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2015 | OriginalPaper | Buchkapitel

20. Composite Goodness-of-Fit Tests for Left-Truncated Loss Samples

verfasst von : Anna Chernobai, Svetlozar T. Rachev, Frank J. Fabozzi

Erschienen in: Handbook of Financial Econometrics and Statistics

Verlag: Springer New York

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Abstract

In many financial models, such as those addressing value at risk and ruin probabilities, the accuracy of the fitted loss distribution in the upper tail of the loss data is crucial. In such situations, it is important to test the fitted loss distribution for the goodness of fit in the upper quantiles, while giving lesser importance to the fit in the low quantiles and the center of the distribution of the data. Additionally, in many loss models the recorded data are left truncated with the number of missing data unknown. We address this gap in literature by proposing appropriate goodness-of-fit tests.

We derive the exact formulae for several goodness-of-fit statistics that should be applied to loss models with left-truncated data where the fit of a distribution in the right tail of the distribution is of central importance. We apply the proposed tests to real financial losses, using a variety of distributions fitted to operational loss and the natural catastrophe insurance claims data, which are subject to the recording thresholds of $1 and $25 million, respectively.

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More accurately, $ \widehat{m} $ should be estimated as $ \widehat{m}=\left\lceil n\frac{z_{\mathrm{H}}}{\left(1-{z}_{\mathrm{H}}\right)}\right\rceil $, but this detail can be ignored for all practical purposes.

It has been shown that the weighting function $ \psi (t)={\left(1,-,t\right)}^{-\beta } $ possesses nice asymptotic properties only for β=[0,2) (Deheuvels and Martynov 2003). For β=2, which is the case considered in this chapter, the asymptotic distribution of the test statistic has infinite mean. This is indeed a concern for very large samples (i.e., the asymptotic case n→∞). Yet, because firms’ operational loss data samples are typically relatively small, the asymptotic distribution of the test statistic should not generate large concerns. Nevertheless, the results of the proposed quadratic class upper-tail Anderson-Darling test should be treated with caution and with consideration of the properties described above.

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Titel: Composite Goodness-of-Fit Tests for Left-Truncated Loss Samples
verfasst von: Anna Chernobai
Svetlozar T. Rachev
Frank J. Fabozzi
Verlag: Springer New York
Buch: Handbook of Financial Econometrics and Statistics
Print ISBN: 978-1-4614-7749-5

Electronic ISBN: 978-1-4614-7750-1

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-1-4614-7750-1_20