On mixed $AR\left(1\right)$ time series model with approximated beta marginal

Abstract

We consider the mixed $AR\left(1\right)$ time series model

$$X_t=\{\begin{array}{cc}\alpha X_{t-1}& \text{w.p. }{\alpha }^p\\ \beta X_{t-1}+{\xi }_t& \text{w.p. }1-{\alpha }^p\end{array}\text{,}\alpha ,\beta \in (0,1)\text{,}$$

when $X_t$ has the two parameter beta distribution $B_2(p,q)$, $p\in (0,1],q>1$. Special attention is given to the case $p=1$ when the marginal distribution is approximated by the power law distribution closely connected with the two parameter Kumaraswamy distribution ${\mathrm{Kum}}_2(p,q),p\in (0,1],q>1$. Using the Laplace transform technique, we prove that for $p=1$ the distribution of the innovation process is uniform discrete. For $p\in (0,1)$, the innovation process has a continuous distribution. We also consider estimation issues of the model.

Introduction

In standard time series analysis one assumes that its marginal distribution is Gaussian. However, a Gaussian distribution will not always be appropriate. In earlier works stationary non-Gaussian time series models were developed for variables with positive and highly skewed distributions. There still remains situations where Gaussian marginals are inappropriate, i.e. where the marginal time-series variable being modeled, although not skewed or inherently positive valued, has a large kurtosis and long-tailed distributions. There are plenty of real situations that cannot be modeled by the Gaussian distribution like in hydrology, meteorology, information theory, economics, etc. Simple models with exponential marginals or mixed exponential marginals are considered in (Gaver and Lewis, 1980, Jevremović, 1990, Lawrence, 1980, Lawrence and Lewis, 1980, Mališić et al., 1987), while other marginals like gamma–(Gaver and Lewis, 1980, Novković, 1997, Sim, 1986), Laplace–(Novković, 1997), uniform–(Chernick, 1981, Ristić and Popović, 2002) and Weibull (Novković, 1997, Sim, 1986) have been discussed. Finally, we point out autoregressive processes $PBAR$ and $NBAR$ constructed in (McKenzie, 1985) for positively and negatively correlated pairs of beta random variables employing certain properties of the $B_2(p,q)$ distribution.

In this paper, we introduce a mixed autoregressive first order time series model with the two parameter beta distribution $B_2(p,q),p\in (0,1],q>1$, whose Laplace transform is approximated when the transformation argument is large. The resulting approximation determines a new distribution and results in a discrete uniform distribution for the innovation process. Special attention is given to the case $p=1$, when the approximation becomes a kind of power law distribution, characterized by the related probability density function (PDF) $f\left(x\right)=q{(1-x)}^{q-1},x\in [0,1]$. This distribution coincides with the two parameter beta distribution $B_2(1,q)$. On the other hand, the $B_2(1,q)$ generates a two parameter Kumaraswamy distribution ${\mathrm{Kum}}_2(p,q),p\in (0,1],q>1$ introduced by Kumaraswamy (1980). This distribution is very important in applications when double bounded random processes arise in practice (Nadarajah, 2007, Nadarajah, 2008), e.g. it has been applied in modeling the storage volume of a reservoir and system design (Fletcher and Ponnambalam, 1996) and turns out to be very important in many hydrological problems. For a complete account of the properties of the Kumaraswamy distribution, consult (Jones, 2009).

A random variable $X_{p,q}$ defined on some standard probability space $(\Omega ,\mathfrak{F},P)$ having the ${\mathrm{Kum}}_2(p,q)$ distribution possesses the PDF (Đorić et al., 2007, pp. 169–170) (in its simplest form):

$$f\left(x\right)=pq{x}^{p-1}{(1-x^p)}^{q-1}\cdot 1_{[0,1]}\left(x\right)\text{.}$$

The cumulative distribution function’s (CDF) nonconstant part is

$$F\left(x\right)=(1-{(1-x^p)}^q)\cdot 1_{[0,1]}\left(x\right)\text{,}$$

where $1_{[0,1]}\left(x\right)$ stands for the indicator function of the closed unit interval. Furthermore, we have

$$X_{p,q}^p\stackrel{d}{=}{Y}_{1,q}\text{,}$$

where $Y_{1,q}$ has two parameter beta distribution $B_2(1,q),q>1$. Note that both ${\mathrm{Kum}}_2(1,1)$ and $B_2(1,1)$ represent the uniform $\mathcal{U}(0,1)$ distribution.

The technical part of the research begins with the derivation of the Laplace transform (LT) of the mixed autoregressive model. The resulting LT function is expressed in terms of the Wright hypergeometric function $\Phi$. Therefore we are faced with extremely hard calculations in inverting the Laplace transform. So, we approximate the derived LT of the model, obtaining the inverse LT mutatis mutandis the distribution of the innovation sequence $\{{\xi }_t:t\in \mathbb{Z}\}$. The resulting approximate distribution will be referred to as the approximated beta ($A{B}_{p,q}$) and as the approximated power law ($APL$) for $p=1$. Therefore, considering initially a time series model with $A{B}_{p,q}$, where the latter is in fact approximated $B_2(p,q),p\in (0,1),q>1$, we arrive at a new model called $A{B}_{p,q}AR\left(1\right)$, which becomes $APLAR\left(1\right)$ for $p=1$.

Random numbers from $A{B}_{p,q}$ were generated using an algorithm described in Ridout (2009) such that instead of the exact LT we will use its approximation given by (3). Since those random numbers are i.i.d. The PDF of $A{B}_{p,q}$ will be estimated using the kernel density method (Silverman, 1986).

In the last section the parameters of the $A{B}_{p,q}AR\left(1\right)$ model are estimated by the so-called minimal neighbors quotient estimator which turns out to be consistent.