# The inverted bivariate and multivariate loss functions : properties and applications

 Author: Tsoi, Sze Leong Title: The inverted bivariate and multivariate loss functions : properties and applications Degree: M.Phil. Year: 2013 Subject: StatisticsHong Kong Polytechnic University -- Dissertations Department: Dept. of Applied Mathematics Pages: xiv, 220 p. : ill. ; 30 cm. Language: English InnoPac Record: http://library.polyu.edu.hk/record=b2639076 URI: http://theses.lib.polyu.edu.hk/handle/200/7137 Abstract: The purpose of this research project is to develop a loss function by inverting the Dirichlet distribution. The motive is to explore the possibility of developing loss functions of higher dimension. Spiring (1993), Spiring and Yeung (1998) and Spiring and Leung (2002) developed a series of Inverted Probability Loss Functions (IPLFs), while most of them are univariate. A multivariate IPLF, which in theory, should have the capability of depicting financial loss when more than one quality characteristics are concerned. The Dirichlet distribution is chosen because of its addictiveness of random variables in nature and its ease to include any number of variables. To depict the long term financial loss of manufacturing a product, it is desired to calculate the expected loss by E[L] = Ω L(x)g(x)dx. L(x) is the loss function and g(x) is the conjugate function, which describes the measurement of the quality characteristic which the manufacturer is interested in. This research project focuses on developing a multivariate inverted probability loss function and some discussion will be made about the conjugate function. The prime objective of this research is to develop an inverted multivariate probability loss function. For this purpose, extending the conjugate function into higher dimension will also be discussed. Although it is highly desired that real data can be collected to form the conjugate function, sometimes it is not feasible to do so and simulation may be the solution to this situation. Three common random variate generation methods, namely the conditional approach, acceptance and rejection approach and gamma factors approach, are used to invert the Dirichlet distribution. The major distinction between the three aforementioned approaches lays in the method in obtaining the estimators, respectively, the method of moments, maximum likelihood estimation and a modified percentile matching method. The theory and the procedure underlying the methods in obtaining estimators are discussed in details and the sets of estimates obtained are then evaluated by the Chi-square goodness-of-fit test to compare the acceptability of the three estimation techniques. Cells arrangement is also discussed and verified by Peacock's two-dimensional Kolmogorov-Smirnov goodness-of-fit test, which is known for its conservativeness. Properties and various probable conjugate distributions are examined to compare the univariate IPLF with the multivariate IPLF. At last, the bivariate inverted normal loss function, introduced by Spiring (1993) and the inverted Dirichlet loss function are applied to data collected in solder paste composition and dimensions of fasteners, as an application of the proposed technique in depicting the financial loss due to deviation in production, and ultimately, to show the feasibility of such bivariate loss functions.

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