On the Use of Second and Third Moments for the Comparison of Linear Gaussian and Simple Bilinear White Noise Processes

The linear Gaussian white noise process (LGWNP) is an independent and identically distributed (iid) sequence with zero mean and finite variance with distribution ( ) 2 0, N σ . Some processes, such as the simple bilinear white noise process (SBWNP), have the same covariance structure like the LGWNP. How can these two processes be distinguished and/or compared? If 1 2 , , , n X X X  is a realization of the SBWNP. This paper studies in detail the covariance structure of , ; 1, 2,3 d t X t Z d ∈ = . It is shown from this study that; 1) the covariance structure of 2 , t X t Z ∈ is non-normal with distribution equivalent to the linear ARMA(2, 1) model; 2) the covariance structure of , t X t Z ∈ is iid; 3) the variance of , t X t Z ∈ can be used for comparison of SBWNP and LGWNP.

σ < ∞ , and the series is called a linear Gaussian white noise process with the following properties [1]- [7].
( ) where R(k) is the autocovariance function at lag k, ρ k is the autocorrelation function at lag k and kk φ is the partial autocorrelation function at lag k.
In other words, a stochastic process , t X t Z ∈ is called a linear Gaussian white noise if , t X t Z ∈ is a sequence of independent and identically distributed (iid) random variables with finite mean and finite variance.Under the assumption that the sample 1 2 , , , n X X X  is an iid sequence, we compute the sample autocorrelations as where The iid hypothesis is always tested with the Ljung and Box [8] statistic where ( ) LB Q m is asymptotically a chi-squared random variable with m degree of freedom.
Several values of m are often used and simulation studies suggest that the choice of ( ) ln m n ≈ provides better power performance [9].
If the data are iid, the squared data 2   , , , n X X X  are also iid [10].Another portmanteau test formulated by Mcleod and Li [10] is based on the same statistic used for the Ljung and Box [8] ( ) ( ) ( ) where the sample autocorrelations of the data are replaced by the sample autocorrelations of the squared data, ( ) As noted by Iwueze et al. [11], a stochastic process , t X t Z ∈ may have the covariance structure (1.1) through (1.5) even when it is not the linear Gaussian white noise process.Iwueze et al. [11] provided additional properties of the linear Gaussian white noise process for proper identification and characterization from other processes with similar covariance structure (1.1) through (1.5).
Let , 1, 2,3,  where , t X t Z ∈ , be the linear Gaussian white noise process, the mean ( ) ( ) were obtained to be [11] ( ) ( ) where ( ) ( ) and [11] concluded that all powers of a linear Gaussian white noise process are iid but not normally distributed.
Using the coefficient of symmetry and kurtosis, Iwueze et al. [11] confirmed the non-normality of , 2,3, 1 gives the mean, variance, the coefficient of symmetry ( 1 β ) and kurtosis ( 2 β ) defined as follows ( ) ( ) ( ) where  ( ) ( ) ( ) Using the standard deviations when  , Iwueze et al. [11] determined the optimal value of d to be three ( Hence, for effective comparison of the linear Gaussian white noise process with any stochastic process with similar covariance structure, , Y X d = = must be used.The most commonly used white noise process is the linear Gaussian white noise process.The process is one of the major outcomes of any estimation procedure which is used in checking the adequacy of fitted models.The linear Gaussian white noise process also plays significant role as a basic building block in the construction of linear and non-linear time series models.However, the major problem is that there are many non-linear processes that exhibit the same covariance structure (Equation (1.1) through Equation (1.5)) as the linear Gaussian white noise process.One of such non-linear models is the bilinear models.
The study of bilinear models was introduced by Granger and Andersen [12] and Subba Rao [13].Granger and Andersen [14] established that all series generated by the simple bilinear model appear to be second order white noise where β is a constant and , t e t Z ∈ is an independent identically distributed normal random variable with ( ) [15] studied the existence problem of a simple bilinear process , Martins [16] obtained the autocorrelation function of the process  [17], Malinski and Figwer [18] and Malinski [19].Iwueze [20] studied the more general bilinear white noise model where , t e t Z ∈ is as defined in (1.19).Iwueze [20] was able to show the following.
1) The series , t X t Z ∈ satisfying (1.21) is strictly stationary, ergodic and unique.
2) The series , 3) The series , t X t Z ∈ satisfying (1.21) has the same covariance structure as the linear Gaussian white noise processes.
4) Obtained the covariance structure of (1.21) to be ( ) 5) The series satisfying (1.21) is invertible if For the simple bilinear model (1.19), it follows that ( ) and the invertibility condition is It is worthy to note that the stationarity condition [19] for model (1.19) and our study in this paper will concentrate on model (1.20).The purpose of this paper is to meet the following goals for the simple bilinear model satisfying (1.20).2) Determine the covariance structure of , 2,3 3) Determine for what values of β the simple bilinear white noise process will be identified as a Linear Gaussian white noise process.

4)
Determine for what values of β the simple bilinear model will be nor- mally distributed.
This paper is further divided into four sections in order to establish and achieve these goals.Section 2 discusses the covariance structure of , 1, 2,3 , t e iid N σ , Section 3 presents the methodology, Section 4 is the results and discussion while, Section five is the conclusion.( )

Covariance
t t Y X t Z = ∈ has the same covariance structure as the linear ARMA(2, 1) process (2.4) where t a is the sequence of independent and identically distributed random variable with By the assumption of stationarity, ( ) We have shown that ( ) With this result, it is clear that when , has the same covariance structure as the linear ARMA(2, 1) process.Its linear equivalence is where t a is the purely random process with ( ) The covariance structure of 3 , is that of the linear white noise process. Proof: Let ( ) ) Generally, Therefore, given

Normality Checking
The Jarque-Bera (JB) test [21] [22] [23] will be used to determine for which values of β a simple bilinear model (1.20) is normally distributed or not.The JB test statistic is ( ) where ( ) ( ) n is the sample size while, 1 γ and 2 γ are the sample skewness and kurtosis coefficients.The asymptotic null distribution of JB is 2 χ with 2 degrees of freedom.

White Noise Test
The modified Ljung-Box test statistic [11] given by X are jointly iid.That will help determine the values of β for which the simple bilinear model (1.20) is not distinguishable from the li- near Gaussian white noise process (LGWNP).Then, the hypothesis of iid data is rejected at level α if the observed ( )

Use of Chi-Square Test for Comparison of the Simple Bilinear White Noise Process and the Linear Gaussian White Noise Process
From Theorem 2.3, the third power of the simple bilinear process is iid.
The chi-square test [24] [25] can be used to perform the test.The chi-square test statistic is

Results and Discussion
One thousand random digits , t e t Z ∈ that met the condition  3. The results obtained using the JB, Q* and the chi-square test indicated , t e t Z ∈ as a LGWNP at 5% level of significance.The LGWNP were used to simulate the SBWNP  H ) show that the SBWNP is iid and can be identified as a LGWNP for some β values.The values of β where the SBWNP will be identified as an LGWNP are summarized in Table 5.

Conclusion
We have been able to establish the covariance structure for , 1, 2,3;

2
simulated using Minitab 16 series software.Only one random digit, shown in Appendix I, was used for demonstration in the study because of want of space.The estimates of the descriptive statistics (mean, variance, skewness ( 1 γ ) and kurtosis ( 2 γ )) and other tests (Jarque Bera (JB) test, modified Ljung Box test (Q*) and chi-square calculated test statistic) for the powers , shown in Table . The estimates of the descriptive statistic and that for the test statistic (JB, Q* and the chi-square calculated test statistic) are shown in Table 4.The values of the JB statistic show that the SBWNP are normally distributed for the values of Q* and the chi-square calculated test statistic ( 0
confirm that the simple bilinear process (1.20) is not a linear Gaussian white noise process (LGWNP).For the LGWNP , A test is C. O. Arimie et al.DOI: 10.4236/ojs.2018.83037575 Open Journal of Statistics needed to the true variance of the simple bilinear process(1.20)andn is the number of observations of the series.The null hypothesis is rejected at level α .degree of freedom.It should be noted that this test works well when the underlying original population , t X t Z ∈ is normally distributed.

Table 3 .
Descriptive Statistics and estimate of the test statistic for rejecting the null hypothesis of equality of the variance of higher moment for the simulated series,

Table 4 .
Descriptive statistics and estimate of the test statistic for simulated bilinear series

Table 5 .
Values of β for comparison of SBWNP as a LGWNP at 0.05 and 0.10 α le- vels.satisfying(1.20).We have also determined the values of β for which the sim- ple bilinear model (1.20) is normally distributed and in which the process can be determined as a LGWNP or not.We recommend that for proper comparison of SBWNP with LGWNP, the SBWNP should be considered for normality, white noise test and test of equality of variance of its third moment being equivalent to the theoretical values of the LGWNP.