NUMERICAL INVERSION OF A CHARACTERISTIC FUNCTION: AN ALTERNATIVE TOOL TO FORM THE PROBABILITY DISTRIBUTION OF OUTPUT QUANTITY IN LINEAR MEASUREMENT MODELS |
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| Viktor Witkovský |
- Abstract:
- The exact (statistical) inference frequently require evaluation of the probability density function (PDF), the cumulative distribution function (CDF), and/or the quantile function (QF) of a random variable from its (known) characteristic function (CF), which is defined as a Fourier transform of its probability distribution function. Working with CFs provides an alternative (frequently more simple) route, than working directly with PDFs and/or CDFs. In particular, derivation of the CF of a weighted sum of independent random variable is a very simple and trivial task (given the CFs of the random variables). However, the analytical derivation of the PDF and/or CDF by using the Fourier transform is available only in special cases. Thus, in most practical situation, a numerical derivation of the PDF/CDF from the CF is an indispensable tool. In metrological applications, such approach can be used to form the probability distribution for the output quantity of a measurement model of additive, linear or generalized linear form. In this paper we shall present a brief overview of some efficient approaches for numerical inversion of the characteristic function, which are especially suitable for typical metrological applications. The suggested numerical approaches are based on the Gil-Pelaez inverse formula and on the approximation by discrete Fourier transform (DFT) and the FFT algorithm for computing PDF/CDF of (univariate) continuous random variables. We also present a sketch of the MATLAB implementation, together with several examples to illustrate its applicability.
- Keywords:
- characteristic function, probability density function, numerical inversion, Fast Fourier Transform (FFT), Gil-Pelaez inverion formula, GUM
- Download:
- IMEKO-WC-2015-TC21-424.pdf
- DOI:
- -
- Event details
- Event name:
- XXI IMEKO World Congress
- Title:
Measurement in Research and Industry
- Place:
- Prague, CZECH REPUBLIC
- Time:
- 30 August 2015 - 04 September 2015