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Location-scale distributions : Linear estimation and probability plotting

Rinne, Horst


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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.