A stochastic process , where is called a white noise or purely random process, if with finite mean and finite variance, all the autocovariances are zero except at lag zero. In many applications, is assumed to be normally distributed with mean zero and variance, , and the series is called a linear Gaussian white noise process with the following properties  -  .
where R(k) is the autocovariance function at lag k, rk is the autocorrelation function at lag k and is the partial autocorrelation function at lag k.
In other words, a stochastic process is called a linear Gaussian white noise if is a sequence of independent and identically distributed (iid) random variables with finite mean and finite variance. Under the assumption that the sample is an iid sequence, we compute the sample autocorrelations as
The iid hypothesis is always tested with the Ljung and Box  statistic
where 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 provides better power performance  .
If the data are iid, the squared data are also iid  . Another portmanteau test formulated by Mcleod and Li  is based on the same statistic used for the Ljung and Box 
where the sample autocorrelations of the data are replaced by the sample autocorrelations of the squared data, .
As noted by Iwueze et al.  , a stochastic process may have the covariance structure (1.1) through (1.5) even when it is not the linear Gaussian white noise process. Iwueze et al.  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 where , be the linear Gaussian white noise process, the mean , the variance , autocovariances were obtained to be 
It is clear from (1.12) that when are iid, the powers of are also iid. Iwueze et al.  also showed the probability density function (pdf) of to be the pdf of a gamma distribution with parameters
. That is, .
when and  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.  confirmed the non-normality of . Table 1 gives the mean, variance, the coefficient of symmetry ( ) and kurtosis ( ) defined as follows
Table 1. Mean, Variance, Coefficient of symmetry ( ) and kurtosis ( ) for , ,when .
Source: Iwueze et al. (2017).
Using the standard deviations when and the kurtosis of , Iwueze et al.  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, 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  and Subba Rao  . Granger and Andersen  established that all series generated by the simple bilinear model
appear to be second order white noise where is a constant and is an independent identically distributed normal random variable with , . Guegan  studied the existence problem of a simple bilinear process satisfying
Martins  obtained the autocorrelation function of the process for the simple bilinear model defined by (1.19) when is iid with a Gaussian distribution. Again, Martins  studied the third order moment structure of (1.19) with non-independent shocks. Recently, properties of the simple bilinear model (1.19) were addressed by Malinski and Bielinska  , Malinski and Figwer  and Malinski  . Iwueze  studied the more general bilinear white noise model
where is as defined in (1.19). Iwueze  was able to show the following.
1) The series satisfying (1.21) is strictly stationary, ergodic and unique.
2) The series satisfying (1.21) is invertible.
3) The series 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
is structure (k, n) independent  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).
1) Determine for the simple bilinear model (1.20).
2) Determine the covariance structure of , when satisfies (1.20).
3) Determine for what values of the simple bilinear white noise process will be identified as a Linear Gaussian white noise process.
4) Determine for what values of the simple bilinear model will be normally distributed.
This paper is further divided into four sections in order to establish and achieve these goals. Section 2 discusses the covariance structure of when , , Section 3 presents the methodology, Section 4 is the results and discussion while, Section five is the conclusion.
2. Covariance Structure of , When ,
Let be the linear Gaussian white noise process with and . Suppose there exists a stationary and invertible process satisfying for every for some constant , then has the following properties:
has the same covariance structure as the linear ARMA(2, 1) process (2.4)
where is the sequence of independent and identically distributed random variable with and .
By the assumption of stationarity,
We have shown that
With this result, it is clear that when is defined by (1.20), has the same covariance structure as the linear ARMA(2, 1) process. Its linear equivalence is
where is the purely random process with and . Table 2 compares with its linear ARMA(2, 1) equivalence.
Let be the linear Gaussian white noise process with and . Suppose there exists a stationary and invertible process satisfying for every and some constant , then the mean and variance of are
Table 2. Covariance analysis of when , and its linear ARMA(2, 1) equivalence.
The covariance structure of is that of the linear white noise process.
, when .
, when .
Generally, , when .
Therefore, given , and , the following are true .
The covariance structure of identifies the process as linear white noise.
3.1. Normality Checking
The Jarque-Bera (JB) test    will be used to determine for which values of a simple bilinear model (1.20) is normally distributed or not. The JB test statistic is
n is the sample size while, and are the sample skewness and kurtosis coefficients. The asymptotic null distribution of JB is with 2 degrees of freedom.
3.2. White Noise Test
The modified Ljung-Box test statistic  given by
is used to test the iid hypothesis for for the simple bilinear model (1.20). It is important to note from Theorem 2.1 that has ARMA(2, 1) structure while from Theorem 2.2, is iid. This test will look for values where both and are jointly iid. That will help determine the values of for which the simple bilinear model (1.20) is not distinguishable from the linear Gaussian white noise process (LGWNP). Then, the hypothesis of iid data is rejected at level if the observed is larger than the quartile of the distribution, where  .
3.3. 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. A test is needed to confirm that the simple bilinear process (1.20) is not a linear Gaussian white noise process (LGWNP). For the LGWNP ; , and . To show that the simple bilinear process (1.20) is not LGWNP, we need to test the hypothesis;
against the alternative hypothesis
The chi-square test   can be used to perform the test. The chi-square test statistic is
where is the sample variance of that follows (1.20), is an estimate of the true variance of the simple bilinear process (1.20) and n is the number of observations of the series. The null hypothesis is rejected at level if the observed value of is larger than quartile of the chi-square distribution with .degree of freedom. It should be noted that this test works well when the underlying original population is normally distributed.
4. Results and Discussion
One thousand random digits that met the condition were 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 ( ) and kurtosis ( )) and other tests (Jarque Bera (JB) test, modified Ljung Box test (Q*) and chi-square calculated test statistic) for the powers of the random digit are shown in Table 3. The results obtained using the JB, Q* and the chi-square test indicated as a LGWNP at 5% level of significance.
The LGWNP were used to simulate the SBWNP , for satisfying the existence of using Fortran 77 program. 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 . Similarly, the values of Q* and the chi-square calculated test statistic ( ) show that the SBWNP is iid and can be identified as a LGWNP for some values. The values of where the SBWNP will be identified as an LGWNP are summarized in Table 5.
We have been able to establish the covariance structure for
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, , as linear Gaussian white noise process.
Table 4. Descriptive statistics and estimate of the test statistic for simulated bilinear series , and .
Table 5. Values of for comparison of SBWNP as a LGWNP at 0.05 and 0.10 levels.
satisfying (1.20). We have also determined the values of for which the simple 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.
Simulated Random Digits; (Read Across).
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