Asymptotic Approximations to the Bias and Variance of a Kernel-type Estimator of the Intensity of the Cyclic Poisson Process with the Linear Trend

From the previous research, a kernel-type estimator of the intensity ofthe cyclic Poisson process with the linear trend has been constructed using a singlerealization of the Poisson process observed in a bounded interval. This proposedestimator has been proved to be consistent as the size of the observation intervaltends to innity. In this paper, asymptotic approximations to its bias, variance andMSE (Mean-Squared-Error) are computed. Asymptotically optimal bandwidth isalso derived.DOI : http://dx.doi.org/10.22342/jims.17.1.8.1-9


Introduction
We consider a Poisson process N in [0, ∞) with (unknown) intensity function λ, which is assumed to be locally integrable and consists of two components, namely a periodic (cyclic) component with period τ > 0 and a linear trend component. Hence, for each s ∈ [0, ∞), the intensity function λ can be written as λ(s) = λ c (s) + as, (1) where λ c (s) is a periodic function with period τ and a is the slope of the linear trend. We do not assume any parametric form of λ c except it is periodic, that is the equation holds true for all s ∈ [0, ∞) and all k ∈ Z, where Z denotes the set of integers. We consider a Poisson process in [0, ∞) instead of the one in R, due to the intensity function λ has to satisfy (1) and must be nonnegative. By a similar reason, we restrict our attention to the case a > 0. Throughout we also assume that the period τ is known, but the slope a and the function λ c on [0, τ ) are both unknown. 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 (cf. (1)) is observed in an interval [0, n].
A consistent kernel-type estimator of λ c at a given point s ∈ [0, ∞) using a single realization N (ω) of the Poisson process N observed in interval [0, n] has been constructed in [5]. Our aim in this paper is to compute asymptotic approximations to the bias, variance and MSE (mean-squared error) of this estimator.
Throughout this paper we assume that s is a Lebesgue point of λ, that is we have lim h↓0 g. see [6], pp. 107-108), which automatically means that s is a Lebesgue point of λ c as well.
Since λ c is a periodic function with period τ , the problem of estimating λ c at a given point s ∈ [0, ∞) can be reduced to the problem of estimating λ c at a given point s ∈ [0, τ ). Therefore, thoughout we assume that s ∈ [0, τ ).
The rest of the paper is organized as follows. Our main results are presented in section 2. The proofs of these main results are presented in section 3 and section 4.

Main Results
Suppose that K : R → R is a function, called kernel, which satisfies the following conditions: (K1) K is a probability density function, (K2) K is bounded, and (K3) K has support in [−1, 1]. Let also h n , which is called bandwidth, be a sequence of positive real numbers converging to 0, that is h n ↓ 0 as n → ∞.
An estimator of the slope a has been proposed in [1], which is given bŷ The estimator of λ c at a given point s ∈ [0, τ ) constructed in [5], is given bŷ .
Note that the estimator given in (4), which is using a general kernel function K, is a generalization of the one proposed in [1], which only consider a uniform kernel. A kernel-type estimator of the intensity of a cyclic Poisson process without trend has been proposed and studied in [2] and [3].
In [1], statistical properties ofâ n given in (3) has been proved. We present again these results in the following lemma, due to they are needed in the proofs of our main results. Lemma 2.1. Suppose that the intensity function λ satisfies (1) and is locally integrable. Then we have and V ar (â n ) = 2a Consistency ofλ c,n,K (s) has been established in [5]. Main results of this paper are presented in Theorems 2.2 and 2.3. In the first theorem, an asymptotic approximation to the expectation of the estimator is presented, while an asymptotic approximation to its variance is given in the second theorem. (1) and is locally integrable. If, in addition, the kernel function K is symmetrical around 0 and satisfies conditions (K1), (K2), (K3), h n ↓ 0, h 2 n ln n → ∞, and λ c has finite second derivative λ ′′ c at s, then

Theorem 2.2. Suppose that the intensity function λ satisfies
as n → ∞.

Theorem 2.3. Suppose that the intensity function λ satisfies (1) and is locally integrable. If, in addition, the kernel function K satisfies conditions
We note that, the results presented in Theorems 2. By (7) and (8) as n → ∞. Now we consider the r.h.s. of (9). By minimizing the sum of its first and second terms, one can obtain an asymptotically optimal bandwidth, which is given by If this optimal bandwidth is used, then the MSE ofλ c,n,K (s) will converge to 0 of rate O((ln n) −4/5 ), as n → ∞.

Proof of Theorem 2.2
Expectation ofλ c,n,K (s) can be computed as follows Eâ n . (10) First we consider the first term on the r.h.s. of (10). This term can be written as where I denotes the indicator function. By a change of variable and using (2), the r.h.s. of (11) can be written as The first term on the r.h.s. of (12) can be written as By a change of variable, the quantity in (13) can be expressed as Since as n → ∞, the quantity in (14) can be written as as n → ∞. By a Taylor expansion, λ c (s + xh n ) can be written as as n → ∞. Substituting (17) to the r.h.s. of (16), we obtain λ c (s) as n → ∞. Since ∫ 1 −1 K(x)dx = 1 and the second term of (18) is equal to zero, we find the first term on the r.h.s. of (12) is equal to as n → ∞.
Next we consider the second term on the r.h.s. of (12). This term can be written as By (15), the quantity in (20) can be written as Since the kernel K is bounded and ∫ 1 −1 xdx = 0, the first term of (21) is equal to zero. A simple calculation shows that the second term of (21) is equal to as n → ∞. The third term of (21) is equal to an/(ln n) + O((ln n) −1 ) as n → ∞. Hence, the second term on the r.h.s. of (12) is equal to as n → ∞. Combining (19) and (22), we obtain that the first term on the r.h.s. of (10) is equal to as n → ∞. Now we consider the second term on the r.h.s. of (10). By (5) of Lemma 2.1, this quantity can be computed as follows as n → ∞. By the assumption h 2 n ln n → ∞, we have O((ln n) −1 ) = o(h 2 n ), as n → ∞. Combining the results in (23) and (24) we obtain as n → ∞. This completes the proof of Theorem 2.2. 2

Proof of Theorem 2.3
Variance ofλ c,n,K (s) can be computed as follows First, we consider the first term on the r.h.s. of (25). Since h n ↓ 0 as n → ∞, then for sufficiently large n, interval [s + kτ − h n , s + kτ + h n ] and [s + jτ − h n , s + jτ + h n ] are disjoint for all k ̸ = j. This implies, for all k ̸ = j, are independent. Hence, the variance in the first term on the r.h.s. of (25) can be computed as follows By a change of variable and using (2), the r.h.s. of (26) can be written as The first term on the r.h.s. of (27) can be written as Now we see that as n → ∞. Since s is a Lebesque point of λ c then as n → ∞. Since the kernel function K is bounded and having support [−1, 1], by (29) and (30), we obtain that the first term on the r.h.s. of (28) is equal to o((ln n) −2 (h n ) −1 )) = o((h n ln n) −1 ), as n → ∞. By a similar argument, the second term on the r.h.s. of (28) is equal to O(1/(h n ln n) 2 ) = o((h n ln n) −1 ), as n → ∞. This last result due to the assumption h n ln n → ∞ as n → ∞.
Next we consider the second term on the r.h.s. of (27). This term can be written as By (29), the first term of (31) can be simplified as as n → ∞. By a change of variable, the second term of (31) can be expressed as as n → ∞. Hence we have that the first term on the r.h.s. of (25) is equal to aτ h n ln n as n → ∞.
Next we consider the second term on the r.h.s. of (25). By (6) of Lemma 2.1, this quantity can be computed as follows ( s + n ln n ) 2 V ar (â n ) = ( s 2 + n 2 (ln n) 2 + 2sn ln n as n → ∞. Now we consider the third term on the r.h.s. of (25). Since the first term on the r.h.s. of (25) is O((h n ln n) −1 ) and its second term is o((h n ln n) −1 ) as n → ∞, by Cauchy-Schwarz inequality, we have that the third term on the r.h.s. of (25)  = aτ h n ln n as n → ∞. This completes the proof of Theorem 2.3. 2