Abstract
In the article, hypothesis test for coefficients in high dimensional regression models is considered. I develop simultaneous test statistic for the hypothesis test in both linear and partial linear models. The derived test is designed for growing p and fixed n where the conventional F-test is no longer appropriate. The asymptotic distribution of the proposed test statistic under the null hypothesis is obtained.
Highlights
Some high dimensional data, such as gene expression datasets in microarray, exhibits the property that the number of covariates greatly exceeds the sample size
The discovery of “large p, small n” paradigm brings challenges to many traditional statistical methods, and the asymptotic properties of various estimators when p goes to infinity much faster than n have been discussed
I have seen a limitation with the F-test defined in Equation (3): it can not be applied to large p and small n
Summary
Some high dimensional data, such as gene expression datasets in microarray, exhibits the property that the number of covariates greatly exceeds the sample size. Reference [3] proposed a two sample test on high dimensional means. Both of these aforementioned articles considered testing under “large p, small n” without a regression structure, which the present article concentrates on. Zhong and Chen in [4] proposed a test statistic for testing the regression coefficients in linear models when p/n → ρ in (0,1). The partial linear models have been extensively studied They have a wide range of applications, from statistics to biomedical sciences. I apply a difference based estimation method in the partial linear models. I will discuss the efficiency of ridge estimator and propose a new test statistic for “large p, fixed n” setting.
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