The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h（.） can be viewed by a graphic method. That is, the plot with the fitted response y on the horizonta...The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h（.） can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h（.）. Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.展开更多
<p align="justify"> <span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>In 1686, Newton discovered the laws of gravitation [&...<p align="justify"> <span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>In 1686, Newton discovered the laws of gravitation [<a href="#ref1">1</a>] and predicted the universal gravitational constant <img alt="" src="Edit_8cc6927a-fa86-44a2-a4e4-c2b809cba958.png" />. In 1798, with a torsion balance, Cavendish [<a href="#ref2">2</a>] measured <img alt="" src="Edit_f51d8d12-e299-4f0f-918d-d4b7cb9d5b9b.png" />. Due to the low intensity of gravitation, it is difficult to obtain reliable results because they are disturbed by surrounding masses and environmental phenomena. Modern physics is unable to link <i>G</i> with other constants. However, in a 2019 article [<a href="#ref3">3</a>], with a new cosmological model, we showed that <i>G</i> seams related to other constants, and we obtained a theoretical value of <img alt="" src="Edit_a2b7158e-b2db-4c33-bab7-898a8cbe0cad.png" />. Here, we want to show that our theoretical value of <i>G</i> is the right one by interpreting measurements of <i>G</i> with the help of a new technique using cubic splines. We make the hypothesis that most <i>G</i> measurements are affected by an unknown systematic error which creates two main groups of data. We obtain a measured value of <img alt="" src="Edit_d447fba6-cde2-4b05-8b67-d1bdbacd412b.png" /><span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>. Knowing that our theoretical value of <i>G</i> is in agreement with the measured value, we want to establish a direct link between <i>G</i> and as many other constants as possible to show, with 33 equations, that <i>G</i> is probably linked with most constants in the universe. These equations may be useful for astrophysicists who work in this domain. Since we have been able to link <i>G</i> with Hubble parameter <em>H<sub>0</sub></em> (which evolve since its reverse gives the apparent age of the universe), we deduce that <i>G</i> is likely not truly constant. It’s value probably slowly varies in time and space. However, at our location in the universe and for a relatively short period, this parameter may seem constant. </p>展开更多
基金Supported by the National Natural science Foundation of China(10701035)ChenGuang Project of Shang-hai Education Development Foundation(2007CG33)a Special Fund for Young Teachers in Shanghai Universities(79001320)
文摘The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h（.） can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h（.）. Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.
基金supported by a NSF grant from National Natural Science Foundation of China(10701035)a special fund for young teachers in Shanghai universities(79001320).
文摘<p align="justify"> <span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>In 1686, Newton discovered the laws of gravitation [<a href="#ref1">1</a>] and predicted the universal gravitational constant <img alt="" src="Edit_8cc6927a-fa86-44a2-a4e4-c2b809cba958.png" />. In 1798, with a torsion balance, Cavendish [<a href="#ref2">2</a>] measured <img alt="" src="Edit_f51d8d12-e299-4f0f-918d-d4b7cb9d5b9b.png" />. Due to the low intensity of gravitation, it is difficult to obtain reliable results because they are disturbed by surrounding masses and environmental phenomena. Modern physics is unable to link <i>G</i> with other constants. However, in a 2019 article [<a href="#ref3">3</a>], with a new cosmological model, we showed that <i>G</i> seams related to other constants, and we obtained a theoretical value of <img alt="" src="Edit_a2b7158e-b2db-4c33-bab7-898a8cbe0cad.png" />. Here, we want to show that our theoretical value of <i>G</i> is the right one by interpreting measurements of <i>G</i> with the help of a new technique using cubic splines. We make the hypothesis that most <i>G</i> measurements are affected by an unknown systematic error which creates two main groups of data. We obtain a measured value of <img alt="" src="Edit_d447fba6-cde2-4b05-8b67-d1bdbacd412b.png" /><span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>. Knowing that our theoretical value of <i>G</i> is in agreement with the measured value, we want to establish a direct link between <i>G</i> and as many other constants as possible to show, with 33 equations, that <i>G</i> is probably linked with most constants in the universe. These equations may be useful for astrophysicists who work in this domain. Since we have been able to link <i>G</i> with Hubble parameter <em>H<sub>0</sub></em> (which evolve since its reverse gives the apparent age of the universe), we deduce that <i>G</i> is likely not truly constant. It’s value probably slowly varies in time and space. However, at our location in the universe and for a relatively short period, this parameter may seem constant. </p>
