Many statistical analysis procedures require a good estimator for a high-dimensional covariance matrix or its inverse, the precision matrix. the null hypothesis, and its theoretical power is studied. Numerical examples demonstrate the effectiveness of our testing procedure. (= 1, . . . , are is a lower triangular matrix with ones on Vemurafenib the diagonal and = diag(= (= > 1, are independent and normally distributed with mean zero, and the covariance matrix of the is and and small and obtained through fitting these regression equations may not work well, and so it is common to impose some kind of regularization on and (Huang et al., 2006; Levina et Vemurafenib al., 2008). When the variables of interest have a natural order and the banded assumption on holds, Rothman et al. (2010) showed that the band size of is is also banded with band size still has the same sparse structure as after reordering the variables using a perfect elimination order. Hence, under this banded assumption on , the regression equations can be rewritten Vemurafenib as > 1 with (? ? and in this large setting. Given estimators of and denotes the band size parameter, whose true value Vemurafenib is is a prespecified positive number smaller than ? 4. Let by and define as and the estimator for as = 0 for every Rabbit Polyclonal to PAK5/6 (phospho-Ser602/Ser560). is sufficiently large. To have an accurate testing procedure, the exact or asymptotic distribution of under the null hypothesis for + 1 may be different from one another, and hence the derivation of the asymptotic distribution of under Vemurafenib the null hypothesis can be challenging. In 23 we propose an improved test statistic and study its asymptotic null distribution. 23. Improved test statistic Under the normality assumption, given has the conditional normal distribution for every > is follows the ? degrees of freedom, follows the degrees of freedom, as to by standardizing each has the distribution, so the mean and variance of are ? 2) and var(?1)/(? 2)2(? 4). After standardization, we get the test statistic (? 1), as , the asymptotic distribution of Lf is standard normal. From Theorem 1 we know that for a significance level also holds as with an arbitrary fixed sample size > + 4. Next, we study the power of the test using < and is denoted by < when (? 1) is true. If < is large enough, many hypothesis testing methods with high power may exist; so here we only study the power of the statistic when the < and tend to . To this end, define (? 1) and , pr(|and go to . It allows < and < (? 1) is equivalent to = 0 for all < < is zero (Wu & Pourahmadi, 2003). However, conditions in Theorem 2 allow the partial correlation coefficients to tend to zero, which implies that all for < tend to zero. This suggests that testing whether is zero using the statistic may not have high power. 24. Band size detection procedure In general, the true band size is a constant smaller than ? 4. Here we perform a series of hypothesis tests to identify is true only when is larger than the actual band size : is false}. {Hence we estimate for which is rejected.|We estimate for which is rejected Hence.} In order to identify the actual band size, the following algorithm can be used. Algorithm 1 Step 1 Initialization: fix the overall significance level = where is a prespecified upper bound on the band size. Step 2 If = 0, {stop and output here.|output and stop here.} Let > where is the significance level for = needs to be chosen to ensure that the overall significance level of the procedure is no larger than is rejected for some = = 1, . . . , here. Let as if ? + 1) for all = 1, . . . , (1 = 005 and use the proposed test.