Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

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• 作者：Cristina Butucea ; Rania Zgheib ; ^{rania.zgheib@u-pem.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：62G10 ; 62H15 ; 62G20 ; 62H10
• 刊名：Journal of Multivariate Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：146
• 期：Complete
• 页码：164-176
• 全文大小：492 K

文摘

We observe a sample of 7259X1500216X&_mathId=si29.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=711f4af062438dbb8950e51f1a732080" title="Click to view the MathML source">n$n$ independent 7259X1500216X&_mathId=si30.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=97bc9dbdee073c52bb4659c0be131673" title="Click to view the MathML source">p$p$-dimensional Gaussian vectors with Toeplitz covariance matrix 7259X1500216X&_mathId=si31.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=82236a1d4ccd1aa87b1d1f627c5ed853" title="Click to view the MathML source">Σ=[σ_∣i−j∣]_1≤i,j≤p$\Sigma ={\left[{\sigma }_{\mid i-j\mid }\right]}_{1\le i,j\le p}$ and 7259X1500216X&_mathId=si32.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=fc0d05d44c45e25e5c39daff3167a72f" title="Click to view the MathML source">σ₀=1${\sigma }_0=1$. We consider the problem of testing the hypothesis that 7259X1500216X&_mathId=si17.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=674c5e2494c791768d97972a61be6de0" title="Click to view the MathML source">Σ$\Sigma$ is the identity matrix asymptotically when 7259X1500216X&_mathId=si34.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=b7d92b194a693edaa14145999d6ecac0" title="Click to view the MathML source">n→∞$n\to \infty$ and 7259X1500216X&_mathId=si35.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=a162e2d099e7deef10b156941af744b5" title="Click to view the MathML source">p→∞$p\to \infty$. We suppose that the covariances 7259X1500216X&_mathId=si36.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=8cab67926759a8c2f79ea67cec4b7236" title="Click to view the MathML source">σ_k${\sigma }_k$ decrease either polynomially (7259X1500216X&_mathId=si37.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=e8470088dd22ec8fd7fcb33e9641c611">7259X1500216X-si37.gif">${\sum }_{k\ge 1}{k}^{2\alpha }{\sigma }_k^2\le L$ for 7259X1500216X&_mathId=si38.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=051462131fcbe579c1749ff88f54de1c" title="Click to view the MathML source">α>1/4$\alpha >1/4$ and 7259X1500216X&_mathId=si39.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=4e0dc479e23ca3ec05de5e69a20391be" title="Click to view the MathML source">L>0$L>0$) or exponentially (7259X1500216X&_mathId=si40.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=0ea8fbb8bebaf0d6f47791701a15d648">7259X1500216X-si40.gif">${\sum }_{k\ge 1}{e}^{2Ak}{\sigma }_k^2\le L$ for 7259X1500216X&_mathId=si41.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=4ea6ea2ccdb4c39565d005f50d118bc2" title="Click to view the MathML source">A,L>0$A,L>0$).

We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen as solution of an extremal problem. We give the asymptotic normality of the test statistic under the null hypothesis for fixed 7259X1500216X&_mathId=si29.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=711f4af062438dbb8950e51f1a732080" title="Click to view the MathML source">n$n$ and 7259X1500216X&_mathId=si43.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=6db9803e8b96fe5656566f7585b9e1f5" title="Click to view the MathML source">p→+∞$p\to +\infty$ and the asymptotic behavior of the type I error probability of our test procedure. We also show that the maximal type II error probability, either tend to 7259X1500216X&_mathId=si44.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=fe47b4054483fc6b9eee685c992b80b1" title="Click to view the MathML source">0$0$, or is bounded from above. In the latter case, the upper bound is given using the asymptotic normality of our test statistic under alternatives close to the separation boundary. Our assumptions imply mild conditions: 7259X1500216X&_mathId=si45.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=6e33098eb60ef11a9ce4384e911048bb" title="Click to view the MathML source">n=o(p^2α−1/2)$n=o\left(p^{2\alpha -1/2}\right)$ (in the polynomial case), 7259X1500216X&_mathId=si46.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=8a526295451b9f419da687b81a0bcaea" title="Click to view the MathML source">n=o(e^p)$n=o\left(e^p\right)$ (in the exponential case).

We prove both rate optimality and sharp optimality of our results, for 7259X1500216X&_mathId=si47.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=2e3502f8d9134d45aa8c851d6a5fda35" title="Click to view the MathML source">α>1$\alpha >1$ in the polynomial case and for any 7259X1500216X&_mathId=si48.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=dad1b18eaef2aa36f3f15fad48066a3c" title="Click to view the MathML source">A>0$A>0$ in the exponential case.

A simulation study illustrates the good behavior of our procedure, in particular for small 7259X1500216X&_mathId=si29.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=711f4af062438dbb8950e51f1a732080" title="Click to view the MathML source">n$n$, large 7259X1500216X&_mathId=si30.gif&_user=111111111&_pii=S0047259X1500216X&_rdoc=1&_issn=0047259X&md5=97bc9dbdee073c52bb4659c0be131673" title="Click to view the MathML source">p$p$.