Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. At the end, a simulation is provided where the suggested estimator is compared with the estimators for the precision matrix proposed in the literature. The optimal shrinkage estimator shows significant improvement and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the generalized inverse of the sample covariance matrix. The abstract was updated accordingl

SInC: An accurate and fast error-model based simulator for SNPs, Indels and CNVs coupled with a read generator for short-read sequence data

We report SInC (SNV, Indel and CNV) simulator and read generator, an open-source tool capable of simulating biological variants taking into account a platform-specific error model. SInC is capable of simulating and generating single- and paired-end reads with user-defined insert size with high efficiency compared to the other existing tools. SInC, due to its multi-threaded capability during read generation, has a low time footprint. SInC is currently optimised to work in limited infrastructure setup and can efficiently exploit the commonly used quad-core desktop architecture to simulate short sequence reads with deep coverage for large genomes. Sinc can be downloaded from https://sourceforge.net/projects/sincsimulator/

Adjusted Empirical Likelihood for Long-memory Time Series Models

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, computing profile empirical likelihood function involving constrained maximization does not always have a solution which leads to several drawbacks. In this paper, we propose an adjusted empirical likelihood procedure to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model. It guarantees the existence of a solution to the required maximization problem as well as maintains same asymptotic properties obtained by Yau (2012). Simulations have been carried out to illustrate that the adjusted empirical likelihood method for different long-time series models provides better confidence regions and coverage probabilities than the unadjusted ones, especially for small sample sizes

TO STUDY THE EFFICACY OF KANCHNAR GUGGULU AND DASHMOOL MATRA BASTI IN THE MANAGEMENT OF VATASHTHEELA W.S.R TO BENIGN PROSTATIC HYPERPLASIA

Benign prostatic hyperplasia is a common senile disease. It is a non-cancerous [benign] enlargement of the prostate gland, also called as fibromyoadenoma. It is involuntary hyperplasia due to disturbance of the ratio and quantity of circulating androgens and estrogens. As the age advances the serum testosterone levels slowly but significantly decreases, but levels of oestrogenic steroids are not decreased equally. According to this theory the prostate enlarges because of increased oestrogenic effects. It is likely that the secretion of intermediate peptide growth factors plays a part in the development of BPH. Vatashtheela as described in Ayurveda, closely resembles benign prostatic hyperplasia of modern medicine in its signs and symptoms. In the Vatashtheela, Mutravaha Srotodushti and vitiation of Vata and Kapha Doshas are involved. So Vata Kapha pacifying drugs along with Matra Basti considered to be helpful in reducing the size of prostate and enhancing the tonicity of urinary bladder. This study was conducted to find out the potency of Ayurvedic regimen of Kanchnar Gugglu orally and Dashmool Tail Matra Basti for the treatment of Vatashtheela as the Kanchnar Gugglu is having Granthihar property and Dashmool Tail pacifies Vata and having Tridoshghana, Deepana, Anulomana, Shothghana and Shoolghana properties. The drug Kanchnar Gugglu was given to 30 patients of group -1, with the dose of 500mg TDS before meals for 30 days. And Dashmool Tail Matra Basti was given to 30 patients of group – 2, with the dose of 50ml once a day for 15 days