Mellin’s Transform and Application to Some Time Series Models

This paper uses the Mellin transform to establish the means, variances, skewness, and kurtosis of fuzzy numbers and applied them to the random coefficient autoregressive (RCA) time series models. We also give a close form expression to the moment generating function related to fuzzy numbers. It is shown that the results of the proposed time series models are consistent with those of the conventional time series models and that the developed concepts are straightforward and easily implemented. 1. Introduction The use of fuzzy set theory as a methodology for modeling and analyzing certain financial problems is of particular interest to a number of researchers due to fuzzy set theory’s ability to quantitatively and qualitatively model those problems which involve vagueness and imprecision. Fuzzy time series models provide a new avenue to deal with subjectivity observed in most financial time series models. Most of the fuzzy financial models developed so far have, generally, been confined to modeling parameters through some form of defuzzification or linear type of fuzzy numbers such as Trapezoidal Fuzzy Number (Tr.F.N.) or Triangular Fuzzy Number (T.F.N.). There are some connections between fuzzy numbers based on convolution principle and some integral transformations such as the Mellin transform. The Mellin transform and its inverse are related to the two-sided Laplace transform. The Mellin transform is used often in applied sciences such as physics, engineering, and computer science because of its scale invariance property. In fact, this scale invariance property is analogous to the Fourier transform’s shift invariance property. In particular, this transform is used for the analysis of linear time-invariant systems. This transform is considered as a transformation from the time-domain in which inputs and outputs are functions of time. Thus, the Mellin transform has significant applications in probability theory, Markov chains, renewal theory, and time series. The objective of this paper is to use the Mellin transform to establish the means, variances, skewness, and kurtosis of fuzzy numbers and then to apply to the random coefficient autoregressive (RCA) time series models. We also provide a close form expression to the moment generating function related to fuzzy numbers. We summarize the preliminaries and notations in Section 1. The remainder of the paper is organized as follows. Section 2 introduces the Mellin transform to obtain statistical moments of any order. The useful Mellin transforms for fuzzy numbers are derived in Section 3. We also define