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Quantile-Based Reliability Analysis pp 59–103Cite as

Birkhäuser

Quantile Function Models

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

Abstract

One of the objectives of quantile-based reliability analysis is to make use of quantile functions as models in lifetime data analysis. Accordingly, in this chapter, we discuss the characteristics of certain quantile functions known in the literature. The models considered are the generalized lambda distribution of Ramberg and Schmeiser, the generalized Tukey lambda family of Freimer, Kollia, Mudholkar and Lin, the four-parameter distribution of van Staden and Loots, the five-parameter lambda family and the power-Pareto model of Gilchrist, the Govindarajulu distribution and the generalized Weibull family of Mudholkar and Kollia.

The shapes of the different systems and their descriptive measures of location, dispersion, skewness and kurtosis in terms of conventional moments, L-moments and percentiles are provided. Various methods of estimation based on moments, percentiles, L-moments, least squares and maximum likelihood are reviewed. Also included are the starship method, the discretized approach specifically introduced for the estimation of parameters in the quantile functions and details of the packages and tables that facilitate the estimation process.

In analysing the reliability aspects, one also needs various functions that describe the ageing phenomenon. The expressions for the hazard quantile function, mean residual quantile function, variance residual quantile function, percentile residual life function and their counter parts in reversed time given in the preceding chapters provide the necessary tools in this direction. Some characterization theorems show the relationships between reliability functions unique to various distributions. Applications of selected models and the estimation procedures are also demonstrated by fitting them to some data on failure times.

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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

Authors
  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 Function Models. In: Quantile-Based Reliability Analysis. Statistics for Industry and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-8361-0_3

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