# Location-scale distributions : Linear estimation and probability plotting

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Statistical distributions can be grouped into families or systems. Such groupings are describedin JOHNSON/KOTZ/KEMP (1992, Chapter 2), JOHNSON/KOTZ/BALAKRISHNAN(1994, Chapter 12) or PATEL(KAPADIA/OWEN (1976, Chapter 4). The most popularfamilies are those of PEARSON, JOHNSON and BURR, the exponential, the stable and theinfinitely divisible distributions or those with a monotone likelihood ratio or with a monotonefailure rate. All these categories have attracted the attention of statisticians and theyare fully discussed in the statistical literature. But there is one family, the location scalefamily, which hitherto has not been discussed in greater detail. To my knowledge this bookis the first comprehensive monograph on one dimensional continuous location scale distributionsand it is organized as follows.Chapter 1 goes into the details of location scale distributions and gives their propertiesalong 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. Wewill only consider the ln transformation. Location scale distributions easily lend themselvesto an assessment by graphical methods. On a suitably chosen probability paper thecumulative distribution function of the universe gives a straight line and the cumulativedistribution of a sample only deviates by chance from a straight line. Thus we can realizean informal goodness of fit test. When we fit the straight line free hand or by eye we mayread off the location and scale parameters as percentiles. Another and objective methodis to find the straight line on probability paper by a least squares technique. Then, theestimates 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 asa prerequisite a detailed representation of the theory of order statistics. Probabilityplotting is a graphical assessment of statistical distributions. To see how this kind ofgraphics 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 estimatingthe location and scale parameters. The methods to be implemented depend on the typeof sample, i.e. grouped or non grouped, censored or uncensored, the type of censoringand also whether the moments of the order statistics are easily calculable or are readilyavailable in tabulated form or not. In the latter case we will give various approximationsto the optimal method of general least squares.Applications of the exact or approximate linear estimation procedures to a great number oflocation 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 pertinentprobability 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 thirdparameter which has to be pre estimated before applying probability plotting and the linearestimation 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 thefunction files to be used by the reader when he wants to do inference on location scaledistributions. Hints how to handle the menu driven program LEPP and how to organizethe data input will be given in Chapter 6 as well as in the comments in the files on the CD.