The ranking method used for testing the equivalence of two distributions has been studied for decades and is widely adopted for its simplicity. However, due to the complexity of calculations, the power of the test is either estimated by normal approximation or found when an appropriate alternative is given. Here, via a Finite Markov chain imbedding (FMCI) technique, we are able to establish the marginal and joint distributions of the rank statistics considering the shift and scale parameters, respectively and simultaneously, under two continuous distribution functions. Furthermore, the procedures of distribution equivalence tests and their power functions are discussed. Numerical results of a joint distribution of two rank statistics under the standard normal distribution and the powers for a sequence of alternative normal distributions with mean from -20 to 20 and standard deviation from 1 to 9 and their reciprocals are presented. In addition, we discuss the powers of the rank statistics under the Lehmann alternatives. Wallenstein et. al. (1993, 1994) discussed power via combinatorial calculations for the scan statistic against a pulse alternative; however, unless certain proper conditions are given, computational difficulties exist. Our work extends their results and provides an alternative way to obtain the distribution of a scan statistic under various alternative conditions. An efficient and intuitive expression for the distribution as well as the power of the scan statistic are introduced via the FMCI. The numerical results of the exact power for a discrete scan statistic against various conditions are presented. Powers through the finite Markov chain imbedding method and a combinatorial algorithm for a continuous scan statistic against a pulse alternative of a higher risk for a disease on a specified subinterval time are also discussed and compared.

The purpose of this thesis is to investigate how the differences between the study and policy sites impact the performance of benefit function transfer. For this purpose, simulated data are created where all information necessary to conduct the benefit function transfer is available. We consider the six cases of difference between the study and policy sites- scale parameter, substitution possibilities, observable characteristics, population preferences, measurement error in variables, and a case of preference heterogeneity at the study site and fixed preferences at the policy site. These cases of difference were considered one at time and their impact on quality of transfer is investigated. RUM model based on reveled preference was used for this analysis. Function estimated at the study site is transferred to the policy site and willingness to pay for five different cases of policy changes are calculated at the study site. The willingness to pay so calculated is compared with true willingness to pay to evaluate the performance of benefit function transfer. When the study and policy site are different only in terms of scale parameter, equality of estimated and true expected WTP is not rejected for 89.7% or more when the sample size is 1000. Similarly, equality of estimated preference coefficients and true preference coefficients is not rejected for 88.8% or more. In this study, we find that benefit transfer performs better only in one direction. When the function is estimated at lower scale and transferred to the policy site with higher scale, the transfer error is less in magnitude than those which are estimated at higher scale and transferred to the policy site with lower scale. This study also finds that transfer error is less when the function from the study site having more site substitutes is transferred to the policy site having less site substitutes whenever there is difference in site substitution possibilities. Transfer error is magnified when measurement error is involved in any of the variables. This study do not suggest function transfer whenever the study site's model is missing one of the important variable at the policy site or whenever the data on variables included in study site's model is not available at the policy site for benefit transfer application. This study also suggests the use of large representative sample with sufficient variation to minimize transfer error in benefit transfer. / Master of Science

Η παρούσα διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής Θεωρίας Αποφάσεων και ειδικότερα στην (σημειακή) εκτίμηση παραμέτρου κλίμακας. Το κλασικό αποτέλεσμα του Strawderman [1974, Ann. Statist., 2, 190–198] για την εκτίμηση της διασποράς κανονικής κατανομής επεκτείνεται σε κατανομές με παράμετρο κλίμακας και μία άλλη άγνωστη («ενοχλητική») παράμετρο για την εκτίμηση της παραμέτρου κλίμακας και του αντιστρόφου της παραμέτρου κλίμακας ως προς την τετραγωνική συνάρτηση ζημίας και τη συνάρτηση ζημίας εντροπίας. Η μέθοδος απόδειξης των αποτελεσμάτων, παρά το γεγονός ότι διατηρεί το «σκελετό» της μεθόδου του Strawderman [1974, Ann. Statist., 2, 190–198], διαφέρει (αναπόφευκτα) τεχνικά από αυτήν επειδή ο Strawderman [1974, Ann. Statist., 2, 190–198] βασίζεται σε ειδικά χαρακτηριστικά της κανονικής κατανομής. Η εφαρμογή αυτών των γενικών αποτελεσμάτων στην εκθετική κατανομή παρέχει νέες ικανές συνθήκες – δηλαδή, διαφορετικές από τις υπάρχουσες στη βιβλιογραφία – για τη βελτίωση των αντίστοιχων καλύτερων αναλλοίωτων ως προς μετασχηματισμούς θέσης-κλίμακας εκτιμητών. Επίσης, κατασκευάζονται νέες κλάσεις εκτιμητών που ικανοποιούν τις νέες συνθήκες. Πέραν της δικής τους αξίας, τα παραπάνω αποτελέσματα είναι χρήσιμα (ουσιαστικά, απαραίτητα) για την κατασκευή εκτιμητών τύπου Strawderman [1974, Ann. Statist., 2, 190–198] για το λόγο των παραμέτρων κλίμακας δύο ανεξάρτητων πληθυσμών. Συγκεκριμένα, κατασκευάζονται νέες κλάσεις εκτιμητών, τύπου Strawderman [1974, Ann. Statist., 2, 190–198], για το λόγο των διασπορών δύο κανονικών κατανομών καθώς και το λόγο των παραμέτρων κλίμακας δύο εκθετικών κατανομών ως προς την τετραγωνική συνάρτηση ζημίας και τη συνάρτηση ζημίας εντροπίας. Η μέθοδος της απόδειξης δεν είναι η τυπική για αυτού του είδους τα προβλήματα, η οποία απαιτεί την επέκταση αποτελεσμάτων από έναν πληθυσμό σε δύο πληθυσμούς. Αντιθέτως, εφαρμόζεται η μεθοδολογία των Iliopoulos and Kourouklis [1999, J. Multivariate Anal., 68, 176-192] που ανάγει το πρόβλημα εκτίμησης του λόγου των παραμέτρων κλίμακας σ2/σ1 σε δύο προβλήματα ενός πληθυσμού, ένα αυτό της εκτίμησης του σ2 και, το άλλο, αυτό της εκτίμησης του 1/σ1. / This PhD thesis deals with the study of the problem of point estimation of a scale parameter from the decision theoretic point of view. Strawderman’s [1974. Ann. Statist., 2, 190–198] result for estimating the variance of a normal distribution is extended to estimating a general scale parameter and the reciprocal of a general scale parameter in the presence of a nuisance parameter under both quadratic and entropy losses. The method of proof for these results, although it retains the "skeleton" of Strawderman’s [1974. Ann. Statist., 2, 190–198] method, differs (inevitably) technically from that since Strawderman [1974. Ann. Statist., 2, 190–198] relies on special features of the normal distribution. Application of these general results to the exponential distribution gives new sufficient conditions, i.e., different from those available in the literature, for improving upon the respective best affine equivariant estimators. Furthermore, new classes of estimators satisfying the above conditions are constructed. Apart from their own value, these results are also useful (essentially, necessary) for the construction of Strawderman [1974. Ann. Statist., 2, 190–198]-type estimators for the ratio of scale parameters of two independent populations. Specifically, new classes of improved Strawderman [1974. Ann. Statist., 2, 190–198]-type estimators for the ratio of the variances of two normal distributions as well as the ratio of the scale parameters of two exponential distributions are constructed under both quadratic and entropy losses. The method of proof is not the typical one for this kind of problem which requires a two-sample extension of respective one-sample arguments. In contrast, the methodology of Iliopoulos and Kourouklis [1999, J. Multivariate Anal., 68, 176-192] is employed which reduces the two-sample problem of estimating the ratio of scale parameters σ2/σ1 to two one-sample problems, namely, one of estimating σ2 and another of estimating 1/σ1.