The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application
In this paper, we investigate the CUSUM-type estimator of mean change-point models based on <i>m</i>-asymptotically almost negatively associated (<i>m</i>-AANA) sequences. The family of <i>m</i>-AANA sequences contains AANA, NA, <i>m</i>-NA, and indepe...
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MDPI AG
2020-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/12/2113 |
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| author | Saisai Ding Xiaoqin Li Xiang Dong Wenzhi Yang |
| author_facet | Saisai Ding Xiaoqin Li Xiang Dong Wenzhi Yang |
| author_sort | Saisai Ding |
| collection | DOAJ |
| description | In this paper, we investigate the CUSUM-type estimator of mean change-point models based on <i>m</i>-asymptotically almost negatively associated (<i>m</i>-AANA) sequences. The family of <i>m</i>-AANA sequences contains AANA, NA, <i>m</i>-NA, and independent sequences as special cases. Under some weak conditions, some convergence rates are obtained such as <inline-formula><math display="inline"><semantics><mrow><msub><mi>O</mi><mi>P</mi></msub><mrow><mo>(</mo><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msub><mi>O</mi><mi>P</mi></msub><mrow><mo>(</mo><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mo form="prefix">log</mo><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>O</mi><mi>P</mi></msub><mrow><mo>(</mo><msup><mi>n</mi><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>. Our rates are better than the ones obtained by Kokoszka and Leipus (Stat. Probab. Lett., 1998, 40, 385–393). In order to illustrate our results, we do perform simulations based on <i>m</i>-AANA sequences. As important applications, we use the CUSUM-type estimator to do the change-point analysis based on three real data such as Quebec temperature, Nile flow, and stock returns for Tesla. Some potential applications to change-point models in finance and economics are also discussed in this paper. |
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| spelling | doaj.art-92118311d2ac4d3e8917c6658ce4741a2023-11-20T22:23:36ZengMDPI AGMathematics2227-73902020-11-01812211310.3390/math8122113The Consistency of the CUSUM-Type Estimator of the Change-Point and Its ApplicationSaisai Ding0Xiaoqin Li1Xiang Dong2Wenzhi Yang3School of Mathematical Sciences, Anhui University, Hefei 230601, ChinaSchool of Mathematical Sciences, Anhui University, Hefei 230601, ChinaSchool of Life Sciences, Anhui University, Hefei 230601, ChinaSchool of Mathematical Sciences, Anhui University, Hefei 230601, ChinaIn this paper, we investigate the CUSUM-type estimator of mean change-point models based on <i>m</i>-asymptotically almost negatively associated (<i>m</i>-AANA) sequences. The family of <i>m</i>-AANA sequences contains AANA, NA, <i>m</i>-NA, and independent sequences as special cases. Under some weak conditions, some convergence rates are obtained such as <inline-formula><math display="inline"><semantics><mrow><msub><mi>O</mi><mi>P</mi></msub><mrow><mo>(</mo><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msub><mi>O</mi><mi>P</mi></msub><mrow><mo>(</mo><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mo form="prefix">log</mo><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>O</mi><mi>P</mi></msub><mrow><mo>(</mo><msup><mi>n</mi><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>. Our rates are better than the ones obtained by Kokoszka and Leipus (Stat. Probab. Lett., 1998, 40, 385–393). In order to illustrate our results, we do perform simulations based on <i>m</i>-AANA sequences. As important applications, we use the CUSUM-type estimator to do the change-point analysis based on three real data such as Quebec temperature, Nile flow, and stock returns for Tesla. Some potential applications to change-point models in finance and economics are also discussed in this paper.https://www.mdpi.com/2227-7390/8/12/2113CUSUM estimatorchange-pointfinancial time seriesnegatively associated sequences |
| spellingShingle | Saisai Ding Xiaoqin Li Xiang Dong Wenzhi Yang The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application Mathematics CUSUM estimator change-point financial time series negatively associated sequences |
| title | The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application |
| title_full | The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application |
| title_fullStr | The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application |
| title_full_unstemmed | The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application |
| title_short | The Consistency of the CUSUM-Type Estimator of the Change-Point and Its Application |
| title_sort | consistency of the cusum type estimator of the change point and its application |
| topic | CUSUM estimator change-point financial time series negatively associated sequences |
| url | https://www.mdpi.com/2227-7390/8/12/2113 |
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