Comparing the marginal densities of two strictly stationary linear processes
A nonparametric test is proposed in this paper to compare the marginal densities of two independent and strictly stationary linear time series. This topic is not only of theoretical interest but of relevance in several scientific areas that use time series to analyze their own topics. Let's see a brief introduction to the mathematics of the paper. Let \((\Omega,F,P)\) be a probability space, for each \(t\in\mathbb{Z}\) let \(X_{t}:\Omega\to\mathbb{R}\), \(Y_{t}:\Omega\to\mathbb{R}\) be random variables such that \(X=(X_{t})_{t\in\mathbb{Z}}\) and \(Y=(Y_{t})_{t\in\mathbb{Z}}\) are strictly stationary time series having finite second order moments. Let \((\alpha_{j})_{j\in\mathbb{Z}}\) and \((\beta_{j})_{j\in\mathbb{Z}}\) be sequences of positive real numbers such that \[ \begin{aligned} \alpha_{j}&/|j|^{\delta_{X}-1} \xrightarrow[j\to\infty]{} c_{\alpha}>0\text{ with }0<\delta_{X}<1/2, \\ \beta_{j}&/|j|^{\delta_{Y}-1} \xrightarrow[j\to\infty]{} c_{\beta}>0\text{ with }0<\delta_{Y}<1/2. \end{aligned} \] In this paper it is assumed that \[ X_{t}=\sum_{j<t}\alpha_{t-j}\varepsilon_{j},\quad Y_{t}=\sum_{j<t}\beta_{t-j}\eta_{j},\quad t\in\mathbb{Z}, \] where \((\varepsilon_{j})_{j\in\mathbb{Z}}\) and \((\eta_{j})_{j\in\mathbb{Z}}\) are sequences of identically distributed independent random variables such that \[ E(\varepsilon_{j})=0=E(\eta_{j}),\quad j\in\mathbb{Z}. \] That is \(X\) and \(Y\) are long-range-dependent (LRD) time series. It is assumed, for simplicity's sake, that \(\delta_{Y}\leq\delta_{X}\). Let \(f_{X}\) and \(f_{Y}\) be densities of \(X_0\) and \(Y_0\) respectively, with respect to the Lebesgue measure on \(\mathbb{R}\), say \(\lambda\). These functions are not assumed to be known, but it is assumed that \[ \int f_{X}^2\cdot d\rho<\infty\text{ and }\int f_{Y}^2\cdot d\rho<\infty, \] where \(\rho\) is a given probability over \(\mathbb{R}\), which has a density with respect to \(\lambda\). Note that since \(X\) and \(Y\) are strictly stationary \(f_{X}\) and \(f_{Y}\) are densities of \(X_{t}\) and \(Y_{t}\) respectively, for all \(t\in\mathbb{Z}\). In this paper a new test of \[ H_0:f_{X}=f_{Y}\quad \text{vs}\quad H_1:f_{X}\neq f_{Y}\tag{1} \] is proposed, based on samples \(\{Z_1,\dots,Z_{n}\}\) from the \(Z=(X,Y)\) process. Let's see such a test. Let \((Q_{j})_{j\geq 0}\) be an orthonormal basis of \(L^2(\mathbb{R},\rho)\). \[ f_{X}=\sum_{j\geq 0}a_{j}Q_{j},\quad f_{Y}=\sum_{j\geq 0}b_{j}Q_{j}. \] Then (1) is equivalent to \[ H_0:a_{j}=b_{j}\text{ for each }j\geq 0\quad \text{vs}\quad H_1:a_{j}\neq b_{j}\text{ for some }j\geq 0. \] For all \(j\neq 0\) be \[ \tilde{Q}_{j}=gQ_{j}, \] where \(g\) is a density of \(ν\) with respect to \(\lambda\). If \(k\geq 1\) and \(s=1,\dots,n\) be \[ V_{s}(k)=(V_{s}^{(1)},\dots,V_{s}^{(k)}),\quad V_{s}^{(j)}=Q_{j}(X_{s})-Q_{j}(Y_{s}), \] and \[ U_{n}(k)=(U_{n}^{(1)},\dots,U_{n}^{(k)})=(1/(u_{n}))\sum_{s=1}^nV_{s}(k), \] where \((u_{n})_{n\geq 1}\) is a suitable positive norming sequence with \(\lim_{n\to\infty}u_{n}=\infty\). For each \(n\geq 1\) be \[ N_{n}(k)=\|U_{n}(k)\|^2,\quad k=1,\dots,d(n), \] where \(\lim_{n\to\infty}d(n)=\infty\), in practice \(d(n)\leq 10\). \(H_0\) is rejected if \(K_{n}\neq 1\) at a given confidence level, with \[ K_{n}=\min\{K\mid N_{n}(K)-K\log(n)\geq N_{n}(k)-k\log(n),\, 1\leq k\leq d(n)\}. \] This test has been proposed by \textit{P. Doukhan} et al. [J. Multivariate Anal. 139, 147--165 (2015; Zbl 1328.62263)], but assuming different dependence relationships on the \(Z\) process than those assumed in this one. The underlying reasons for proposing such a test are explained heuristically. Section 4 analyzes the asymptotic behavior of the proposed test under the null hypothesis. The results obtained are also evaluated in the case of some transformations involving the \(X\) and \(Y\) processes. The consistency of the test for some alternative hypotheses against \(H_0\) is shown. In the following section, estimators of the parameters that allow the test to be implemented are proposed. A comprehensive and detailed simulation analysis with different autoregressive fractionally integrated moving average (ARFIMA) processes shows the performance of the test with empirically set critical levels of 5\%. Section 7 describes the results achieved by applying the test to a real data set. The authors say that the proofs of some of the results stated can be found in supplementary material, but no indication is given as to where this material is to be found. However, it is a paper that can be followed without much difficulties.
scientific article; zbMATH DE number 7350850
scientific article