Bitte beziehen Sie sich beim Zitieren dieses Dokumentes immer auf folgende
URL: http://geb.uni-giessen.de/geb/volltexte/2010/7607/
Location-scale distributions : Linear estimation and probability plotting
pdf-Format: | Dokument 1.pdf (8.194 KB) | |
zip gepackt: | Dokument1.zip (170 KB) |
Die zip-Datei enthält die Daten- und Programmdateien zum Buch sowie 2 readme-Texte zu den Daten und Programmen. |
Universität | Justus-Liebig-Universität Gießen | |
Fachgebiet: | Wirtschaftswissenschaften | |
DDC-Sachgruppe: | Statistik | |
Dokumentart: | Buch (Monographie) | |
Sprache: | Englisch | |
Erstellungsjahr: | 2010 | |
Publikationsdatum: | 17.05.2010 | |
Kurzfassung auf Deutsch: | Statistical distributions can be grouped into families or systems. Such groupings are described in JOHNSON/KOTZ/KEMP (1992, Chapter 2), JOHNSON/KOTZ/BALAKRISHNAN (1994, Chapter 12) or PATEL(KAPADIA/OWEN (1976, Chapter 4). The most popular families are those of PEARSON, JOHNSON and BURR, the exponential, the stable and the infinitely divisible distributions or those with a monotone likelihood ratio or with a monotone failure rate. All these categories have attracted the attention of statisticians and they are fully discussed in the statistical literature. But there is one family, the location–scale family, which hitherto has not been discussed in greater detail. To my knowledge this book is the first comprehensive monograph on one–dimensional continuous location–scale distributions and it is organized as follows. Chapter 1 goes into the details of location–scale distributions and gives their properties along with a short list of those distributions which are genuinely location–scale and which — after a suitable transformation of its variable — become member of this class. We will only consider the ln–transformation. Location–scale distributions easily lend themselves to an assessment by graphical methods. On a suitably chosen probability paper the cumulative distribution function of the universe gives a straight line and the cumulative distribution of a sample only deviates by chance from a straight line. Thus we can realize an informal goodness–of–fit test. When we fit the straight line free–hand or by eye we may read off the location and scale parameters as percentiles. Another and objective method is to find the straight line on probability paper by a least–squares technique. Then, the estimates of the location and scale parameters will be the parameters of that straight line. Because probability plotting heavily relies on ordered observations Chapter 2 gives — as a prerequisite — a detailed representation of the theory of order statistics. Probability plotting is a graphical assessment of statistical distributions. To see how this kind of graphics fits into the framework of statistical graphics we have written Chapter 3. A first core chapter is Chapter 4. It presents the theory and the methods of linear estimating the location and scale parameters. The methods to be implemented depend on the type of sample, i.e. grouped or non–grouped, censored or uncensored, the type of censoring and also whether the moments of the order statistics are easily calculable or are readily available in tabulated form or not. In the latter case we will give various approximations to the optimal method of general least–squares. Applications of the exact or approximate linear estimation procedures to a great number of location–scale distributions will be presented in Chapter 5, which is central to this book. For each of 35 distributions we give a warrant of arrest enumerating the characteristics, the underlying stochastic model and the fields of application together with the pertinent probability paper and the estimators of the location parameter and the scale parameter. Distributions which have to be transformed to location–scale type sometimes have a third parameter which has to be pre–estimated before applying probability plotting and the linear estimation procedure. We will show how to estimate this third parameter. The calculations and graphics of Chapter 5 have been done using MATLAB,1 Version 7.4 (R2007a). The accompanying CD contains the MATLAB script M–file LEPP and all the function–files to be used by the reader when he wants to do inference on location–scale distributions. Hints how to handle the menu–driven program LEPP and how to organize the data input will be given in Chapter 6 as well as in the comments in the files on the CD. |