RAPTT: An Exact Two-Sample Test in High Dimensions Using Random Projections

In high dimensions, the classical Hotelling's $T^2$ test tends to have low power or becomes undefined due to singularity of the sample covariance matrix. In this paper, this problem is overcome by projecting the data matrix onto lower dimensional subspaces through multiplication by random matrices. We propose RAPTT (RAndom Projection T-Test), an exact test for equality of means of two normal populations based on projected lower dimensional data. RAPTT does not require any constraints on the dimension of the data or the sample size. A simulation study indicates that in high dimensions the power of this test is often greater than that of competing tests. The advantage of RAPTT is illustrated on high-dimensional gene expression data involving the discrimination of tumor and normal colon tissues

AR(1) sequence with random coefficients: Regenerative properties and its application

Let $\{X_n\}_{n\ge0}$ be a sequence of real valued random variables such that $X_n=\rho_n X_{n-1}+\epsilon_n,~n=1,2,\ldots$, where $\{(\rho_n,\epsilon_n)\}_{n\ge1}$ are i.i.d. and independent of initial value (possibly random) $X_0$. In this paper it is shown that, under some natural conditions on the distribution of $(\rho_1,\epsilon_1)$, the sequence $\{X_n\}_{n\ge0}$ is regenerative in the sense that it could be broken up into i.i.d. components. Further, when $\rho_1$ and $\epsilon_1$ are independent, we construct a non-parametric strongly consistent estimator of the characteristic functions of $\rho_1$ and $\epsilon_1$

Apparent ice accumulation rate in East Antarctica: Relation with temperature and thinning pattern

We present here formal evidence of a strong linkage between temperature and East Antarctic ice accumulation over the past eight hundred kiloyears, after accounting for thinning. The conclusions are based on statistical analysis of a proposed empirical model based on ice core data from multiple locations with ground topography ranging from local peaks to local valleys. The method permits adjustment of the apparent accumulation rate for a very general thinning process of ice sheet over the ages, is robust to any misspecification of the age scale, and does not require delineation of the accumulation rate from thinning. Records show 5% to 8% increase in the accumulation rate for every 1${}^\circ$C rise in temperature. This is consistent with the theoretical expectation on the average rate of increase in moisture absorption capacity of the atmosphere with rise in temperature, as inferred from the Clausius-Clapeyron equation. This finding reinforces indications of the resilience of the Antarctic Ice Sheet to the effects of warming induced by climate change, which have been documented in other studies based on recent data. Analysis of the thinning pattern of ice revealed an exponential rate of thinning over several glacial cycles and eventual attainment of a saturation level