A NOTE ON ESTIMATION OF THE GLOBAL INTENSITY OF A CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR TREND

A NOTE ON ESTIMATION OF THE GLOBAL INTENSITY OF A CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR TREND I WAYAN MANGKU Deprtment of Mthemtics, Fculty of Mthemtics nd Nturl Sciences, Bogor Agriculturl University Jl. Mernti, Kmpus IPB Drmg, Bogor, 668 Indonesi Abstrct. We construct nd investigte consistent kernel-type nonprmetric estimtor of the globl intensity of cyclic Poisson process in the presence of liner trend. It is ssumed tht only single reliztion of the Poisson process is observed in bounded window. We prove tht the proposed estimtor is consistent when the size of the window indefinitely expnds. The symptotic bis nd vrince of the proposed estimtor re computed. Bis reduction of the estimtor is lso proposed. 99 Mthemtics Subject Clssifiction: 6G55, 62G5, 62G2. Keywords nd Phrses: cyclic Poisson process, globl intensity, liner trend, nonprmetric estimtion, consistency, bis, vrince.. Introduction Let X be Poisson point process on [, with bsolutely continuous σ-finite men mesure µ w.r.t. Lebesgue mesure ν nd with unknown loclly integrble intensity function λ, i.e., for ny bounded Borel set B we hve µb ENB λsds <. Furthermore, λ is ssumed to consist of two components, nmely periodic or cyclic component with period > nd unknown liner trend component. In other words, for ny point s [,, we cn write the intensity function λ s B λs λ c s + s. where λ c s is periodic function with period nd denotes the slope of the liner trend. In the present pper, we do not ssume ny prmetric form of λ c except tht it is periodic. Tht is we ssume

2 I WAYAN MANGKU tht the equlity λ c s + k λ c s.2 holds for ll s [, nd k Z. Here we consider Poisson point process on [, insted of, for instnce, on R becuse λ hs to stisfy. nd must be non negtive. For the sme reson we lso restrict our ttention to the cse >. Furthermore, let W, W 2,... be sequence of intervls [, W n ], n, 2,..., such tht the size or the Lebesgue mesure νw n W n of W n is finite for ech fixed n N, but W n,.3 s n. Suppose now tht, for some ω Ω, single reliztion Xω of the Poisson process X defined on probbility spce Ω, F,P with intensity function λ cf.. is observed, though only within bounded intervl, clled window W [,. Our gol in this pper is to construct consistent non-prmetric estimtor of the globl intensity θ µ[,] λ c sds.4 of the cyclic component λ c of λ in., using only single reliztion Xω of the Poisson process X observed in W : W n. We lso compute the symptotic bis nd vrince of the proposed estimtor. The present pper ims t extending previous work for the purely cyclic cse, i.e., cf. [2] to the more generl model.. Prllel to this pper, Helmers nd Mngku [3] consider the problem of estimting the cyclic component λ c t given point s [, of the intensity given in. of cyclic Poisson process in the presence of liner trend. In fct, the estimtor ˆθ n,b given in 3.2 is used in [3] for correcting the bis of the estimtor of λ c. Estimtion of the intensity function λ c t given point s [, of purely cyclic Poisson process, tht is Poisson proces hving intensity given in. with, hs been investigted, mong others, in [4], [5], [6], [8], nd [9]. There re mny prcticl situtions where we hve to use only single reliztion for estimting intensity of cyclic Poisson process. A review of such pplictions cn be seen in [4], nd number of them cn lso be found in [], [7], [9], [] nd [2]. In section 2 we present the estimtors nd some preliminry results. These results re evluted in section 3, by Monte Crlo simultion. This evlution leds to bis corrected estimtor. Proofs of ll theorems re given in section 4. 2. The estimtors nd preliminry results In this pper, we focus to the cse when the period is known, but the slope nd the function λ c on [, re both unknown. Note lso

JMA, VOL. 4, NO.2, DESEMBER, 25,- 3 tht, in mny prcticl pplictions, we know the period, for instnce: one dy, one week, one month, one yer, etc. In this sitution we my define estimtors of respectively nd θ s follows nd ˆθ n : ln Wn k â n : 2XW n W n 2, 2. X[k, k + ] W n k To obtin the estimtor â n of it suffices to note tht EXW n 2 W n 2 + O W n, â n 2 + W n.2.2 ln Wn s n, which directly yields the estimtor given in 2.. Note lso tht if X were Poisson process with intensity λs s, then â n would be the mximum likelihood estimtor of cf. []. Next, we describe the ide behind the construction of the estimtor ˆθ n of θ. For ny k N, we cn write θ k+ k λ c sds. 2.3 Let L n : k k Ik W n. Then, by 2.3, we cn write θ L n k L n k L n L n k k k k k k+ k k+ k k+ k k λ c sis W n ds λs sis W n ds λsis W n ds s + kis + k W n ds EX[k, k + ] W n L n k k s L n k Is + k W nds k Is + k W n ds. 2.4 L n k By noting tht k k Is + k W n L n + O L n nd s ds 2 /2, we see tht the second term on the r.h.s. of 2.4 is

4 I WAYAN MANGKU /2. We lso will use the fct tht Is + k W n ds Wn + ζ n L n L n k W n L n + ζ n L n W n L n cf. definition of ζ n in line below 2., where ζ n for ll n. By these pproximtions nd by pproximting the expecttion in the first term on the r.h.s. of 2.4 with its stochstic counterprt, we obtin θ EX[k, k + ] W n L n k 2 + W n. 2.5 L n k From the in 2.5 nd noting tht L n ln W n / s n, we see tht EX[k, k + ] W n θ n ln Wn k 2 + W n,2.6 k ln Wn cn be viewed s n estimtor of θ, provided both the period nd the slope of the liner trend re ssumed to be known. If is unknown, we replce by â n cf. 2. nd one obtins the estimtor of θ given in 2.2. In Helmers nd Mngku [3] hs been proved the following lemm. Lemm 2.. Suppose tht the intensity function λ stisfies. nd is loclly integrble. Then we hve E â n + 2θ W n + O 2.7 W n 2 nd V r â n 2 W n + O 2 W n 3 2.8 s n. Hence, by.3, â n is consistent estimtor of ; its men-squred error MSE is given by MSEâ n 4θ 2 +2 W n 2 + O W n 3 s n. Consistency of ˆθ n is estblished in Theorem 2.2. In Theorem 2.3 we compute the symptotic pproximtions to respectively the bis nd vrince of the estimtor ˆθ n. Theorem 2.2. Suppose tht the intensity function λ stisfies. nd is loclly integrble. Then we hve ˆθ n p θ, 2.9

JMA, VOL. 4, NO.2, DESEMBER, 25,- 5 s n. In other words, ˆθ n is consistent estimtor of θ. In ddition, the MSE of ˆθ n converges to, s n. Theorem 2.3. Suppose tht the intensity function λ stisfies. nd is loclly integrble. Then we hve Eˆθ n θ 2 γθ γ/2 + ζ n + o 2. ln W n / ln W n s n, where where γ.577.. is Euler s constnt nd ζ n k Ix+k W ndx W n / nd ζ n for ll n. In ddition, we lso hve V r ˆθn ln W n / + θ/ + /2π2 /6 2 γ ln W n / 2 +o 2. ln W n 2 s n. 3. Simultions nd bis reduction For the simultions, we consider the intensity function { } 2πs λs λ c s + s A exp ρ cos + φ + s, tht is., where λ c is the intensity function discussed in Vere-Jones [4]. We chose ρ, 5, φ nd.5. With this choice of the prmeters, we hve { } 2πs λs A exp cos +.5s. 3. In our simultions we consider three vlues of θ, which is determined by the choice of A, nmely i smll vlue of θ, i.e. θ.266 A, ii moderte vlue of θ, i.e. θ 2.5322 A 2 nd iii lrge vlue of θ, i.e. θ 5.644 A 4 cf. Remrk 3.4. We use W n [, ]. Exmple 3.. In this exmple we study the performnce of the estimtor ˆθ n in 2.2, in the cse tht the intensity function λs is given by 3.. i For the cse smll vlue of θ, θ.266, by 2. nd 2., we obtin the symptotic pproximtions to respectively the bis nd the vrince of ˆθ n s follows: Bisˆθ n.3736 nd V rˆθ n.232. From the simultion, using M 4 independent reliztions of the process X observed in the W n [, ], we obtin respectively ˆBisˆθ n.3793 nd ˆV rˆλ n.22, where ˆBisˆθ n is the smple men minus the true vlue θ nd ˆV rˆθ n is the smple vrince. Summrizing, we hve Bisˆθ n

6 I WAYAN MANGKU ˆBisˆθ n.3736.3793.57 nd V rˆθ n ˆV rˆλ n.232.22.. ii For θ 2.5322, by 2. nd 2., nd from the simultion M 4 we obtin respectively Bisˆθ n ˆBisˆθ n.737.733.66 nd V rˆθ n ˆV rˆλ n.38.364.7. iii For θ 5.644, by 2. nd 2., nd from the simultion M 4 we obtin respectively Bisˆθ n ˆBisˆθ n.3938.42.272 nd V rˆθ n ˆV rˆλ n.677.634.43. From Exmple 3., we see tht the symptotic pproximtions to the bis nd vrince given in 2. nd 2. predict quite well the vrince nd bis of the estimtor ˆθ n in finite smples. However, we see tht the bis of ˆθ n is quite big. We cn reduce this bis by dding n estimtor of the second term on the r.h.s. of 2. into ˆθ n. By employing this ide, we obtin bis corrected estimtor of θ s follows ˆθ n,b : ˆθ n + 2 γˆθ n γ/2 + ζ n â n. 3.2 ln W n / Theorem 3.2. Suppose tht the intensity function λ stisfies. nd is loclly integrble. Then we hve Eˆθ n,b θ + o, 3.3 ln W n s n, nd V r s n. ˆθn,b ln W n / + θ/ + /2π2 /6 + 2 γ ln W n / 2 +o, 3.4 ln W n 2 Exmple 3.3. In this exmple we study the performnce of the estimtor ˆθ n,b in 3.2, in the cse tht the intensity function λs is given by 3.. i For θ.266, from the simultion M 4 nd by 3.4, we obtin respectively ˆBisˆθ n,b.9 nd V rˆθ n,b ˆV rˆθ n,b.283.354.7. ii For θ 2.5322, from the simultion M 4 nd by 3.4, we obtin respectively ˆBisˆθ n,b.256 nd V rˆθ n,b ˆV rˆθ n,b.43.578.47. iii For θ 5.644, from the simultion M 4 nd by 3.4, we obtin respectively ˆBisˆθ n,b.3993 nd V rˆθ n,b ˆV rˆθ n,b.728.5.323.

JMA, VOL. 4, NO.2, DESEMBER, 25,- 7 It is cler tht the bis of the estimtor ˆθ n,b is much smller thn the bis of the originl estimtor ˆθ n. So the bis reduction proposed in 3.2 works. Remrk 3.4. A cutionry remrk on the rnge of vlidity of Theorems 2.3 nd 3.2 in prcticl pplictions is importnt here. The reminder terms in 2., 2., 3.3 nd 3.4 will depend on the vlues of the prmeters involved, such s θ,, nd. In order to hve the pproximtions in 2. nd 3.3 to be vlid, θ nd should not too big compred to ln W n /. The second order pproximtions in 2. nd 3.4 is vlid provided θ nd re not too big compred to ln W n / 2. Note tht, in cse iii of Exmples 3. nd 3.3 we hve θ 5.644 which is lmost the sme s the vlue of ln W n / 5.2983. For this reson, we do not consider the cse where the vlue of θ is lrger thn tht in cse iii of the current exmples. To conclude this section, we remrk tht the bis corrected estimtor ˆθ n,b is to be preferred in prcticl pplictions. We first prove Theorem 2.3. Proof of Theorem 2.3 4. Proofs First we prove 2.. Note tht Eˆθ n ln W n / k 2 + W n ln W n / The first term on the r.h.s. of 4. is equl to ln W n / ln W n / ln W n / k k k k + ln W n / + ln W n / k+ k λ c x x EX[k, k + ] W n k Eâ n. 4. λxix W n dx λ c x + k + x + kix + k W n dx k k k Ix + k W ndx k Ix + k W ndx Ix + k W n dx, 4.2 k

8 I WAYAN MANGKU where we hve used. nd.2. Note tht k Ix + k W n ln W n / + γ + o 4.3 k s n uniformly in x [,] cf. [3], p.5. Using 4.3, simple clcultion shows tht the first term on the r.h.s. of 4.2 is equl to θ + θγ ln W n / + o ln W n 4.4 s n, nd its second term is equl to 2 + γ/2 ln W n / + o ln W n 4.5 s n. Clerly Ix + k W n dx W n / + ζ n 4.6 k cf. definition of ζ n in line below 2.. By 4.6, the third term on the r.h.s. of 4.2 reduces to W n ln W n / + ζ n ln W n /. 4.7 Using 2.7, the second term on the r.h.s. of 4. reduces to 2 W n ln W n / 2θ ln W n / + O W n 4.8 s n. Combining 4.4, 4.5, 4.7 nd 4.8, we obtin 2.. Next we prove 2.. Let A n nd B n denote respectively the first nd second term on the r.h.s. of 2.2. In other words, we write ˆθ n A n B n. Then we cn compute the vrince of ˆθ n s follows V r ˆθn V r A n + V r B n 2Cov A n,b n. 4.9 Note tht, for ny j k, j,k, 2,..., we hve X[j, j +] W n nd X[k, k + ] W n re independent. Then V ra n cn be

computed s follows JMA, VOL. 4, NO.2, DESEMBER, 25,- 9 V r A n ln W n / 2 k 2 2V r X[k,k + ] W n k 2 ln W n / 2 2 ln W n / 2 + 2 ln W n / 2 + ln W n / 2 k k 2 λ c x x k k λ c x + x + kix + k W n dx k k 2Ix + k W ndx k 2Ix + k W ndx k Ix + k W ndx. 4. Here we hve used. nd.2. Note tht k k 2Ix + k W n π2 + o 4. 6 s n, uniformly in x [,] cf. [3], p.34. Using 4., simple clcultion shows tht the first term on the r.h.s. of 4. is equl to θ/π 2 /6 ln W n / + o 4.2 2 ln W n 2 s n, nd its second term is equl to /2π 2 /6 ln W n / + o 2 ln W n 2 4.3 s n. By 4.3, the third term on the r.h.s. of 4. reduces to ln W n / + γ ln W n / + o 4.4 2 ln W n 2 s n. Combining 4.2, 4.3 nd 4.4, we obtin V r A n s n. ln W n / + θ/ + /2π2 /6 + γ ln W n / 2 + o ln W n 2 4.5

I WAYAN MANGKU Next we consider the second nd third term on the r.h.s. of 4.9. By 2.8, we obtin 2 W n 2 V rb n /2 + ln W n / W n + O 2 W n 3 2 ln W n / + O 4.6 2 W n ln W n / s n. Next we compute CovA n,b n s follows CovA n, B n W n ln Wn + /2 2 W n 2 ln Wn k Cov X[k,k + ] W n, XW n k 2 W n ln Wn + 2 W n 2 ln Wn k V r X[k,k + ] W n k 2 W n ln Wn + 2 W n 2 ln Wn λ c x + x k Ix + k W ndx k 2 + W n ln Wn + 2 W n 2 ln Wn Ix + k W n dx. k 4.7 By 4.3, we see tht the first term on the r.h.s. of 4.7 is of order O W n ln W n, s n. By 4.6, the second term on the r.h.s. of 4.7 reduces to 2ln W n 2 + O W n ln W n, s n. Hence, the third term on the r.h.s. of 4.9 is equl to 4 2CovA n,b n ln W n / + O 2 W n ln W n / 4.8 s n. Combining 4.5, 4.6 nd 4.8, we obtin 2.. This completes the proof of Theorem 2.3. Proof of Theorem 2.2 By 2. nd the ssumption.3, we obtin Eˆθ n θ + o 4.9

JMA, VOL. 4, NO.2, DESEMBER, 25,- s n, while 2. nd ssumption.3 imply V r ˆθn o 4.2 s n. Together 4.9 nd 4.2, imply 2.9. This completes the proof of Theorem 2.2. Proof of Theorem 3.2 First we prove 3.3. To do this, we first rewrite the estimtor ˆθ n,b in 3.2 2 γ ˆθ n,b : + ˆθ n γ/2 + ζ n ln W n / ln W n / ân. 4.2 By 2., we see tht the expecttion of the first term on the r.h.s. of 4.2 is equl to θ + γ/2 + ζ n + o 4.22 ln W n / ln W n s n. By 2.7, the expecttion of the second term on the r.h.s. of 4.2 reduces to γ/2 + ζ n + o 4.23 ln W n / ln W n s n. Combining 4.22 nd 4.23 we obtin 3.3. Next we prove 3.4. Using 4.2, V rˆθ n,b cn be computed s follows V rˆθ n,b + 2 2 2 γ V rˆθ n + γ/2 + ζ n 2 2 ln W n / ln W n / 2V râ n + 2 γ ln W n / γ/2 + ζn ln W n / Covˆθ n,â n. 4.24 By 2., we see tht the first term on the r.h.s. of 4.24 is equl to ln W n / + θ/ + /2π2 /6 2 γ 22 γ + ln W n / 2 ln W n / 2 +o ln W n 2 ln W n / + θ/ + /2π2 /6 + 2 γ + o ln W n / 2 ln W n 2 4.25 s n. Note tht, the 2 γ on the r.h.s. of 2. is replced by +2 γ on the r.h.s. of 4.25. By 2.8, it esily seen tht the second term on the r.h.s. of 4.24 is of order O W n 2 ln W n 2, which is oln W n 2 s n. Finlly, Cuchy-Schwrz inequlity

2 I WAYAN MANGKU shows tht the third term on the r.h.s. of 4.24 is of negligible order oln W n 2 s n. Hence, we hve 3.4. This completes the proof of Theorem 3.2. References [] J. D. Dley nd D. Vere-Jones 988, An Introduction to the Theory of Point Processes. Springer, New York. [2] R. Helmers nd I W. Mngku 2, Sttisticl Estimtion of Poisson intensity functions. Proceedings of the SEAMS - GMU Interntionl Conference on Mthemtics nd Its Applictions, Yogykrt, July 26-29, 999, p. 9-2. [3] R. Helmers nd I W. Mngku 25, Estimting the intensity of cyclic Poisson process in the presence of liner trend, submitted to Ann. Inst. Sttist. Mth. [4] R. Helmers, I W. Mngku, nd R. Zitikis 23, Consistent estimtion of the intensity function of cyclic Poisson process. J. Multivrite Anl. 84, 9-39. [5] R. Helmers, I W. Mngku, nd R. Zitikis 25, Sttisticl properties of kernel-type estimtor of the intensity function of cyclic Poisson process. J. Multivrite Anl., 92, -23. [6] R. Helmers nd R. Zitikis 999, On estimtion of Poisson intensity functions. Ann. Inst. Sttist. Mth. 5 2, 265-28. [7] A. F. Krr 99, Point Processes nd their Sttisticl Inference. Second Edition, Mrcel Dekker, New York. [8] Y. A. Kutoynts 984, On nonprmetric estimtion of intensity function of inhomogeneous Poisson Processes. Probl. Contr. Inform. Theory, 3, 4, 253-258. [9] Y. A. Kutoynts 998, Sttisticl Inference for Sptil Poisson Processes. Lecture Notes in Sttistics, Volume 34, Springer, New York. [] I W. Mngku, I. Widiystuti, nd I G. P. Purnb 26, Estimting the intensity in the form of power function of n inhomogeneous Poisson process. Submitted for publiction. [] R. D. Reiss 993, A Course on Point Processes. Springer, New York. [2] D. R. Snyder nd M. I. Miller 99, Rndom Point Processes in Time nd Spce. Springer, New York. [3] E. C. Titchmrsh 96, The Theory of Functions. Oxford Univeristy Press, London. [4] D. Vere-Jones 982, On the estimtion of frequency in point-process dt. J. Appl. Probb., 9A, 383-394.