Projection Onto A Simplex

This mini-paper presents a fast and simple algorithm to compute the projection onto the canonical simplex $\triangle^n$. Utilizing the Moreau's identity, we show that the problem is essentially a univariate minimization and the objective function is strictly convex and continuously differentiable. Moreover, it is shown that there are at most n candidates which can be computed explicitly, and the minimizer is the only one that falls into the correct interval

Sensor Selection Based on Generalized Information Gain for Target Tracking in Large Sensor Networks

In this paper, sensor selection problems for target tracking in large sensor networks with linear equality or inequality constraints are considered. First, we derive an equivalent Kalman filter for sensor selection, i.e., generalized information filter. Then, under a regularity condition, we prove that the multistage look-ahead policy that minimizes either the final or the average estimation error covariances of next multiple time steps is equivalent to a myopic sensor selection policy that maximizes the trace of the generalized information gain at each time step. Moreover, when the measurement noises are uncorrelated between sensors, the optimal solution can be obtained analytically for sensor selection when constraints are temporally separable. When constraints are temporally inseparable, sensor selections can be obtained by approximately solving a linear programming problem so that the sensor selection problem for a large sensor network can be dealt with quickly. Although there is no guarantee that the gap between the performance of the chosen subset and the performance bound is always small, numerical examples suggest that the algorithm is near-optimal in many cases. Finally, when the measurement noises are correlated between sensors, the sensor selection problem with temporally inseparable constraints can be relaxed to a Boolean quadratic programming problem which can be efficiently solved by a Gaussian randomization procedure along with solving a semi-definite programming problem. Numerical examples show that the proposed method is much better than the method that ignores dependence of noises.Comment: 38 pages, 14 figures, submitted to Journa

A modified Corrado test for assessing abnormal security returns

Event studies typically use the methodology developed by Fama et al. [19699. Fama , E. , Fisher , L. , Jensen , M. and Roll , R. 1969 . The adjustment of stock prices to new information . International Economic Review , 10 ( 1 ) : 1 – 21 . [CrossRef] View all references. The adjustment of stock prices to new information. International Economic Review 10, no. 1: 1–21] to segregate a stock's return into expected and unexpected components. Moreover, conventional practice assumes that abnormal returns evolve in terms of a normal distribution. There is, however, an increasing tendency for event studies to employ non-parametric testing procedures due to the mounting empirical evidence which shows that stock returns are incompatible with the normal distribution. This paper focuses on the widely used non-parametric ranking procedure developed by Corrado [19896. Corrado , C. 1989 . A nonparametric test for abnormal security price performance in event studies . Journal of Financial Economics , 23 ( 2 ) : 385 – 95 . [CrossRef], [Web of Science ®] View all references. A nonparametric test for abnormal security price performance in event studies. Journal of Financial Economics 23, no. 2: 385–95] for assessing the significance of abnormal security returns. In particular, we develop a consistent estimator for the variance of the sum of ranks of the abnormal returns, and show how this leads to a more efficient test statistic (as well as to less cumbersome computational procedures) than the test originally proposed by Corrado (19896. Corrado , C. 1989 . A nonparametric test for abnormal security price performance in event studies . Journal of Financial Economics , 23 ( 2 ) : 385 – 95 . [CrossRef], [Web of Science ®] View all references). We also use the theorem of Berry [19413. Berry , A. 1941 . The accuracy of the Gaussian approximation to the sum of independent variates . Transactions of the American Mathematical Society , 49 ( 1 ) : 122 – 36 . [CrossRef] View all references. The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society 49, no. 1: 122–36] and Esseen [19458. Esseen , C. 1945 . Fourier analysis of distribution functions: A mathematical study of the Laplace–Gaussian law . Acta Mathematica , 77 ( 1 ) : 1 – 125 . [CrossRef] View all references. Fourier analysis of distribution functions: A mathematical study of the Laplace–Gaussian law. Acta Mathematica 77, no. 1: 1–125] to demonstrate how the distribution of the modified Corrado test statistic developed here asymptotically converges towards the normal distribution. This shows that describing the distributional properties of the sum of the ranks in terms of the normal distribution is highly problematic for small sample sizes and small event windows. In these circumstances, we show that a second-order Edgeworth expansion provides a good approximation to the actual probability distribution of the modified Corrado test statistic. The application of the modified Corrado test developed here is illustrated using data for the purchase and sale by UK directors of shares in their own companies