The complexity of decoding the standard Reed-Solomon code is a well known open prob-lem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum es...The complexity of decoding the standard Reed-Solomon code is a well known open prob-lem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance).展开更多
We propose a semiparametric Wald statistic to test the validity of logistic regression models based on case-control data. The test statistic is constructed using a semiparametric ROC curve estimator and a nonparametri...We propose a semiparametric Wald statistic to test the validity of logistic regression models based on case-control data. The test statistic is constructed using a semiparametric ROC curve estimator and a nonparametric ROC curve estimator. The statistic has an asymptotic chi-squared distribution and is an alternative to the Kolmogorov-Smirnov-type statistic proposed by Qin and Zhang in 1997, the chi-squared-type statistic proposed by Zhang in 1999 and the information matrix test statistic proposed by Zhang in 2001. The statistic is easy to compute in the sense that it requires none of the following methods: using a bootstrap method to find its critical values, partitioning the sample data or inverting a high-dimensional matrix. We present some results on simulation and on analysis of two real examples. Moreover, we discuss how to extend our statistic to a family of statistics and how to construct its Kolmogorov-Smirnov counterpart.展开更多
文摘The complexity of decoding the standard Reed-Solomon code is a well known open prob-lem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance).
基金the 11.5 Natural Scientific Plan (Grant No. 2006BAD09A04)Nanjing UniversityStart Fund (Grant No. 020822410110)
文摘We propose a semiparametric Wald statistic to test the validity of logistic regression models based on case-control data. The test statistic is constructed using a semiparametric ROC curve estimator and a nonparametric ROC curve estimator. The statistic has an asymptotic chi-squared distribution and is an alternative to the Kolmogorov-Smirnov-type statistic proposed by Qin and Zhang in 1997, the chi-squared-type statistic proposed by Zhang in 1999 and the information matrix test statistic proposed by Zhang in 2001. The statistic is easy to compute in the sense that it requires none of the following methods: using a bootstrap method to find its critical values, partitioning the sample data or inverting a high-dimensional matrix. We present some results on simulation and on analysis of two real examples. Moreover, we discuss how to extend our statistic to a family of statistics and how to construct its Kolmogorov-Smirnov counterpart.
