Estimating the intensity of a cyclic Poisson process in the presence of additive and multiplicative linear trend

In this paper we survey some results on estimation of the intensity function of a cyclic Poisson process in the presence of additive and multiplicative linear trend. We do not assume any parametric form for the cyclic component of the intensity function, except that it is periodic. Moreover, we consider the case when there is only a single realization of the Poisson process is observed in a bounded interval. The considered estimators are weakly and strongly consistent when the size of the observation interval indefinitely expands. Asymptotic approximations to the bias and variance of those estimators are presented.


Introduction
In this paper we survey some results on estimation of the intensity function of a cyclic Poisson process in the presence of additive and multiplicative linear trend. The problems of estimating intensity of a cyclic Poisson process using a single realization of the process arises frequently in many diverse areas, including: communications, hydrology, meteorology, insurance, reliability, medical sciences, and seismology. Some existing methods using parametric approach are as follows. Rathbun and Cressie [14] investigated maximum likelihood estimators and Bayes estimators for regular parametric models with unknown finite dimensional parameter.
Helmers and Zitikis [4] consider a uniform kernel type estimator for λ(s) in the case where λ is a parametric function of spatial location. These authors focus their attention to the case that X is a Poisson process on [0, ∞) with intensity function λ(s) = exp α + βs + γs 2 + K 1 sin(ω 0 s) + K 2 cos(ω 0 s) , s > 0, where α, β, γ, K 1 , and K 2 are unknown parameters, and ω 0 is a known 'frequency'.
In this paper, we consider the problem of estimating the intensity λ of a cyclic Poisson process (with and without trend) without assuming any parametric form of λ, except that it is periodic.
The rest of the paper is organized as follows. In section 2 we rewiew some results on estimation of the intensity of a (purely) cyclic Poisson process. Some results on estimation of the intensity of a cyclic Poisson process in the presence of additive linear trend are presented in section 3. In section 4 we present some results on estimation of the intensity of a cyclic Poisson process in the presence of multiplicative linear trend. Some conclusions are given in section 5.

Estimating intensity of a cyclic Poisson process
In this section we rewiew some results on estimation of the intensity of a cyclic Poisson process. Let N c be a Poisson process on [0, ∞) with (unknown) locally integrable intensity function λ. We assume that λ is a periodic function with (known) period τ . We do not assume any parametric form of λ, except that it is periodic. That is, for each point s ∈ [0, ∞) and all k ∈ Z, with Z denotes the set of integers, we have λ(s + kτ ) = λ(s).
Suppose that, for some ω ∈ Ω, a single realization N c (ω) of the Poisson process N c defined on a probability space (Ω, F, P) with intensity function λ is observed, though only within a bounded interval [0, n].
Note that, since λ is a periodic function with period τ , the problem of estimating λ at a given point s ∈ [0, ∞) can be reduced into a problem of estimating λ at a given point s ∈ [0, τ ). Hence, for the rest of this paper, we will assume that s ∈ [0, τ ).
We also note that, the meaning of the asymptotic n → ∞ in this paper is somewhat different from the classical one. Here n does not denote our sample size, but it denotes the length of the interval of observations. The size of our samples is a random variable denoted by N c ([0, n]).
Let K : R → R be a real valued function, called kernel, which satisfies the following conditions: (K1) K is a probability density function, (K2) K is bounded, and (K3) K has (closed) support [−1, 1]. Let also h n be a sequence of positive real numbers converging to 0, that is, h n ↓ 0, as n → ∞.
The estimator of λ at a given point s ∈ [0, τ ) is given bŷ For the rest of this section we assume that the intensity function λ is periodic and locally integrable, the kernel K satisfies conditions (K1), (K2), (K3), and h n ↓ 0 as n → ∞.
Our first result is as follows. If nh n → ∞, then as n → ∞, provided s is a Lebesgue point of λ. In other words,λ n,K (s) is a consistent estimator of λ(s). In addition, the Mean-Squared-Error (MSE) ofλ n,K (s) converges to 0, as n → ∞, that The as n → ∞, provided s is a Lebesgue point of λ.
Under slightly stronger assumption, we also have strong consistency ofλ n,K (s) as follows. If as n → ∞, provided s is a Lebesgue point of λ. In other words,λ n,K (s) is a strongly consistent estimator of λ(s).
Asymptotic approximation to the bias ofλ n,K (s) can be described as follows. If the intensity function λ has finite second derivative λ at s, the kernel K is symmetric and nh 2 n → ∞, then as n → ∞.
Asymptotic approximation to the variance ofλ n,K (s) is given by as n → ∞, provided s is a Lebesgue point of λ.
From (5) and (6) we obtain asymptotic approximation to the MSE ofλ n,K (s), which is given by as n → ∞.
By minimizing the main terms for the variance and the squared bias on the r.h.s. of (7), we then obtain the optimal choice of h n , which is given by With this choice of h n , the optimal rate of decrease of M SE(λ n,K (s)) is of order O(n −4/5 ) as n → ∞.
Mathematical details of the results presented in this section can be found in [5] and [6]. Some related work can be found in [3], [10], and [15].
where λ c (s) is a periodic function with (known) period τ and a denotes the slope of the linear trend.
Suppose now that, for some ω ∈ Ω, a single realization N (ω) of the Poisson process N defined on a probability space (Ω, F, P) with intensity function λ (cf. (8)) is observed, though only within a bounded interval [0, n].
An estimator of a is given byâ The expected value and variance ofâ n are given by respectively as n → ∞, where θ = τ −1 τ 0 λ c (s)ds, the global intensity of the periodic component λ c . Hencê a n is a consistent estimator of a. Its MSE is given by as n → ∞.
The estimator of λ c , the cyclic component of the intensity function λ, is given bŷ .
If we are interested in estimating λ(s) at a given point s, then λ(s) can be estimated bŷ λ n,K (s) =λ c,n,K (s) +â n s.
For the rest of this section, we assume that the intensity function λ satisfies (8) and is locally integrable, the kernel K satisfies conditions (K1), (K2), (K3), and h n ↓ 0 as n → ∞. Asymptotic approximation to the bias ofλ c,n,K (s) can be described as follows. If the intensity function λ has finite second derivative λ at s and h 2 n ln n → ∞, then as n → ∞.
Asymptotic approximation to the variance ofλ c,n,K (s) is given by V ar λ c,n,K (s) = aτ 2h n ln n 1 −1 as n → ∞, provided s is a Lebesgue point of λ c .
From the asymptotic approximations to the bias and variance ofλ c,n,K (s) we can obtain asymptotic approximations to the MSE ofλ c,n,K (s), which is given by M SE λ c,n,K (s) = aτ 2h n ln n 1 −1 as n → ∞.
By minimizing the main terms for the variance and the squared bias of M SE(λ c,n,K (s)), we then obtain the optimal choice of h n , which is given by With this choice of h n , the optimal rate of decrease of M SE(λ c,n,K (s)) is of order O((ln n) −4/5 ) as n → ∞.
Mathematical details of the results presented in this section can be found in [2], [11], and [12]. Some related work can be found in [7].

Estimating the intensity obtained as the product of a periodic function with the linear trend
In this section, we survey some results on estimation of the intensity of a cyclic Poisson process in the presence of multiplicative linear trend. Let N * be a Poisson process on [0, ∞) with (unknown) locally integrable intensity function λ. The intensity function λ is assumed to be a periodic function multiplied by the linear trend. That is, for any given point s ∈ [0, ∞), the intensity function λ can be written as where λ * c (s) is a periodic function with known period τ and a > 0 denotes the slope of the linear trend.
Since aλ * c is also a periodic function with period τ , then, without loss of generality, the intensity function λ given in (9) can also be written as where λ c (s) = aλ * c (s).
Suppose that, for some ω ∈ Ω, a single realization N * (ω) of the Poisson process N * defined on a probability space (Ω, F, P) with intensity function λ given by (10) is observed, though only within a bounded interval [0, n].
The estimator of λ c at a given point s ∈ [0, τ ) is given bŷ For the rest of this section we assume that the intensity function λ satisfies (10) and is locally integrable, the kernel K satisfies conditions (K1), (K2), (K3), and h n ↓ 0 as n → ∞.
Our first result of this section is as follows. If n 2 h n / ln n → ∞, then as n → ∞, provided s is a Lebesgue point of λ. Hence,λ * c,n,K (s) is a consistent estimator of λ c (s). In addition, the MSE ofλ * c,n,K (s) converges to 0, as n → ∞, that is we have M SE(λ * c,n,K (s)) → 0, as n → ∞.
Asymptotic approximation to the bias ofλ * c,n,K (s) can be described as follows. If λ c has finite second derivative λ c at s, the kernel K is symmetric and nh 2 n → ∞, then Eλ * c,n,K (s) = λ c (s) + as n → ∞.
Asymptotic approximation to the variance ofλ * c,n,K (s) is given by From asymptotic approximations to the bias and variance ofλ * c,n,K (s) we obtain an asymptotic approximation to the MSE ofλ * c,n,K (s), which is given by as n → ∞.
By minimizing the sum of the main terms for the variance and the squared bias of M SE(λ * c,n,K (s)), we then obtain the optimal choice of h n , which is given by With this choice of h n , the optimal rate of decrease of M SE(λ * c,n,K (s)) is of order O((n 2 / ln n) −4/5 ) as n → ∞.
Mathematical details of the results presented in this section can be found in [8]. Some related work can be found in [1], [9], and [13].

Conclusion
In this paper we consider Poisson processes with three models of intensity functions, namely: (i) λ(s) = λ c (s) (purely cyclic Poisson process), (ii) λ(s) = λ c (s) + as (cyclic Poisson process in the presence of additice linear trend) , and (iii) λ(s) = (λ c (s))as (cyclic Poisson process in the presence of multiplicatice linear trend). Estimators of λ c (s) in these three cases are denoted respectively by (i)λ n,K (s), (ii)λ c,n,K (s) and (iii)λ * c,n,K (s).
First we obtain that all these three estimators are having the same asymptotic approximations to the bias, that is as n → ∞. However, those three estimators have different asymptotic approximations to the variance, which are as follows.