Statistički problemi ocenjivanja količnika disperzija i visokih kvantila raspodela

Abstract

Glavni sadr·zaj doktorske teze odnosi se na predlog novih, transformisanih in- tervala poverenja za koli·cnik disperzija dva uzorka. Naime, u literaturi su do sada predlo·zene metode bazirane na F statistici. Med¹utim, nedostatak raz- matranih intervala jeste velika osetljivost u odnosu na pretpostavke o para- metrima raspodele. Predlo·zena statistika u literaturi mo·ze biti modi¯kovana. U tezi je nad¹en Edgeworthov razvoj t-statistike i na bazi toga su upored¹ivani intervali. Takod¹e je ukazano na osnovu simulacija, da interval baziran na Johnsonovoj transformaciji daje bolji rezultat u smislu verovatno¶ce pokri- vanja u odnosu na F interval i interval baziran na Hallovoj transforma- ciji. U radu je posve¶cena pa·znja i ve¶c poznatim intervalima poverenja za matemati·cko o·cekivanje i disperziju za problem jednog uzorka, kao i za problem dva uzorka. Posebno je razmatran problem razlike proporcija dva uzorka, sa numeri·ckim rezultatima i podacima iz oblasti osiguranja. Pored pomenutog, u tezi su prikazane postoje¶ce metode za ocenu indeksa ek- stremne vrednosti i visokih kvantila. Posebno je razmatrana direktno simuli- rana ocena kvantila i verovatno¶ca pokrivanja njenog odstupanja od ta·cne vrednosti, za slu·caj Pareto i Gama raspodele, kao i uop·steni slu·caj uop·stene Pareto raspodele. Rezultati su dobijeni na osnovu teorijskih rezultata teorije velikog odstupanja i dato je njeno uop·stenje na topolo·skim prostorima. Uz teoriju verovatno¶ce i elementarnih principa klasi·cne analize, u istra·zivanju su kori·s¶cene i metode teorijske statistike i statisti·ckog zaklju·civanja.

The main goal of the thesis is the development of a new suggested trans- form con¯dence intervals for the ratio of the variances of the two samples. Since now, the methods based on the F statistic have been suggested in the literature. However, the defect of that intervals is the huge sensitivity in re- lation with assumption of parameters distribution. Suggested statistic could be modi¯ed. Edgeworth expansion of the t-statistic has found the place in the thesis and based on that intervals have been compared. Also, on the base of the simulation it was point out that Johnsons transformation give better result in the sense of probability covering in regard to F interval and interval based on Halls transformation. Moreover, the con¯dence intervals for the mean and variances for the one and two sample problems have been considered in the dissertation. Especially, the problem of the di®erence of the proportions for the two samples, with the numerical results and data from the insurance. In addition, the existing methods for the estimation of the extreme value index and the high quantiles have been reviewed. Particularly, the direct simulation estimation of the quantile and probability covering of its deviation from the rights value, for Pareto and Gamma distributions, and also for general Pareto distribution have been discussed. The results were obtained by large deviation theory and their generalization on the topological spaces is stated. In this research, beside the probability theory and elemen- tary principles of the classical analysis, methods of the statistical theory and statistical conclusions have been applied.