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Abstract
The dissertation deals with the estimation of covariance and precision
matrices for high-dimensional time series with long-memory.
In Chapter 2, we generalize part of the results of [\cite{SN19}] (i) from the spectral norm
to the general vector norm induced matrix $\ell _{v,w}$ norm $\left\Vert
\cdot \right\Vert _{(v,w)}$ for any $v,w\in \lbrack 1,\infty ]$, (ii) from
the Frobenius norm to the general entrywise matrix $L^{v,w}$ norm $%
\left\Vert \cdot \right\Vert _{L^{v,w}}$ for any $v,w\in \lbrack 1,\infty ]$%
, and (iii) from $p\geq n^{c}$ for some constant $c>0$\ to $p\geq
(n/g_{2})^{c}$ for some constant $c>0$, where $g_{2}$ is an upper bound of $%
\max_{1\leq k\leq p}\left\Vert (\rho _{\lbrack k]}^{ij})_{n\times
n}\right\Vert _{2}$. We also generalize their minimax result by removing the
restriction of $p\geq n^{\beta }$ for some $\beta >1$. In particular, we
obtain the minimax result for the convergence rate of the precision matrix
estimator proposed by [\cite{CLL11}].
In Chapter 3, based on the results of [%
\cite{SN19}], we investigate the joint estimation of multiple precision
matrices. We generalize the results of [\cite{LL15}] from i.i.d. data to
long-memory data. Especially, we obtain the estimation of the entrywise $%
L^{1}$ norm and the Frobenius norm of risk, and their expectations. Our
numerical experiment results support our theory analysis.
In Chapter 4, based on the results of [\cite{SN19}], we investigate the joint estimation
of weighted multiple precision matrices. We generalize the results of [\cite%
{CPM}] from i.i.d. data to long-memory data. Especially, we obtain the
estimation of the entrywise $L^{1}$ norm and the Frobenius norm of risk, and
their expectations. Our numerical experiment results support our theory
analysis.
In Chapter 5, based on the results of [\cite{SN19}], we introduce
a new assumption to investigate the joint estimation of multiple precision
matrices, and generalize the results of [\cite{LL15}] from i.i.d. data to
long-memory data. Especially, we obtain the estimation of the vector norm
induced matrix $\ell _{1}$ norm and the Frobenius norm of risk, and their
expectations.
matrices for high-dimensional time series with long-memory.
In Chapter 2, we generalize part of the results of [\cite{SN19}] (i) from the spectral norm
to the general vector norm induced matrix $\ell _{v,w}$ norm $\left\Vert
\cdot \right\Vert _{(v,w)}$ for any $v,w\in \lbrack 1,\infty ]$, (ii) from
the Frobenius norm to the general entrywise matrix $L^{v,w}$ norm $%
\left\Vert \cdot \right\Vert _{L^{v,w}}$ for any $v,w\in \lbrack 1,\infty ]$%
, and (iii) from $p\geq n^{c}$ for some constant $c>0$\ to $p\geq
(n/g_{2})^{c}$ for some constant $c>0$, where $g_{2}$ is an upper bound of $%
\max_{1\leq k\leq p}\left\Vert (\rho _{\lbrack k]}^{ij})_{n\times
n}\right\Vert _{2}$. We also generalize their minimax result by removing the
restriction of $p\geq n^{\beta }$ for some $\beta >1$. In particular, we
obtain the minimax result for the convergence rate of the precision matrix
estimator proposed by [\cite{CLL11}].
In Chapter 3, based on the results of [%
\cite{SN19}], we investigate the joint estimation of multiple precision
matrices. We generalize the results of [\cite{LL15}] from i.i.d. data to
long-memory data. Especially, we obtain the estimation of the entrywise $%
L^{1}$ norm and the Frobenius norm of risk, and their expectations. Our
numerical experiment results support our theory analysis.
In Chapter 4, based on the results of [\cite{SN19}], we investigate the joint estimation
of weighted multiple precision matrices. We generalize the results of [\cite%
{CPM}] from i.i.d. data to long-memory data. Especially, we obtain the
estimation of the entrywise $L^{1}$ norm and the Frobenius norm of risk, and
their expectations. Our numerical experiment results support our theory
analysis.
In Chapter 5, based on the results of [\cite{SN19}], we introduce
a new assumption to investigate the joint estimation of multiple precision
matrices, and generalize the results of [\cite{LL15}] from i.i.d. data to
long-memory data. Especially, we obtain the estimation of the vector norm
induced matrix $\ell _{1}$ norm and the Frobenius norm of risk, and their
expectations.