In many areas of applied statistics, confidence intervals for the mean of the population are of interest. Confidence intervals are typically constructed as-suming normality although non-normally distributed data are a...In many areas of applied statistics, confidence intervals for the mean of the population are of interest. Confidence intervals are typically constructed as-suming normality although non-normally distributed data are a common occurrence in practice. Given a large enough sample size, confidence intervals for the mean can be constructed by applying the Central Limit Theorem or by the bootstrap method. Another commonly used method in practice is the back-transformation method, which takes on the following three steps. First, apply a transformation to the data such that the transformed data are normally distributed. Second, obtain confidence intervals for the transformed mean in the usual manner, which assumes normality. Third, apply the back- transformation to obtain confidence intervals for the mean of the original, non-transformed distribution. The parametric Wald method and a small sample likelihood-based third order method, which can address non-normality, are also reviewed in this paper. Our simulation results suggest that common approaches such as back-transformation give erroneous and misleading results even when the sample size is large. However, the likelihood-based third order method gives extremely accurate results even when the sample size is small.展开更多
This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling the...This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s 】 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.展开更多
文摘In many areas of applied statistics, confidence intervals for the mean of the population are of interest. Confidence intervals are typically constructed as-suming normality although non-normally distributed data are a common occurrence in practice. Given a large enough sample size, confidence intervals for the mean can be constructed by applying the Central Limit Theorem or by the bootstrap method. Another commonly used method in practice is the back-transformation method, which takes on the following three steps. First, apply a transformation to the data such that the transformed data are normally distributed. Second, obtain confidence intervals for the transformed mean in the usual manner, which assumes normality. Third, apply the back- transformation to obtain confidence intervals for the mean of the original, non-transformed distribution. The parametric Wald method and a small sample likelihood-based third order method, which can address non-normality, are also reviewed in this paper. Our simulation results suggest that common approaches such as back-transformation give erroneous and misleading results even when the sample size is large. However, the likelihood-based third order method gives extremely accurate results even when the sample size is small.
基金supported by National Natural Science Foundation of China (Grant No.11071152)Natural Science Foundation of Guangdong Province (Grant Nos. 05008289, 32038)the Doctoral Foundation of Guangdong Province (Grant No. 04300917)
文摘This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s 】 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.
