The Median-Unbiased Estimation Of Quasilinear Autoregressive Model

Comparing with other methods of parameter estimation,median-unbiased estimation has its own advantage, for example, it's free from the influence of outliers from sample. Therefore it is necessary to deepen to research this kind of of method. The papers in which are about the median-unbiased estimation of time series model are Hurwicz (1950),Andrews (1993),Zielinski (1999),Ning-zhong Shi and Dehui Wang(2003),Luger(2003) etc. This paper is conducted to introduce the application of the method of median-unbiased estimation for some models in both point-estimation and interval estimation. Firstly I introduced a lemma being used as its important basis which isLemma Letξ₁,ξ₂,…,ξ_N be random variables, let c be a constant such that (C1) P{ξ_j≤c} = 1/2 for all j = 1,2,…,N ; (C2) for every choice i₁, i₂,…, i_m(1≤i₁2<…m≤N), m = 1, 2,…, N of integers, and for every x₁,…, x_m-1 P{ξ_im≤c |ξ_i1 = x₁,…ξ_<sub>i_m-1 = x_m-1} = 1/2, Then P{Med(ξ₁,ξ₂,…,ξ_N)≤c}=1/2 Secondly, I used this method to estimate the parameter of quasilinear au-toregressive model as X_t =α₀ +α₁g₁ (X_t-1,…, X_t-p)+…+α_sg_s(X_t-1,…, X_t-p) +ε_t and then get a theoremTheorem Let {ε_t} be random variables, P{g(X_t,X_t+1) = 0} = 0, t = 1,2,…, n-1; P{ε_t≤0} = P{ε_t≥0} = 1/2, t = 1,2,…, n. So the Hurwicz estimation ofαis (α|^)_HUR, it is median-unbiased in which (α|^)_HUR = Med{ξ₁,…,ξ_n-2},ξ_i = X_i+2/g(X_i,X_i+1), i = 1,2,…, n - 2.Thirdly, I used median-unbiased estimation to estimate the parameter of a nonlinear time series model which was extended from ARCH(0,2) model. The model is in which {ε_t} is i.i.d.: Eε_t = 0, Eε_t² =σ² <∞.ε_t and {X_s, s < t} are mutually independent.ε_1/2is the median ofε_t², it's alone and independent ofα, g(x) is a known function which make the model stationary. whenα=α₀, g(X_t-1, X_t-2) = 1 +α₁/α₀X_t-1 +α₂/α₀X_t-2 ,α₁/α₀ andα₂/α₀ are known , this model is called ARCH(0,2). therefore , this model could be considered as the extension of ARCH(0,2) model, and we get thatTheorem Let {ε_t} be random variables, which satisfies(1)P{g(X_t, X_t+1) = 0} = 0, t = 1,2,…, n - 1;(2)P{ε_t²â‰¤Îµ(1/2)} = P{ε_t²â‰¥Îµ(1/2)} = 1/2, t = 1,2,…, n. then, the Hurwicz estimation ofαis (α|^)_HUR it is median-unbiased that P{(α|^)_HUR≤α} = 1/2,α∈(-1,1) in which (α|^)_HUR = Med{ξ₃,…,ξ_n},ξ_i = X_i²/ε(1/2)g(X_i-1,X_i-2),i = 3,4,…,n.When it comes to interval estimation, there are two methods(1)One is to estimate the parameter by geometric meaning of binomial distribution and the idea of median-unbiased estimation, we get the result thatThan we can get the confidence interval and confidence upper (lower) limit ofαby selecting different i and j.As far as quasilinear autoregressive model (2.1.2) is concernded, let where Z_(t) is the order statistic of Z_t.As far as model (2.2.6) is concernded, let(2)The other one use the characteristic function and empirical distribution function to estimate confidence interval and confidence upper (lower) limit by solving a equation which is if a is ture , S_N^*(α)would obey asymptotic normal distribution. Then the confidence interval ofαis (α_L,φ/2^*,α_R,φ/2^*) whose confidence level is 1 -φ, in which S_N^*(α_L,φ/2)^* = -c_φ/2, S_N^*(α_R,φ/2)^* = c_φ/2, c_φ/2 is theφ/2 quantile of a standard normal distribution.Finally, I used stochastic simulation to check up results by random number come from normal distribution and t-distribution respectively. Then I compared the estimate value with the true value to get the result.