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Quantile Functions

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Quantile-Based Reliability Analysis

Part of the book series: Statistics for Industry and Technology ((SIT))

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Abstract

A probability distribution can be specified either in terms of the distribution function or by the quantile function. This chapter addresses the problem of describing the various characteristics of a distribution through its quantile function. We give a brief summary of the important milestones in the development of this area of research. The definition and properties of the quantile function with examples are presented. In Table 1.1, quantile functions of various life distributions, representing different data situations, are included. Descriptive measures of the distributions such as location, dispersion and skewness are traditionally expressed in terms of the moments. The limitations of such measures are pointed out and some alternative quantile-based measures are discussed. Order statistics play an important role in statistical analysis. Distributions of order statistics in quantile forms, their properties and role in reliability analysis form the next topic in the chapter. There are many problems associated with the use of conventional moments in modelling and analysis. Exploring these, and as an alternative, the definition, properties and application of L-moments in describing a distribution are presented. Finally, the role of certain graphical representations like the Q-Q plot, box-plot and leaf-plot are shown to be useful tools for a preliminary analysis of the data.

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Notes

  1. 1.

    There are different choices for these plotting points and recently Balakrishnan et al. [52] discussed the determination of optimal plotting points by the use of Pitman closeness criterion.

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Authors and Affiliations

  1. Department of Statistics, Cochin University of Science and Technology, Cochin, Kerala, India

    N. Unnikrishnan Nair & P. G. Sankaran

  2. Department of Mathematics and Statistics, McMasters University, Hamilton, ON, Canada

    N. Balakrishnan

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  1. N. Unnikrishnan Nair
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  2. P. G. Sankaran
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  3. N. Balakrishnan
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Nair, N.U., Sankaran, P.G., Balakrishnan, N. (2013). Quantile Functions. In: Quantile-Based Reliability Analysis. Statistics for Industry and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-8361-0_1

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