The Numerical Simulation for Asymptotic Normality of the Intensity Obtained as a Product of a Periodic Function with the Power Trend Function of a Nonhomogeneous Poisson Process

Doi: 10.24042/djm.v3i3.6374 In this article, we provided a numerical simulation for asymptotic normality of a kernel type estimator for the intensity obtained as a product of a periodic function with the power trend function of a nonhomogeneous Poisson Process. The aim of this simulation is to observe how convergence the variance and bias of the estimator. The simulation shows that the larger the value of power function in intensity function, it is required the length of the observation interval to obtain the convergent of the estimator.


INTRODUCTION
There are many real phenomena that can be modeled using stochastic processes. This kind of model uses opportunity rules that describe the behavior of a system that is not known in the future. For example, the process of coming customers to a service center such as bank, post office, supermarket, etc ((Mahmud et al., n.d.;Sumiati et al., 2019;Tse, 2014)). The Poisson process is one of the continuous stochastic processes, namely the enumeration process where the number of events at time intervals spreads with the Poisson distribution.
The Poisson process is divided into two, namely the homogeneous Poisson process and the nonhomogeneous Poisson process. In a homogeneous Poisson process, the intensity function (a function of expectation value) is a constant function (not dependent on time), whereas the intensity function in the nonhomogeneous Poisson process depends on time. Periodic Poisson process is a nonhomogeneous Poisson process whose intensity function is a periodic function. This process is one of the special forms of stochastic processes with a continuous time that have many benefits to modeling various kinds of real phenomena related to opportunity rules. Some applications of this periodic Poisson process can be applied in the fields of communication, hydrology, meteorology, insurance, and seismology, (Sumiati et al., 2019), Suppose N is a nonhomogeneous Poisson process at interval [0, ] ( is an integer number) with an unknown function of intensity λ and assumed that the intensity function is locally integrated. It is also assumed that the function of this intensity is the product of the periodic component and the power trend component with b>0 is known, so that for all s∈[0,∞] the function of intensity λ (s) can be expressed as (1) with * ( ) is a periodic function with a known period. The constant is the slope of the trend with > 0. Equation (1) can also be written as ( ) = ( * ( )) , (2) with * ( )is also a periodic function. Suppose ( ) = * ( ), then Equation (2) becomes ( ) = ( ( )) .
Furthermore, because ( ) is a periodic function, then ( ) = ( + ), where is the period and is the natural number. Let : ℝ → ℝ is a real value function. The function is called the kernel if it fulfills the properties: (K1) is an opportunity density function, (K2) is bounded, and (K3) has a domain in [−1,1], (Helmers et al., 2003).
The proposed estimator ( ) at ∈ [0, ] has been studied in (Erliana et al., 2014) by only using a single realization ( ) from a Poisson process with the intensity function ( ) as in Equation (3) observed in interval [0, ]. The estimator is as follows: . (4) In the previous article, (Erliana et al., 2014), also has been given the bias, variance, and optimal bandwidth for all values b. Strong consistency of the estimator has also been proven in (Maulidi et al., 2015). Other similar studies can be seen in (Mangku, 2011), (Erliana et al., 2014) and (Mangku et al., 2013).

METHOD
This research started from the Theorems that analitically prove the normal distribution of the estimator for all cases of values.
The normality asymptotic of the estimator is a useful tool in developing a prediction model. This is because by knowing the asymptotic normality, the interval confident of the estimator can be obtained. Here we provide the theorems for the asymptotic normality of the estimator that has been formulated, Equation (4).
Theorem 1 (Asymptotic Normality of ̂, , ( )for < < 1) Suppose that the intensity satisfies (1) and locally integrable. The kernel function satisfies (K1), (K2), (K3), has a bounded second derivative around of The proofs of theorems above can be obtained through a rough analysis, (Maulidi, 2015). However, the basic theory to proof these theorems can be studied in (Dudley, 1989), (Hogg et al., 2005), and (Serfling, 1980). The kernel function that has been used in this simulation was a uniform kernel function ( ) = 1 2 simulation algorithms are given as follows: a. Generate the realization of periodic Poisson process at interval [0, ] and period . b. Generate the estimator at a point with the bandwidth ℎ and uniform kernel. c. Determine mean of the estimator, mean of the theory, variance of the estimator, variance of the theory, bias, and the difference between the variance of the estimator and variance of the theory.

RESULTS AND DISCUSSION
The results of the simulation for each case of the value with s=3 are shown in Table 1, Table 2, and Table 3 as follows:  Table 1 shows the bias values and the difference between the variance of the theory and variance of the simulation. It can be seen that the bias in Table 1 converges to zero faster than the bias in Table 2 and Table 3. The value of = 0.5 From this simulation, the larger values of requires a large interval of observation to determine the convergence of the estimator. Unlike the bias, the variance values in Table 1 are slower to converge to zero when compared to Table 2 and Table 3. So that the greater value of , than the smaller variance of the estimator. it can be seen that the greater the value of , then bias, which in this case represents the difference between the red points and the graph lines of intensity, is greater for the observation interval [0,100].

CONCLUSIONS AND SUGGESTIONS
From the simulation that has been done to see the convergence rate of bias and variance of the estimator, the results show that the greater the value , then the larger observation interval will be needed to achieve the convergence of estimators. This results should be proved theoretically by using estimator proposed and the Theorems related.
The suggestions for the next simulation are to provide a numerical simulation for each case of Theorem 1, Theorem 2, and Theorem 3 specifically. It could be done by taken the bandwidth function which satisfies the given conditions.