Parameter Estimation And Application Of The Fractional Compound Poisson Process

Recently the main experimentally observed features of anomalous kinetic phenom-ena in complex systems are non-exponential time, which is caused by long-run memory effects in complex systems. Therefore, the fractional Poisson process(FPP)have been studied by several authors, such as Laskin, Repin, Mainardi et al. In this process the waiting times distribution has been described in terms of functions of Mittag-Leffler, Wright or other types.Parameter estimation issues have also been addressed in the literature. Sapozh-nikov discussed the estimation of parameters of a compound Poisson process. Bi-han gave the nonparametric estimation for compound Poisson process on compact Lie groups. Cahoy provided an algorithm for simulation of FPP waiting times, derived method-of-moments estimators for parameters and showed asymptotic normality of the estimators. Parameter estimation for other fractional processes can be found in other literatures. Here we try to consider method-of-moments estimators and quantile estimator for parameter in the fractional compound Poisson process(FCPP). And we gave the application of the FCPP, that is, the fractional Poisson surplus process(FPSP), we also provided the pdf of the FPSP and the pdf of the ruin time T of the FPSP with Wright function.The paper is organized as follows. Firstly, we recall some properties of the FPP and FCPP. Secondly, we give estimators for two parameters in the FCPP based on the first and second order moments and the corresponding asymptotic normality. Thirdly, quantile estimators for two parameters in the FCPP are given. Finally, the application of the the FCPP, that is, the fractional Poisson surplus process.Firstly,We recall concept and some properties of the FPP and FCPP.Let{T}i∞be a sequence of independent identically distributed and non-negative random variables interpreted as waiting times between subsequent events arriving at random time, obeying the given probability. Here,{tk}k∞=0 are defined by t0= 0, tk= ∑ik=1Ti,then we get a renewal process associated to a counting process{N(t),t> 0} defined as Where N(t) is the random number of events up to time t, i.e., the random renewal number occurring in (0,t], tk is so-called renewal time. When the waiting time distri-butions were described by function of Mittag-Leffler with one-parameter v ∈ [0,1], the above renewal process is the fractional Poisson process, in the limit v= 1,it becomes the Poisson process. For notational convenience, we use{Nv(t),t≥ 0,0< v≤ 1} to represent the fractional Poisson process.Then we obtain the probability mass function: where Ev(z) is Mittag-Leffler function with one-parameter v ∈ [0,1], defined as it is an entire function of order v, and reduces an exp(z) for v= 1.Cahoy gave the following algorithm for simulation of the FPP waiting times. Let U1, U2 and U3 be independent, and uniformly distributed in [0,1]. Then, the fractional Poisson process waiting timesGiven the fractional Poisson process{Nv(t),t≥0,0< v≤ 1}, waiting times {Ti}i∞=1 were distributed by the function of Mittag-Leffler type with one-parameter v ∈ [0,1]. Assume {Xi}i∞=1 were independent identically distributed (real, not nec-essary positive) random variables with cumulated distributed function F(x) and the probability density function f(x), and independent of waiting times{Ti}i∞=11, then is called the compound fractional Poisson process. In a physical context, the Xis represent the jump of a diffusing particle and the resulting random walk model is known as the continuous time random walk, abbreviated as CTRW.In the second place, we derive the method-of-moments estimators for parameters v and μ. Firstly, we calculate the first and second order moment by differentiating the mgf once and twice. The expansion of the Mv(s,t) is differentiating Mv(s, t) with respect to s lead to note that g(0)= 1, then we can get the first and second order moments where di= E{X1i),i= 1,2,...If we have observations Yv,1(t),..., Yv,n(t), then the first and second order sample moments can be given as Equating the moments with the sample moments, we have we have μtv= m1(t)T(v+1)/d1, then substitute it, we obtain where B(v,v) is the Beta function. Thus estimate v can be obtained numerically. After obtaining v, we can estimate μ byLet θ= (v,μ) and θ = (v,μ), The following theorem gives asymptotic normality of the estimator θ.theoreml:Let Yv,1(t),..., Yv,n(t) be observations from the FCPP{Yv(t),t> 0,0< v≤ 1} and｜u4｜< ∞, then (?)n(θ-θ)â†'L N2{0, G∑G’), where and four partial derivatives in G, u3(t) and u4t) are given in the proof.Thirdly, we derive the method of parameter estimation based on the quantiles.Let X1, X2, …, Xn be a random sample from a population has the density function f(x), for the given p ∈ (0,1), f(x) is continue at î–¶, and p-quantile of the population, and ξ> 0. Define k as k= np+o((?)n):then for the fth ordered statistic X(k), it holds thatThus, for given p1,p2 ∈ (0,1), f(x) is continuous at î–¶= (ξp1,ξp2), which is the P = (p1,P2)-quantile of population, and ξp1,ξp2> 0, define k1, k2 as as ki= npi+o((?)n), i= 1,2, and the (k1,k2)-the joint ordered statistics of the sample by (X(k1)),(k2), then X(k)= (X(k),X(k2)) has the asymptotic distribution: where,Let{Yv(t),0< v≤ l,t> 0} be a given compound fractional Poisson process. From the previous derivation, it follows that the mean and variance of the process areif the first and second order moments dl and d2 of random sample Xzs both exist, then the random variable seriesNext, consider the estimation of the parameter p and v by the quantile.Let (Yv ,1(t), Yv ,2(t), ... , Y, ...(t)) be a sample vector of the FCPP evaluated at time t, Yv,(k)(t) = (Yv,(k})(t)X,(k2)(t)), ki = npi + o((?)n),z = 1,2 is the p = (p1,p2)-th quantile vector of the sample, Yv,p(t) = (Yv,p1 (t), Yv,p2 (t)) is the p-quantile vector of population quantiles, zp = (zp1,zp2) is the vector of p-population quantile of standard normal variable Zt.For the standard normal distribution Zt, its distribution function }D(x) is contin-uously derivable on (-∞, +∞). Then we can define the pi-quantile of Zt as using Zt = 1/σ(t)) (Yv(t) - u(t)), we get namely, thus,Using the fact that the pi-quantile of the sample of FCPP {Yv(t), t > 0, 0 ≤ v ≤ 1} is equal to the pi-quantile of the population, we can construct the following equations to estimate u(t) and u(t). Based on them, we can estimate the parameters p and v of the process. Let then the above equations can be formulated in the form of matrix as: the vector of estimator ofθ(t)isθ(t)=A-1(zp)Yv,(k)(t),and we shall obtain where,Let us choose the level p1=p∈(0,0.5),p2=1-p1,then since the normal is even,we shall have zp2=z1-p=-zp,f(zp)=f(zp2).Hence σ12=0,which implies asymptotic independence of the estimators u(t)and σ(t),and thus let therefore then the value of v can be obtained by method of numerical analysis where B(v, v) is Beta function, and similarlyAt last,we give the definition and related properties of the fractional poisson cor-rective risk model and the fractional poisson surplus process.The random process{Yv(t),μ> 0,0< v≤1,t≥ 0} is called the fractional poisson corrective risk model, when it satisfies: where, (i) Nv(t) is the cumulative number of claims in [0,t], and Nv(0)= 0, for the fixedt> 0, Nv(t) is the random variables who can be the non-negative integer, and (ii) Xi(i= 1,2, … ) is the ith the claim size in [0,t], is the independent identicallydistributed random variables, and independent of Nv(t); (iii) for the fixed t> 0, Yv(t) is the total amount of the claim in [0, t], Yv/(0)= 0, andYv(t) is the corrective model.The characteristics variables of the long term corrective risk model are as follows: the expectation is the variance is the moment generating function is where, E[X1], E[X12] is the first moment and the second moment of the variable X1, MX1 (s) is the moment generating function of the variable X1, Ev{z) is the Mittag-Leffle function with one parametere.The random process{U(t),μ≥ 0,0< v≤ 1, t≥ 0} is called the fractional Poisson surplus process, when it satisfies: where,(i) u0 is a constant,u0≥ 0,is the initial reserve; (ii) c is the average (premium) income in unit time, and ct≥ E[Yv(t)]; (iii) {Yv(t), μ>0,0<v≤l,t≥0} is the fractional Poisson corrective risk model. The characteristics variables of the fractional Poisson surplus process are as fol-lows: the expectation is: the variance is: the moment generating function is: where, Eî–¡[X1], E[X12] is the first moment and the second moment of the variable X1, h(s)= E[e-s·X1], Ev(z) is the Mittag-Leffle function with one parametere.Here, we give the theorems about the probability distribution of the fractional Poisson surplus process.theorem2:For the fractional Poisson surplus process{U(t),μ≥0,0<v≤1,t> 0}, Nv(t) is the cumulative number of the claims in [0, t], Xi(i= 1,2, … ) is the ith amount of the claim in [0, t], and the probability density of Xi is g(x), independent of Nu(t), then the probability density function f(x,t) of U(t) is where, δ(x)为Drac b function, g*k(u0 + ct - x) is the kth order convolution of the probability density function g (x) of the variable Xi, E,,,k)(z) is the kth derivative of the Mittag Leffe function with one parameter.The random process IY,(t), M > 0, 0 < u < 1, t > 0} is called the W-fractional poisson corrective risk model, when it satisfies: where, (i) N,, (t) is the cumulative number of claims in (0, t), and N(0) = 0, for the fixedt > 0, Nv (t) is the random variables who can be the non-negative integer, andwhere, W-v,1(z) =∑n=0∞ {zn/(n!Γ(-vn+1))}; (ii) Xi (i = 1, 2, …) is the ith the claim size in [0, t], is the independent identicallydistributed random variables, and independent of Nv(t); (iii) for the fixed t > 0, Y, (t) is the total amount of the claim in [0, t], Y, (0) = 0, andY,(t) is the corrective model.The random process {U(t), IL > 0, 0 < v < 1, t > 0} is called the W-fractional Poisson surplus process, when it satisfies: where, {Yv,(t), μ,c > 0,0 < v < l, t > 0} is the W-fractional Poisson corrective risk model.theorern3: For the W-fractional Poisson surplus process JU(t), p > 0,0 < v < 1, t > 0}, Nv (t) is the cumulative number of the claims in [0, t], Xi(i = 1, 2, … ) is the ith amount of the claim in [0, t], and the probability density of Xi is g(x), independent of Nv(t),then the probability probability density function f(x,t)of U(t) is where,δ(x)为Drac δ function,g*k(u0ct-x)is the kth order convolution of the probability density function g(x)of the variable Xi,W-v,1(x)is the Wright function.theorm4:For the W-fractional Poisson surplus process.{U(t),μ≥0,0<v< 1,t≥0),the claiim sizes Xi～exp(λ),that is then the probability density pr(t)of the ruin time Tis where,fv*n(t) is the n order convolution of the probability density fv(t),fv(t)=μt-1 W-v,0(-1/μtv),W-v,0(1-/μtv)is Wright function.