In statistical analysis, distributional assumptions are needed to apply parametric procedures. Assumptions about underlying distribution should be true for accurate statistical inferences. Goodness-of-fit tests are used for checking the validity of the distributional assumptions. To apply some of the goodness-of-fit tests, the unknown population parameters are estimated. The null distributions of test statistics become complicated or depend on the unknown parameters if population parameters are replaced by their estimators. This will restrict the use of the test. Goodness-of-fit statistics which are invariant to parameters can be used if the distribution under null hypothesis is a location-scale distribution. For location and scale invariant goodness-of-fit tests, there is no need to estimate the unknown population parameters. However, approximations are used in some of those tests. Different types of estimation and approximation techniques are used in this study to compute goodness-of-fit statistics for complete and censored samples from univariate distributions as well as complete samples from bivariate normal distribution. Simulated power properties of the goodness-of-fit tests against a broad range of skew and symmetric alternative distributions are examined to identify the estimation effects in goodness-of-fit tests. The main aim of this thesis is to modify goodness-of-fit tests by using different estimators or approximation techniques, and finally see the effect of estimation on the power of these tests.

Model credibility index is defined to be a sample size under which the power of rejection equals 0.5. It applies goodness-of-fit testing thinking and uses a one-number summary statistic as an assessment tool in a false model world. The estimation of the model credibility index involves a bootstrap resampling technique. To assess the consistency of the estimator of model credibility index, we instead study the variance of the power achieved at a fixed sample size. An improved subsampling method is proposed to obtain an unbiased estimator of the variance of power. We present two examples to interpret the mechanics of building model credibility index and estimate its error in model selection. One example is two-way independent model by Pearson Chi-square test, and another example is multi-dimensional logistic regression model using likelihood ratio test.

The thesis is devoted to the application of a new class of probability metrics, N-distances, introduced by Klebanov (Klebanov, 2005; Zinger et al., 1989), to the problems of verification of the classical statistical hypotheses of goodness of fit, homogeneity, symmetry and independence. First of all a construction of statistics based on N metrics for testing mentioned hypotheses is proposed. Then the problem of determination of the critical region of the criteria is investigated. The main results of the thesis are connected with the asymptotic behavior of test statistics under the null and alternative hypotheses. In general case the limit null distribution of proposed in the thesis tests statistics is established in terms of the distribution of infinite quadratic form of random normal variables with coefficients dependent on eigenvalues and functions of a certain integral operator. It is proved that under the alternative hypothesis the test statistics are asymptotically normal. In case of parametric hypothesis of goodness of fit particular attention is devoted to normality and exponentiality criteria. For hypothesis of homogeneity a construction of multivariate distribution free two-sample test is proposed. Testing the hypothesis of uniformity on hypersphere in more detail S1 and S2 cases are investigated. In conclusion, a comparison of N-distance tests with some classical criteria is provided. For simple hypothesis of goodness of fit in univariate case as a measure for... [to full text] / Disertacinis darbas yra skirtas N-metrikų teorijos (Klebanov, 2005; Zinger et al., 1989) pritaikymui klasikinėms statistinėms suderinamumo, homogeniškumo, simetriškumo bei nepriklausomumo hipotezėms tikrinti. Darbo pradžioje pasiūlytas minėtų hipotezių testinių statistikų konstravimo būdas, naudojant N-metrikas. Toliau nagrinėjama problema susijusi su suformuotų kriterijų kritinės srities nustatymu. Pagrindiniai darbo rezultatai yra susiję su pasiūlytų kriterijaus statistikų asimptotiniu skirstiniu. Bendru atveju N-metrikos statistikų asimptotinis skirstinys esant nulinei hipotezei sutampa su Gauso atsitiktinių dydžių begalinės kvadratinės formos skirstiniu. Alternatyvos atveju testinių statistikų ribinis skirstinys yra normalusis. Sudėtinės suderinamumo hipotezės atveju išsamiau yra analizuojami normalumo ir ekponentiškumo kriterijai. Daugiamačiu atveju pasiūlyta konstrukcija, nepriklausanti nuo skirstinio homogeniškumo testo. Tikrinant tolygumo hipersferoje hipotezę detaliau yra nagrinėjami apskritimo ir sferos atvejai. Darbo pabaigoje lyginami pasiūlytos N-metrikos bei kai kurie klasikiniai kriterijai. Neparametrinės suderinamumo hipotezės vienamačiu atveju, kaip palyginimo priemonė, nagrinėjamas Bahaduro asimptotinis santykinis efektyvumas (Bahadur, 1960; Nikitin, 1995). Kartu su teoriniais rezultatais pasiūlytų N-metrikos tipo testų galingumas ištirtas, naudojant Monte-Karlo metodą. Be paprastos ir sudėtinės suderinamumo hipotezių yra analizuojami homogeniškumo testai... [toliau žr. visą tekstą]

Disertacinis darbas yra skirtas N-metrikų teorijos (Klebanov, 2005; Zinger et al., 1989) pritaikymui klasikinėms statistinėms suderinamumo, homogeniškumo, simetriškumo bei nepriklausomumo hipotezėms tikrinti. Darbo pradžioje pasiūlytas minėtų hipotezių testinių statistikų konstravimo būdas, naudojant N-metrikas. Toliau nagrinėjama problema susijusi su suformuotų kriterijų kritinės srities nustatymu. Pagrindiniai darbo rezultatai yra susiję su pasiūlytų kriterijaus statistikų asimptotiniu skirstiniu. Bendru atveju N-metrikos statistikų asimptotinis skirstinys esant nulinei hipotezei sutampa su Gauso atsitiktinių dydžių begalinės kvadratinės formos skirstiniu. Alternatyvos atveju testinių statistikų ribinis skirstinys yra normalusis. Sudėtinės suderinamumo hipotezės atveju išsamiau yra analizuojami normalumo ir ekponentiškumo kriterijai. Daugiamačiu atveju pasiūlyta konstrukcija, nepriklausanti nuo skirstinio homogeniškumo testo. Tikrinant tolygumo hipersferoje hipotezę detaliau yra nagrinėjami apskritimo ir sferos atvejai. Darbo pabaigoje lyginami pasiūlytos N-metrikos bei kai kurie klasikiniai kriterijai. Neparametrinės suderinamumo hipotezės vienamačiu atveju, kaip palyginimo priemonė, nagrinėjamas Bahaduro asimptotinis santykinis efektyvumas (Bahadur, 1960; Nikitin, 1995). Kartu su teoriniais rezultatais pasiūlytų N-metrikos tipo testų galingumas ištirtas, naudojant Monte-Karlo metodą. Be paprastos ir sudėtinės suderinamumo hipotezių yra analizuojami homogeniškumo testai... [toliau žr. visą tekstą] / The thesis is devoted to the application of a new class of probability metrics, N-distances, introduced by Klebanov (Klebanov, 2005; Zinger et al., 1989), to the problems of verification of the classical statistical hypotheses of goodness of fit, homogeneity, symmetry and independence. First of all a construction of statistics based on N-metrics for testing mentioned hypotheses is proposed. Then the problem of determination of the critical region of the criteria is investigated. The main results of the thesis are connected with the asymptotic behavior of test statistics under the null and alternative hypotheses. In general case the limit null distribution of proposed in the thesis tests statistics is established in terms of the distribution of infinite quadratic form of random normal variables with coefficients dependent on eigenvalues and functions of a certain integral operator. It is proved that under the alternative hypothesis the test statistics are asymptotically normal. In case of parametric hypothesis of goodness of fit particular attention is devoted to normality and exponentiality criteria. For hypothesis of homogeneity a construction of multivariate distribution-free two-sample test is proposed. Testing the hypothesis of uniformity on hypersphere in more detail S1 and S2 cases are investigated. In conclusion, a comparison of N-distance tests with some classical criteria is provided. For simple hypothesis of goodness of fit in univariate case as a measure for... [to full text]

The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general estimation method for continuous univariate distributions. The MSP method, which is closely related to the maximum likelihood (ML) method, can be derived from an approximation based on simple spacings of the Kullback-Leibler information. It is known to give consistent and asymptotically efficient estimates under general conditions and works also in situations where the ML method fails, e.g. for the three parameter Weibull model. In this thesis it is proved under general conditions that MSP estimates of parameters in the Euclidian metric are strongly consistent. The ideas behind the MSP method are extended and a class of estimation methods is introduced. These methods, called generalized MSP methods, are derived from approxima­tions based on sum-functions of rath order spacings of certain information mea­sures, i.e. the ^-divergences introduced by Csiszår (1963). It is shown under general conditions that generalized MSP methods give consistent estimates. In particular, it is proved that generalized MSP methods give L1 consistent esti­mates in any family of distributions with unimodal densities, without any further conditions on the distributions. Other properties such as distributional robust­ness are also discussed. Several limit theorems for sum-functions of rath order spacings are given, for ra fixed as well as for the case when ra is allowed to in­crease to infinity with the sample size. These results provide a strongly consistent nonparametric estimator of entropy, as well as a characterization of the uniform distribution. Further, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives. / digitalisering@umu

This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a Lévy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noisy behavior of the data and a given reference jump measure in terms of so-called coupling distances. After a short introduction to Lévy processes and coupling distances we recall basic statistical approximation results and derive rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study to simulated and paleoclimate data. It indicates the dominant presence of a non-stable heavy-tailed jump Lévy component for some tail index greater than 2.

In this study, the most prominent goodness-of-fit tests for censored samples are reviewed. Power properties of goodness-of-fit statistics of the null hypothesis that a sample which is censored from right, left and both right and left which comes from uniform, normal and exponential distributions are investigated. Then, by a similar argument extreme value, student t with 6 degrees of freedom and generalized logistic distributions are discussed in detail through a comprehensive simulation study. A variety of real life applications are given. Suitable test statistics for testing the above distributions for censored samples are also suggested in the conclusion.

Thesis (M.S.)--Ohio University, August, 2004. / Includes bibliographical references (p. 111-120).

Thesis (M.S.)--Ohio University, August, 2004. / Title from PDF t.p. Includes bibliographical references (p. 111-120)

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Includes bibliographical references (p. 263-266).