Many data problems nowadays carry the structure that the number *p* of covariables may greatly exceed sample sizen, i.e.,p>>n. In such a setting, a huge amount of work has been pursued addressing prediction of a new response variable, estimation of an underlying parameter vector and variable selection, see for example the books by Hastie, Tibshirani and Friedman(2009),Bühlmann and van de Geer(2011) or the more specific review article byFan and Lv(2010). With a few exceptions, the proposed methods and presented mathematical theory do not address the problem of assigning uncertainties, statistical significance or confidence: thus, the area of statistical hypothesis testing and construction of confidence intervals is largely unexplored and underdeveloped. Yet, such significance or confidence measures are crucial in applications where interpretation of parameters and variables is very important. The focus of this paper is the construction of p-values via computational lasso algorithm and corresponding multiple testing adjustment for a high-dimensional linear model which is often very useful in p>>n settings:

$$Y = X\beta^{0} + \epsilon$$

where $Y=(Y_1, \cdots, Y_n)^T$, $X$ is a fixed design $n\times p$ design matrix. $\beta^0$ is the true underlying $p\times 1$ parameter vector and $\epsilon$ is the $n\times 1$ stochastic error vector with $\epsilon_1, \cdots, \epsilon_n$ i.i.d. having $E[\epsilon_i]=0$ and $Var(\epsilon_i)=\sigma^2 < \infty $

*Our paper is in progress and will be out before the end of October* !