points (quantiles) based on the simultaneous-closeness of order statistics to population quantiles. We show that the plotting points so determined are robust over a multitude of symmetric distributions and then demonstrate their usefulness by examining the power properties of a correlation goodness-of-fit test for normality.

Generalized estimating equations (GEE) were proposed for the analysis of correlated data. They are popular because regression parameters can be consistently estimated even if only the mean structure is correctly specified. GEE have been extended in several ways, including regression diagnostics for outlier detection. However, GEE have rarely been used for analyzing controlled clinical trials. The SB-LOT trial, a double-blind placebo-controlled randomized multicenter trial in which the oedema-protective effect of a vasoactive drug was investigated in patients suffering from chronic insufficiency was re-analyzed using the GEE approach. It is demonstrated that the autoregressive working correlation structure is the most plausible working correlation structure in this study. The effect of the vasoactive drug is a difference in lower leg volume of 2.64 ml per week (p=0.0288, 95% confidence interval 0.27-4.99 ml per week), making a difference of 30 ml at the end of the study. Deletion diagnostics are used for identification of outliers and influential probands. After exclusion of the most influential patients from the analysis, the overall conclusion of the study is not altered. At the same time, the goodness of fit as assessed by half-normal plots increases substantially. In summary, the use of GEE in a longitudinal clinical trial is an alternative to the standard analysis which usually involves only the last follow-up. Both the GEE and the regression diagnostic techniques should accompany the GEE analysis to serve as sensitivity analysis.

A new test for the goodness of fit of parametric forms of the drift and volatility functions of interest rate models is proposed. The test is based on a marked empirical process of the residuals. More specifically, a marked empirical process is constructed using estimators of the integrated regression function and the integrated conditional variance function for the drift function and the volatility function, respectively. Distributions of these processes are approximated using bootstrap techniques. This test is then applied to simulated classical financial models and is illustrated in an empirical application to a EURIBOR data set.

The consequences of model misspecification for multinomial data when using minimum ϕ -divergence or minimum disparity estimators to estimate the model parameters are considered. These estimators are shown to converge to a well-defined limit. Two applications of the results obtained are considered. First, it is proved that the bootstrap consistently estimates the null distribution of certain class of test statistics for model misspecification detection. Second, an application to the model selection test problem is studied. Both applications are illustrated with numerical examples.

In this paper, a data-driven test is proposed to compare two independent or dependent stationary time series, in terms of the second order dynamics. We show that the problem of time series comparison is equivalent to a goodness-of-fit test checking if a constant model is adequate. Using the same framework, the proposed test is easily extended to compare multiple time series and time series of different lengths. Different to previous methods, it is based on generalized score statistics in an estimating equation setting, with some weak and flexible conditions. An extensive simulation study illustrates the validity of the asymptotic result and finite sample properties, using the tapered periodogram. The proposed test is found to perform well for many different situations, including time series with heavy-tailed or skewed innovations. An application to damage detection using vibration data is discussed.

The use of trimming procedures constitutes a natural approach to robustifying statistical methods. This is the case of goodness-of-fit tests based on a distance, which can be modified by choosing trimmed versions of the distributions minimizing that distance. The L"2-Wasserstein distance is used to introduce the trimming methodology for assessing when a data sample can be considered mostly normal. The method can be extended to other location and scale models, introducing a robust approach to model validation, and allows an additional descriptive analysis by determining the subset of the data with the best improved fit to the model. This is a consequence of the use of data-driven trimming methods instead of the more classical symmetric trimming procedures.

Longitudinal studies involving categorical responses are extensively applied in many fields of research and are often fitted by the generalized estimating equations (GEE) approach and generalized linear mixed models (GLMMs). The assessment of model fit is an important issue for model inference. The purpose of this article is to extend Pan's (2002a) goodness-of-fit tests for GEE models with longitudinal binary data to the tests for logistic proportional odds models with longitudinal ordinal data. Two proposed methods based on Pearson chi-squared test and unweighted sum of residual squares are developed, and the approximate expectations and variances of the test statistics are easily computed. Four major variants of working correlation structures, independent, AR(1), exchangeable and unspecified, are considered to estimate the variances of the proposed test statistics. Simulation studies in terms of type I error rate and the power performance of the proposed tests are presented for various sample sizes. Furthermore, the approaches are demonstrated by two real data sets.

A goodness of fit test for the Pareto distribution, when the observations are subjected to Type II right censoring is proposed. The test statistic involves transformations of the original data and is based on the nonparametric Nelson-Aalen estimator of the cumulative hazard function. By Monte Carlo simulation, the empirical distribution of the test statistic is obtained and the power of the test is investigated for some alternative distributions. The power is compared with adaptations for Type II censored data of the Cramer-von Mises and Anderson-Darling tests, and a test based on Kullback-Leibler information. For some alternative distributions with monotone decreasing hazard function, the proposed test has higher power. The methodology is illustrated by reanalyzing two published data sets.

The likelihood approach based on the empirical distribution functions is a well-accepted statistical tool for testing. However, the proof schemes of the Neyman-Pearson type lemmas induce consideration of density-based likelihood ratios to obtain powerful test statistics. In this article, we introduce the distribution-free density-based likelihood technique, applied to test for goodness-of-fit. We focus on tests for normality and uniformity, which are common tasks in applied studies. The well-known goodness-of-fit tests based on sample entropy are shown to be a product of the proposed empirical likelihood (EL) methodology. Although the efficiency of test statistics based on classes of entropy estimators has been widely addressed in the statistical literature, estimation of the sample entropy has been not invariantly defined, and hence this estimation produces tests that are difficult to be applied to real data studies. The proposed EL approach defines clear forms of the entropy-based tests. Monte Carlo simulation results confirm the preference of the proposed method from a power perspective. Real data examples study the proposed approach in practice.

A class of goodness-of-fit tests based on the empirical characteristic function is studied. They can be applied to continuous as well as to discrete or mixed data with any arbitrary fixed dimension. The tests are consistent against any fixed alternative for suitable choices of the weight function involved in the definition of the test statistic. The bootstrap can be employed to estimate consistently the null distribution of the test statistic. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation.

This paper proposes a bootstrap goodness of fit test for the Generalized Pareto distribution (GPd) with shape parameter @c. The proposed test is an intersection-union test which tests separately the cases of @c>=0 and @c

Generalizing the Cramer-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L"p-norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L"p-norms should receive more attention. It is shown that, given a distribution function F"0 and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.

The Pearson's chi-squared statistic (X^2) does not in general follow a chi-square distribution when it is used for goodness-of-fit testing for a multinomial distribution based on sparse contingency table data. We explore properties of [Zelterman, D., 1987. Goodness-of-fit tests for large sparse multinomial distributions. J. Amer. Statist. Assoc. 82 (398), 624-629] D^2 statistic and compare them with those of X^2 and compare the power of goodness-of-fit test among the tests using D^2, X^2, and the statistic (L"r) which is proposed by [Maydeu-Olivares, A., Joe, H., 2005. Limited- and full-information estimation and goodness-of-fit testing in 2^n contingency tables: A unified framework. J. Amer. Statist. Assoc. 100 (471), 1009-1020] when the given contingency table is very sparse. We show that the variance of D^2 is not larger than the variance of X^2 under null hypotheses where all the cell probabilities are positive, that the distribution of D^2 becomes more skewed as the multinomial distribution becomes more asymmetric and sparse, and that, as for the L"r statistic, the power of the goodness-of-fit testing depends on the models which are selected for the testing. A simulation experiment strongly recommends to use both D^2 and L"r for goodness-of-fit testing with large sparse contingency table data.

In this paper we derive new tests for goodness of fit based on the fiducial empirical distribution function (EDF) after the probability integral transformation of the sample. Note that the fiducial EDF for a set of given sample observations is a randomized distribution function. By substituting the fiducial EDF for the classical EDF in the Kolmogorov-Smirnov, Cramer-von Mises statistics and so forth, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. It emerges from Monte Carlo simulations that in most cases there exist some of the new tests having better power properties than the corresponding tests based on the classical EDF and Pyke's modified EDF.

Semi-Markov processes have gained popularity as multistate models of disease progression, but have rarely been used in randomised clinical trials (RCTs) when many endpoints are collected over time. Reasons for this include the additional complexity created by censoring, the complexity of the model usually characterised by transition intensities, and difficulties in estimating clinically relevant quantities. In this paper, we show that an indirect approach can address these issues, while accommodating censored data and permitting accurate estimation of survivor functions, hazard ratio and net reductions in risk. The technique combines saddlepoint approximation and likelihood theory to fit a semi-Markov model, specified in terms of its transition probabilities and moment-generating functions (MGFs). A key feature of this approach is that the proportional hazard (PH) assumption commonly assumed for all transition time distributions is no longer needed. Inference for clinically relevant quantities is obtained through bootstrapping or randomisation tests; a simple goodness of fit procedure exploiting the link with the cumulative incidence function for competing risks is also introduced. As illustration, an illness-death model (with transition times between randomisation, non-fatal stroke, and death) for a cardiovascular trial (LIPID) is provided. Evidence of a cumulative benefit of continued treatment, not identified by standard analysis methods in earlier published work, is presented. The technique is flexible enough to be applied extensively in clinical trials.

The generalized estimating equations (GEE) approach has been widely used to analyze repeated measures data. However, in the absence of likelihood ratio tests, model diagnostic checking tools are not well established for the GEE approach, whereas they are for other likelihood-based approaches. Diagnostic checking tools are essential for determining a model's goodness of fit, especially for non-normal data. In this paper, we propose simple residual plots to investigate the goodness of fit of the model based on the GEE approach for discrete data. The proposed residual plots are based on the quantile-quantile (Q-Q) plots of a @g^2-distribution, and are particularly useful for comparing several models simultaneously.

Tests for the Laplace distribution based on the sample skewness and kurtosis coefficients are shown to be related to components of smooth tests of goodness of fit and are compared with a number of tests including the Anderson-Darling test, a new data-driven smooth test, a new empirical characteristic function based test and a new maximum entropy test. This last would be our slight preference as the test of choice for testing for the Laplace distribution.

In areas such as biology, geology and meteorology data often occur as angles. This paper examines the use of Bayesian mixtures of triangular distributions for the semiparametric analysis of circular data. Applications to density estimation, goodness-of-fit testing and semiparametric regression are demonstrated.

This paper considers a class of distributions arising from the difference of two discrete random variables belonging to the Panjer family of distributions. Some distributional properties and computation of probabilities are discussed. Goodness of fit and tests of hypotheses involving the likelihood ratio, score and Wald tests have been considered. As an illustration, an application to paired count data is given.

Statistical models are often based on normal distributions and procedures for testing such distributional assumption are needed. Many goodness-of-fit tests are available. However, most of them are quite insensitive in detecting non-normality when the alternative distribution is symmetric. On the other hand all the procedures are quite powerful against skewed alternatives. A new test for normality based on a polynomial regression is presented. It is very effective in detecting non-normality when the alternative distribution is symmetric. A comparison between well known tests and this new procedure is performed by simulation study. Other properties are also investigated.

In this paper, we consider a family of generalized Birnbaum-Saunders distributions and present a lifetime analysis based mainly on the hazard function of this model. In addition, we carry out maximum likelihood estimation by using an iterative algorithm, which produces robust estimates. Asymptotic inference is also presented. Next, the quality of the estimation method is examined by means of Monte Carlo simulations. We then provide a practical example to illustrate the obtained results. From this example and based on goodness-of-fit methods, we show that the GBS distribution results in a more appropriate model for modeling fatigue data than other models commonly used to model this type of data. Finally, we estimate the hazard function and the critical point of this function.

In this article, statistical inference and prediction analyses for the Weibull process with incomplete observations via classical approach are studied. Specifically, observations in the early developmental phase of a testing program cannot be observed. We derive the closed-form expressions for the maximum likelihood estimates of the parameters in both the failure- and time-truncated Weibull processes. Confidence interval and hypothesis testing for the parameters of interest are considered. In addition, predictive inferences on future failures and the goodness-of-fit test of the model are developed. Two real examples from an engine system development study and a Boeing air-conditioning system development study are presented to illustrate the proposed methodologies.

When continuous predictors are present, classical Pearson and deviance goodness-of-fit tests to assess logistic model fit break down. The Hosmer-Lemeshow test can be used in these situations. While simple to perform and widely used, it does not have desirable power in many cases and provides no further information on the source of any detectable lack of fit. Tsiatis proposed a score statistic to test for covariate regional effects. While conceptually elegant, its lack of a general rule for how to partition the covariate space has, to a certain degree, limited its popularity. We propose a new method for goodness-of-fit testing that uses a very general partitioning strategy (clustering) in the covariate space and either a Pearson statistic or a score statistic. Properties of the proposed statistics are discussed, and a simulation study demonstrates increased power to detect model misspecification in a variety of settings. An application of these different methods on data from a clinical trial illustrates their use. Discussions on further improvement of the proposed tests and extending this new method to other data situations, such as ordinal response regression models are also included.

A number of methods have been proposed to estimate the period of a variable star; e.g., a recent approach uses smoothing spline regression to fit tentative periodic functions (light curves) and selects the period minimizing a robust goodness-of-fit criterion. These methods assume that measurement errors vary independently over time. Empirical evidence, however, indicates substantial temporal dependence, possibly related to changes in observing conditions. Dependence complicates the period analysis in several respects: selection of a ''best'' period among several local optima, estimation of the light curve, and evaluation of uncertainty about period and light curve estimates. This article presents methods designed to accommodate dependent errors. An analysis of several data sets shows that the proposed approach can produce substantially different and arguably better results compared with other methods.

We develop a series of Bayesian statistical models for estimating survival of wild populations monitored using capture-recapture experiments and photoidentification data. The proposed methodology is based on Cormack-Jolly-Seber model [Cormack, R.M., 1964. Estimates of survival from the sighting of marked animals. Biometrika 51, 429-438; Jolly, G.M., 1965. Explicit estimates from capture-recapture data with both death and immigration-stochastic model. Biometrika 52, 225-247; Seber, G.A.F., 1965. A note on the multiple recapture census. Biometrika 52, 249-259]. Besides time effects in capture probabilities, the proposed models allow taking into account heterogeneity in capture probability caused by the existence of different groups of individuals in the population. For that purpose, the capture probabilities are fitted using a logistic model. Additionally, it is also possible to estimate group-specific survival rates. Goodness of fit is evaluated using Bayes factor methodology. The models are applied to an 11-year photoidentification capture-recapture experiment for bowhead whales, Balaena mysticetus. The best model provides an estimate close to the one obtained by Zeh et al. [2002. Survival of bowhead whales, Balaena mysticetus, estimated from 1981-1998 photoidentification data. Biometrics 58, 832-840] using the Jolly-Seber model, but accounting for heterogeneity in capture probabilities improves precision.

An extension of the Shapiro-Wilk test to verify the hypothesis of normality in the presence of nuisance regression and scale has been previously considered. Such a test is typically based on the pair of the maximum likelihood and BLUE estimators of the standard deviation in the linear regression model. It has been shown that the asymptotic null distribution of the test criterion, extended to the regression model, is equivalent to that of the original Shapiro-Wilk test for the location-scale model. A simulation study is performed in order to show that both criteria are close under the normality hypothesis for moderate as well for large data sets. The power of the test against various alternative distributions of the model errors is illustrated. Furthermore, it is shown that the probabilities of errors of both the first and second kinds do not depend on the design matrix or on the parameters of the linear model.

The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. Simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process are presented.

Methods of regression diagnostics for functional regression models are developed which relate a functional response to predictor variables that can be multivariate vectors or random functions. For this purpose, a residual process is defined by subtracting the predicted from the observed response functions. This residual process is expanded into functional principal components (FPC), and the corresponding FPC scores are used as natural proxies for the residuals in functional regression models. For the case of a univariate covariate, a randomization test is proposed based on these scores to examine if the residual process depends on the covariate. If this is the case, it indicates lack of fit of the model. Graphical methods based on the FPC scores of observed and fitted functions can be used to complement more formal tests. The methods are illustrated with data from a recent study of Drosophila fruit flies regarding life-cycle gene expression trajectories as well as functional data from a dose-response experiment for Mediterranean fruit flies (Ceratitis capitata).

Logistic regression models are frequently used in epidemiological studies for estimating associations that demographic, behavioral, and risk factor variables have on a dichotomous outcome, such as disease being present versus absent. After the coefficients in a logistic regression model have been estimated, goodness-of-fit of the resulting model should be examined, particularly if the purpose of the model is to estimate probabilities of event occurrences. While various goodness-of-fit tests have been proposed, the properties of these tests have been studied under the assumption that observations selected were independent and identically distributed. Increasingly, epidemiologists are using large-scale sample survey data when fitting logistic regression models, such as the National Health Interview Survey or the National Health and Nutrition Examination Survey. Unfortunately, for such situations no goodness-of-fit testing procedures have been developed or implemented in available software. To address this problem, goodness-of-fit tests for logistic regression models when data are collected using complex sampling designs are proposed. Properties of the proposed tests were examined using extensive simulation studies and results were compared to traditional goodness-of-fit tests. A Stata ado function svylogitgof for estimating the F-adjusted mean residual test after svylogit fit is available at the author's website http://www.people.vcu.edu/~kjarcher/Research/Data.htm.

We consider testing for an exponential distribution with unspecified rate parameter when it is only possible to observe the counts in groups with boundaries specified before sighting the data. On the basis of a size and power study we recommend that tests of fit for the exponential distribution be based on the Anderson-Darling statistic and the SW2 statistic recommended by Gulati and Neus [(2001). Goodness-of-fit statistics for the exponential distribution when the data are grouped. In: Huber-Carol, C., Balakrishnan, N., Nikulin, M.S., Mesbah, M., (Eds.), Goodness-of-Fit Tests and Validity of Models. Birkhauser, Boston, pp. 113-123 (Chapter 9)]. We also suggest that inference based on one of these be complemented by examination of the components of the X^2 statistic. We illustrate how to use these components to give improved models.

A goodness of fit test for copulas based on Rosenblatt's transformation is investigated. This test performs well if the marginal distribution functions are known and are used in the test statistic. If the marginal distribution functions are unknown and are replaced by their empirical estimates, then the test's properties change significantly. This is shown in detail by simulation for special cases. A bootstrap version of the test is suggested and it is shown by simulation that it performs well. An empirical application of this test to daily returns of German assets reveals that a Gaussian copula is unsuitable to describe their dependence structure. A t"@n-copula with low degrees of freedom such as @n=4 or 5 fits the data in some cases.

This paper presents a two-step procedure using the method of moment or percentile to find initial values and then maximize the numerical log likelihood to fit the appropriate generalized lambda distribution to data. This paper demonstrates the use of this procedure to fit well-known statistical distributions as well as some empirical data. Overall, the use of numerical maximum log likelihood estimation is a valuable alternative among existing methods of fitting. It provides not only convincing results in terms of quantile plots and goodness of fit tests but also has the advantage of a lower variability in its parameter estimation compared to the existing starship (King and MacGillivray, 1999) and method of moment (Karian and Dudewicz, 2000) fitting schemes.

We develop a goodness-of-fit measure with desirable properties for use in the hierarchical logistic regression setting. The statistic is an unweighted sum of squares (USS) of the kernel smoothed model residuals. We develop expressions for the moments of this statistic and create a standardized statistic with hypothesized asymptotic standard normal distribution under the null hypothesis that the model is correctly specified. Extensive simulation studies demonstrate satisfactory adherence to Type I error rates of the Kernel smoothed USS statistic in a variety of likely data settings. Finally, we discuss issues of bandwidth selection for using our proposed statistic in practice and illustrate its use in an example.

The traditional regression analysis is usually applied to homogeneous observations. However, there are several real situations where the observations are not homogeneous. In these cases, by utilizing the traditional regression, we have a loss of performance in fitting terms. Then, for improving the goodness of fit, it is more suitable to apply the so-called clusterwise regression analysis. The aim of clusterwise linear regression analysis is to embed the techniques of clustering into regression analysis. In this way, the clustering methods are utilized for overcoming the heterogeneity problem in regression analysis. Furthermore, by integrating cluster analysis into the regression framework, the regression parameters (regression analysis) and membership degrees (cluster analysis) can be estimated simultaneously by optimizing one single objective function. In this paper the clusterwise linear regression has been analyzed in a fuzzy framework. In particular, a fuzzy clusterwise linear regression model (FCWLR model) with symmetrical fuzzy output and crisp input variables for performing fuzzy cluster analysis within a fuzzy linear regression framework is suggested. For measuring the goodness of fit of the suggested FCWLR model with fuzzy output, a fitting index is proposed. In order to illustrate the usefulness of FCWLR model in practice, several applications to artificial and real datasets are shown.

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.