With the development of the big data age, the demand of modern emerging disciplines for the fine quantitative tail characteristics of financial data analysis is constantly developing, so the research method of tail index change point in the thick tail sequence is particularly important. In this paper, we extend the Ratio statistic proposed by Kim to the test of change point of tail index in infinite variance observations. The null distribution of the statistic and its consistency under alternative hypothesis were obtained. Prior with the least squares estimation method used in previous articles,we used robust M estimation to estimate unknown parameter ,making parameter estimation in the model more accurate.The Monte Carlo numerical simulation shows that our test works well.
In the age of big data, pre-detecting structural changes allows us to better interpret data, predict data more accurately and avoid risk. Therefore, the study of structural change points has attracted wide attention from many scholars. Tail index change point is one of the core contens of change point research, which is widely used in pratical life. Such as finance, hydrology, communication engineering and other fields. Several tail index change-point tests have been proposed in the past decade. Quintos, Fan, and Phllips [1] employed three tests to verify tail index change-points in independent samples and ARCH models. And later Kim and Lee [4] studied the tail index change-point test based on autoregressive residuals.
Reasonable estimates of the parameters in the models are also momentous. The least squares estimation is the most commonly used point estimation method in parameter estimation, but data with anomalies or strong influence points are often encountered in pratical problems, while the least-squares estimator is sensitive to the emergence of outliers in the data. Therefore, in order to eliminate or reduce the effect of outliers, The estimator is required to have a robustness. While M estimation is a commonly used class of estimates in current robust estimation methods, and there have been a series of extensive studies on M estimation. Richard, Keight and Liu [10] used M estimation to study the estimates of autogressive parameters in heavy-tailed sequences, and later Keith Knight proposed the limit theory of M estimation in heavy-tailed sequences.
In this paper, We employ robust M estimation to estimate unkown parameters in the model and obtain the asymptotic behaviour of the estimator. To define the test statistic, we cite the test statistic based on detecting change point in the persistence by Kim[8] to test the change point of heavy index.
This paper is arranged as follows. Section 2 we show the model and assumptions. Section 3 represent test statistic and asymptotic distribution. Section 4 the Monte Carlo simulation is made.Section 5 summarize this paper.
y = u + ε , t = 1,2, ⋯ T
( )ρ = ψ(. );Eψ (η ) = 0 ;E(|ψ(η )|
( )ψ (. ) is Lipschitz continuous; 0 < Eψ
Lemma.1. if the assumption (A ) holds, then
(η ) < ∞ .
Moreover, U (r) =r/ U (
T/ (μ − μ) → ⋅ (() ( )) , t = 1,2, ⋯ [Tτ]
T/ (μ − μ) → ( ()())⋅ ( ( )) , t = [Tτ] + 1, [Tτ] + 2, ⋯ T where μ , μ represents M estimation based on respectively y , y , ⋯ y[ ] and y[]
Proof of Lemma.2.
We consider estimates μ defined by minimizing ∑ ρ(y − μ). Define the process
Z (v) = ∑[ ]
ρ
ε + vT
− ρ(ε ) ,
(
and note that v which minimizes Z is simply T (μ − μ).Then we using the Taylor series expansion of each summand of Z around v = 0,we get
Z (v) = T v ∑[ ] ψ(ε ) + T v ∑[ ] ψ (ε∗), where ε∗ lies between ε and ε + T
v.Using the fact that ψ is Lipschitz-continuous|ψ (ε ) − ψ (ε∗)| ≤ K T v , and reference Keith Knight[11],we can get
[ ] T ∑[ ] ψ(η ) → B(τ),T ∑[ ]ψ (ε ) − E(ψ (ε )) Thus, the finite dimensional distributions of Z (⋅)converge weakly to those of Z(⋅), where Z(v) = vB(τ) + v τE(ψ (ε )),let Z (v) = 0,we get v = T/ (μ − μ) → In the same way, t = Tτ + 1, Tτ + 2, ⋯ T, v = T/ (μ − μ) → specific reference Keith Knight[10].
Now,we take into account the following null hypothesis and alternative hypothesis: ( ()() )⋅ ( ( )) .
() ⋅ ( ( ))
H :k = k I{ ∗} + k I{ ∗}
We consider the Ratio statistic presented by Kim: Ξ (τ) = ([ where ε, and ε, represents M estimation residuals based on respectively y , y , ⋯ y[ ] and , ⋯ , y .According to ratio statistic, a Maximum Chow-type test is proposed Theorem 3.1 Suppose the assumption A − A
holds, then under the null hypothesis,
Ξ = maxΞ (τ), T ∈ (
∈ Ξ (τ) ⇒ ( ) ( ( ) ) And Ξ ⇒ Ξ (τ),G (r) = U (r),r ∈ (0, τ);G (r − τ) = U (r) − U (τ),r ∈ (τ, 1) Proof of Theorem 3.1 According to Lemma.1, Lemma.2, let t = Tr ,we have
(ε − (u − u)) ⇒ U (r) = G (r),so a ∑ ε, = a ∑ [τT] ∑[ ] ∑ ε, ⇒ τ G (r).
Similarly, a ∑ ε, = a ∑ (ε − (u − u)) ⇒ U (r − τ) = G (r − τ). Finally, we get the limit distribution of the statistics unde the null hypothesis Ξ (τ) ⇒ (1 − τ)
G (r − τ) dr τ G (r) dr
Theorem 3.2 If assumption A − A holds, and τ∗ is the break time, then under the alternative
hypothesis, (
∈
(
< k and 0 < ≤ ∗, we have Ξ For τ∗ < τ < 1，we have = maxΞ ∈
(τ) = O (
∈
a = T/ L(T),a = T/
Proof of Theorem 3.2
We first consider the case (i) k > k ,0 < τ ≤ τ∗,similar to Proof of Theorem 3.1, the denominator
) → ∞,As T → ∞,Ξ
→ ∞
(
∑[ ] ∑ ([1 − τ]T)
ε, [ ∗ ] ∑ [ ] = O (a ).The numerator is When k > k ,Ξ (τ) = O (
) → ∞.The case τ∗ < τ < 1, the denominator is [τT] and the numerator is ([1 − τ]T) ∑[ ∗ ] ∑
ε,
= maxΞ (τ) = O ( ∈ ).As T → ∞, Ξ → ∞.
− A
holds, then under the alternative hypothesis, if k
≠ k ,we
Ξ
∗ = max Ξ , Ξ
, As T → ∞, Ξ
∗ → ∞.
(
This section we use the Monte Carlo numerical simulation method to verify the effectiveness of our Ratio test.We obtain the empirical level value and empirical potential function value . Consider the following data generation process: = + , t = 1,2, … T . The sample size is
T = distributed, so we only give the critical value of the original statistic. 200，500，1000. heavy-tailed index {κ = 0.6, 0.8,0.9 ,1.6,1.8,1.9} .The test was repeated 2000 times and significance level α = 0.05.The original statistic and the inverted statistic are identically 0.9 → 0.8 0.9 → 0.6 0.8 → 0.6 0.8 → 0.5 0.9 → 0.8 0.9 → 0.6 0.8 → 0.6 0.8 → 0.5 0.9 → 0.8 0.9 → 0.6 0.8 → 0.6 0.8 → 0.5 = 0.3
In this paper, we investigate the heavy index change point in thick tail sequence, and this theme is closely related to the big data setting and financial market. we adopt M-estimate to estimate unknow parameter, and absort to the statistic proposed by Kim, obtained the limit distribution of the statistics unde the null hypothesis and its consistency under alternative hypothesis.Numerical simulations verify that our statistics work very well.