1. Introduction

Some of the problems of time series are cyclical fluctuation and regime-switching, full range of shape changing predictive distributions (multimodalities), change like behavior and inability to handle cycles traits that were characterized by the data series [2]. Some non-linear time series models such as Threshold Auto-Regressive (TAR), and Self Exciting Threshold Autoregressive (SETAR) were proposed contain the made mentioned stylized properties, but were limited to two regime-shifting [8]. Mixture Autoregressive (MAR) model proposed by [14] captured and modeled the aforementioned stylized traits with strictly Gaussian or normal distributed error term.

The problem of regime-switching model via mixture models are yet to be fully and completely addressed by model with Gaussian random noise [1].

According to [4], structural time-series model has been statistically deficiency due to inadequate to take into consideration multimodalities, large fluctuations, heavy-tail and non-Gaussian distributional random noise that are causal traits of the multimodalities and tenure-changing. However, the MAR model with Gaussian marginal distribution is noted to inadequately capture extreme events that are usually the causal of distortion that leads to multimodalities and tenure-changing [7]. Consequently, the stated problem of MAR model with Gaussian error term would be addressed by Fréchet random noise, being one of the three Extreme Value Distributions (EVDs). In other words, Fréchet Mixture Autoregressive model with Fréchet error term will be developed and derived as a substitute for MAR to capture extreme events (outliers) that are causes of large fluctuations in arising due to abnormal kurtosis and skewness. Hence, this article develops, ascertains and estimates embedded parameters (AR coefficients, mixing weights and Fréchet coefficients) via Expectation-Maximization algorithm of the process with k-regime-switching. In addition, the limiting distribution (asymptotic property) of the model will be ascertained via Gnedenko-Fisher Tippet limiting behavior as well as the ergodic of the FMA process. The mean and variance of the process will be ascertained as well.

2. Literature Review

The history of mixture time series model was originated and propounded by [15] in the late 1900s, when he used normal mixture to model outliers and employ the method of moments to decompose normal mixtures. He deduced that Autoregressive Integrated Moving Average (ARIMA), Self-Exciting Threshold Autoregressive (SETAR) model, and other nonlinear models can only depict a unimodal predictive time series random noise distribution. He affirmed that mixture model would be needed for stylist, cyclical, switching tenure properties to be revealed. Afterwards, [3] predicted with the MAR model and deduced that MAR model possessed the attractive property in such a way that spatial attributes of the conditional density of its prediction contingents upon on the immediate past events of the process. Additionally, he maintained that the multiple steps’ distribution would be a collection (mixture) of normal distributions if and only if the source MAR model is a mixture of normal densities.

However, despite propounded of all these models, no model studied the stationarity and ergodicity traits of MAR related models. It was in line of this deficiency that [8] propounded and affirmed the theoretical advantage of Gaussian mixture autoregressive (GMAR) conditions for stationarity and ergodicity to be met. They claimed that GMAR properties were straightforward to establish. In addition, they maintained that p^th order model can be explicitly expressed in a (Optimal order (p) +1)– lengthy stationary distribution with ascertained constant mixing weights to connote mixture of normal probability distributions.

In advancement, [7] proposed a novel nonlinear Vector Autoregressive (VAR) otherwise called Gaussian Mixture Vector autoregressive (GMVAR) model. They developed and explained its asymptotic theory of maximum likelihood whose usefulness is vital when dealing with bivariate time series settings.

Furthermore, [6] proposed a new nonlinear time series via a vector incorporation into the MAR model called Vector-MAR (VAR) model otherwise known as Gaussian Mixture Vector Autoregressive (GMVAR) model. They affirmed and claimed that the GMVAR model can be categorized as member of the mixture VAR models because of its normally distributed error term, constructed and developed for studying nonlinear models with time-varying series that exhibited tenure changing traits. They maintained that primarily the GMVAR model differs from the variety of VAR models in the meaning of the mingling proportions that regularize the tenure likelihoods.

[11] provided an up-to-date theoretical and methodological developments on finite mixture models due to its flexibility increasingly exploitation and applications. They adopted the semiparametric technique way of parameter estimation via the EM algorithm to carve out the distributional shape changing, group-structure and cluster analysis in the data captured to the model. Proposed models reviewed were formulated to capture, and correct the stylized properties; entire outrange of pattern switching distributions (multimodalities), change points like behavior, regime switching (capable of handling recurring periodical sequence), and time-varying volatilities (conditional means-variances), but no method/model yet to cater for heavy-tailed, long-memory and non-Gaussian MAR model. Hence, a full sketch model will be developed such that its mean and variance will as ascertained as well as its parameter estimation, limiting distribution and ergodic process.

3. Specification of the Fréchet Mixture Autoregressive Model

[10] and [13] defined a k-component of Mixture Autoregressive (MAR) model to be

(1)

Rewriting the k-component of MAR model in (1) in Fréchet Mixture Autoregressive gives

(2)

where;

Otherwise,

(3)

Where, , for . For mixing weights , for is the Cumulative Distribution Function of the standard distributions, where , with mean and variance .

3.1. The Conditional Mean and Variance for FMA Model

The conditional mean and variance for FMA model of given the previous preceding values is as follow:

(4)

since,

The mean of Fréchet is , therefore

(5)

which relies on previous preceding values of the time series & is the PDF of Fréchet.

So,

But, for (previous preceding values of the time series that relies on the current values).

Therefore,

(6)

The expression is a positive quantity. Hence, this meets the non-negative property of variance.

Since the mean and variance of Fréchet PDF are; and respectively

Alternatively, of equation (6) can be written as

(7)

3.2. Parameter Estimation for FMA via EM Algorithm

Adopting the Expectation-Maximization (EM) algorithm propounded by [10] for non-linear and regime-switching. Let for

Additionally, supposing that “S” is the unobserved random variable where is a dimensional vector such that

whose component is

For that is,

Let be the universal space of parameters.

Given , the Fréchet distribution of the complete data is given by

(8)

If is the log-likelihood. Letting be the joint conditional log-likelihood function for large sample size (n) for maximizing the function gives

(9)

(10)

Where First derivatives of each of the parameter gives,

(11)

(12)

(13)

(14)

(15)

For

Then, the second derivatives of individual coefficient estimated from (11) - (15) depends of setting a function to be a random variable at time and counter

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

Adopting the EM algorithm for estimating the parameter space in of (10).

(a) The first step of the Expectation-Maximization algorithm which stands for E-step, If each of the coefficient (parameter) in the parameter space is known, such that the unknown values of the unobserved data is then substituted by enforcing their means on each coefficient on the observed data X_t. Subjecting has the enforced mean on , then the individual probability of is the total probability of the enforced means over individual mean, as given in equation below:

(26)

For

(b) The M-step where the unknown values of S is presumed via guessing and their enforced means are assumed to be the coefficients. The mixing weight per each regime is then estimate via

(27)

For where such that the estimates of parameters of these two steps are iterated until convergence is reached. The criterion for checking convergence was obtained by [12] as,

(28)

for being the j^th component of .

3.3. Limiting Distribution (Asymptotic Property) for FMA Model

The Limit distribution (Asymptotic property) for FMA model would be based on limiting laws for Extreme Value Distributions stated by Gnedenko-Fisher Tippet theorem [9].

The theorem stated that: Let be a sequence of IID random variables of sample size (n) with common Cumulative Density Function with its maximum denoted as such that distribution (Limiting behavior ) of as the sample size (n) approaches infinity is

Investigating the wavering of in a similar vein to that of Central Limit Theorem (CTL) for sum of the random variables , then the distribution of belongs to the maximum domain of non-degenerate distribution if there exist sequences of constants and , such there exist a real value for .

Such that is non-degenerate distribution function, and belongs to any of the

Starting with,

the Cumulative Distribution Function for Fréchet is

(29)

So,

(30)

Where,

(31)

According to [5], the normalizing constants of and for Fréchet distribution are approximately and respectively, where is the set of natural numbers.

(32)

Logarithm of both sides

(33)

Taking exponent of both sides so as to naturalize the logarithm

(34)

Which means that as , the limiting distribution of the FMA asymptotically approaches

3.4. Ergodic Process for FMA Model

Strictly stationary and ergodic processes for FMA model would be ascertained via

Where, are likelihoods and expectations of likelihoods respectively.

(35)

(36)

But,

(37)

From Jensen’s inequality, for every positively integrable X.

(38)

Recall,

(39)

(40)

(41)

since,

(42)

(43)

4. Conclusions

The proposed model of being one of the prominent models for extreme valued mixture autoregressive was devised in order to sieze and objurgate stylized traits; entire greater range of multimodalities, regime-switching, (ability to handle cycles), time-varying volatilities (conditional variances), time-varying mixing weights with Fréchet Probability Density Function (PDF). Long-memory, heavy-tailed distributed observations are the causal that arise to outliers, change points like behavior, abnormal kurtosis, and skewness. The formulated and proposed will be an ideal and substitute mixture autoregressive model whenever heavy-tailed distributed observations are the causal that arise to outliers, abnormal kurtosis, and skewness brings about the regime-switching.

ACKNOWLEDGEMENTS

We wish to acknowledge Africa Union for funding this research as well as reviewers.