Subsampling for Non-stationary Time Series with Long Memory and Heavy Tails Using Weak Dependence Condition

Abstract

Statistical inference for unknown distributions of statistics or estimators may be based on asymptotic distributions. Unfortunately, in the case of dependent data the structure of such statistical procedures is often ineffective. In the last three decades we can observe an intensive development the of so-called resampling methods. Using these methods, it is possible to directly approximate the unknown distributions of statistics and estimators. A problem that needs to be solved during the study of the resampling procedures is the consistency. Their consistency for independent or stationary observations has been extensively studied. Resampling for time series with a specific non-stationarity, i.e. the periodic and almost periodic strong mixing dependence structure also been the subject of research. Recent research on resampling methods focus mainly on the time series with the weak dependence structure, defined by Paul Doukhan, Louhichi et al. and simultaneously Bickel and B$\ddot{u}$hlmann. In the article a time series model with specific features i.e.: long memory, heavy tails (with at least a fourth moment, e.g.: GED, t-Student), weak dependence and periodic structure is presented. and the estimator of the mean function in the above-mentioned time series is investigated. In the article the necessary central limit theorems and consistency theorems for the mean function estimator (for one of the resampling techniques—the subsampling) are proven.

Appendix

In the appendix the proofs of some results are presented.

Proof

Lemma 5

From direct calculations we can obtain the strict formula for the kurtosis of the model $X_t,$ which is:

$$ \frac{E(X_t-EX_t)^4}{(E(X_t-EX_t)^2)^2}=\frac{E\sigma _t^4 EG_t^4}{(E\sigma _t^2)^2 (EG_t^2)^2}=3\frac{\varGamma ({5/\alpha })\varGamma ({1/\alpha })}{\varGamma ^2({3/\alpha })}. $$

If we use the Stirling’s formula for $\varGamma$ function we will obtain the approximation as follows:

$$ kurtosis \approx 3\cdot 1.4\cdot 4.3^{1/\alpha }. $$

The kurtosis is more than 3 for all $\alpha > 0.$    $\square$

Proof

Lemma 6

The mean of $X_{t}$ is $\eta _t,$ so it is periodic. The variance is periodic:

$$ \gamma (t+T,0)= (f_{t+T})^{2}(1+\varphi ^2)\gamma _{G}(t+T,0)= $$

$$ =(f_{t+T})^{2}(1+\varphi ^2)\gamma _{G}(0)=(f_{t})^{2}(1+\varphi ^2)\gamma _{G}(t,0)=\gamma (t,0). $$

The autocovariance also:

$$ \gamma (t+T,h)= |f_{t+T}f_{t+T+h}|\varphi ^2\gamma _{G}(t+T,h)= $$

$$ =|f_{t+T}f_{t+T+h}|\varphi ^2\gamma _{G}(h)=|f_{t}f_{t+h}|\varphi ^2\gamma _{G}(t,h)=\gamma (t,h). $$

The form of the variance and autocovariance follows from the form of the variance of variable with the GED distribution.    $\square$

Proof

Theorem 5

Let us consider a sequence of statistics $B_{k_N}(s),$ for fixed $s=1,2,...,T$ and $k_N$ as in the Theorem 2.

$L_{k_N}(s)(x)=P(B_{k_N}(s)\le x)$ is cumulative distribution function of $B_{k_N}(s)$.

From the assumptions

$$sup_{x\in \mathbb {R}}|L_{k_N}(s)(x)-L(s)(x)|\longrightarrow 0,\ \ k_N\rightarrow \infty $$

For overlapping samples the number of subsamples:

$Y_{b,q}(s)=(X_{s+qT},X_{s+(q+1)T}...,X_{s+(q+b-1)T}),$ $q=0,1,...,k_N-b$ and the number of subsampling statistics:

$B_{k_N,b,q}(s) = \sqrt{b}(\hat{\eta }_{k_N,b,q}(s)-\hat{\eta }_{k_N}(s))$ is $k_N-b+1$.

Above statistics are used to approximate the distributions $L_{k_N}(s)(x)$ by empirical distribution functions: $L_{k_N,b,q}(s)(x)=\frac{1}{k_N-b+1}\sum _{q=0}^{k_N-b}\mathbb {I}_{\{B_{k_N,b,q}(s)\le x\}}$.

Let us define subsampled distribution:

$$ U_{k_N,b,q}(s)(x)=\frac{1}{k_N-b+1}\sum _{q=0}^{k_N-b}\varphi ({\sqrt{b}(\hat{\eta }_{k_N,b,q}(s)-\eta _{k_N}(s))}{\varepsilon _n}). $$

The sequence $\varepsilon _n$ is decreasing to zero and $\varphi$ is the non-increasing continuous function such that $\varphi = 1$ or 0 according to $x \le 0$ or $x \ge 1$ and which is affine between 0 and 1.

From [32] it is known that

$$ \forall x \in \mathbb {R}\ |L_{k_N,b,q}(s)(x)-U_{k_N,b,q}(s)(x)|\mathop {\longrightarrow }\limits ^{p}0. $$

It follows that it is enough to investigate only the variance of $U_{k_N,b,q}(s),$ $s=1,...,T$

It is enough to investigate the variance of $U_{k_N,b_s,p}(s),\ \ s=1,...,T$ (Theorem 3.2.1 ([32])).

$$ VarU_{k_N,b_s,p}(s)(x)=(k_N-b_s+1)^{-2}(\sum _{|h|<k_N-b_s+1}(k_N-b_s+1-|h|)\gamma (h)) $$

here $\gamma (h)=Cov(\varphi (B_{k_N,b,p}(s)/\varepsilon _n),\varphi (B_{k_N,b,p+h}(s)/\varepsilon _n)).$

From the assumption that we have $\lambda$-weak dependence and under condition A1 through A4 and the Lemma 3.1 in ([2])

$$ Cov(\varphi (B_{k_N,b,p}(s)/\varepsilon _n),\varphi (B_{k_N,b,p+h}(s)/\varepsilon _n))\le \sqrt{b_s} \lambda _{h-b+1}/\varepsilon _n $$

It implies that $Var(U_{k_N,b_s,p}(s)(x))$ tends to zero, it proves point 1. of the Theorem 5.

The proof of point 2. and 3. if the 1. holds and under assumption of the model (6) is the same as the proof of 3. in the Theorem 3.2.1 ([32]).    $\square$

Proof

Theorem 6

For any vector of constants $c \in \mathbb {R}^T$ we have the equation for the subsampling version of the characteristic functions of the distributions:

$$ \phi ^*_{B_{k_N,b,q}}(c) = \phi ^*_{c^T B_{k_N,b,q}} (1) \ \ \mathrm{in \ GED \ case} $$

Let $Z_{s+pT} = c_sX_{s+pT},$ where $p = 0, \ldots ,k_N - 1$ and $s = 1, \ldots , T.$ The series $\{Z_t\}$ fulfills the assumptions of Theorem 6, which means that subsampling is consistent for the mean $(\eta _N)_Z.$ By Theorem A in Athreya [3] we have:

in GED case

$$ \phi ^*_{c^T B_{k_N,b,q}} (1)\mathop {\rightarrow }\limits ^{p} \phi _{k_N(\eta ,c^T\varSigma c)} (1)=\phi _{k_N(\eta ,\varSigma )} (c). $$

Moreover

$$ P^* (B_{k_N,b,q} \le x)\mathop {\rightarrow }\limits ^{p} F_{k_N(\eta ,\varSigma )}(x), $$

for any $x \in \mathbb {R}^T,$ where $F_{k_N(\eta ,\varSigma )}(x)$ is the cumulative distribution function of $k_N (\eta , \varSigma ).$

The second point of the thesis of the Theorem 6 follows then from P$\acute{o}$lya’s theorem ([33], p. 447).    $\square$