Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.

Conventional economic theory outlines that developing countries experience economic growth through an undervalued exchange rate and that exchange rate overvaluations has negative long term effects on economic growth. This paper examined the impact of exchange rate movements as well as exchange rate misalignments on economic growth for the Trinidad and Tobago economy over the period 1960 to 2016. We find statistically significant evidence that both exchange rate appreciation and misalignments impact negatively on economic growth in the T&T economy. Drilling deeper we find interestingly that there exist no non-linear effects of exchange rate misalignments on growth. Specifically we find statistically significant evidence that both overvaluations and undervaluations hamper economic growth in the Trinidad and Tobago economy. We attribute this to T&T’s small and underdeveloped manufacturing sector that tends to be overlooked on account of its energy resources, in addition to the fact that its manufacturing sector is highly import oriented. A major policy recommendation would be for the critical reassessment of the rules governing the HSF, as government expenditure was allowed to follow energy revenues due to its current limitations.

In this research paper, we implement the theory of the primorial function, to develop the Supremum and Infimum bounds for the sum of (log(p))². There are, however, considerable computational difficulties related to these bounds. Therefore, from a pragmatic point of view, a set of Upper and Lower bounds had been developed to bypass this issue. Despite the increased estimation error, the Upper and Lower bounds are still considered sufficiently accurate, while facilitating an easy and fast computation of the estimate of the sum.

In this article, we establish sharp trigonometric-polynomial bounds for unnormalized sinc function.

Let G be a graph with n vertices and let 1; 2; : : : ; n be the eigenvalues of Randic matrix. The Randic Estrada index of G is REE(G) = Ón i=1 ei . In this paper, we establish lower and upper bounds for Randic index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

Upper confidence bound multi-armed bandit algorithms (UCB) typically rely on concentration in- equalities (such as Hoeffding’s inequality) for the creation of the upper confidence bound. Intu- itively, the tighter the bound is, the more likely the respective arm is or isn’t judged appropriately for selection. Hence we derive and utilise an optimal inequality. Usually the sample mean (and sometimes the sample variance) of previous rewards are the information which are used in the bounds which drive the algorithm, but intuitively the more infor- mation that taken from the previous rewards, the tighter the bound could be. Hence our inequality explicitly considers the values of each and every past reward into the upper bound expression which drives the method. We show how this UCB method fits into the broader scope of other information theoretic UCB algorithms, but unlike them is free from assumptions about the distribution of the data, We conclude by reporting some already established regret information, and give some numerical simulations to demonstrate the method’s effectiveness.

We show how some concentration inequalities for sampling without replacement can be used for bounding future samples. This process can be extended to bound the sum of future samples from multiple populations, and we analyse an illustrative sample allocation problem.

By using the Sala-i-Martin version of extreme bounds analysis, which is based on the cumulative density function, cross-sectional evidence is presented on the robust determinants of the infection and case mortality rates of Covid-19. The results show that, for most part, different factors determine the infection and mortality rates in developed and developing countries. However, common factors are also apparent, including the risk environment, the number of tests per million and the percentage of population over 65. Notwithstanding the possibility that the official data represent a partial reflection of the truth, it is concluded that the results presented in this study provide useful information that may aid decision-making in the fight against the pandemic.

The current work aims to develop an approximation of the slice of Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space $\mathbb{R}^3$ that, to the best of the author's knowledge, has not been addressed yet. This approximation of the actual slice is in a closed form of an explicit parametric equation in the case that the slice is not passing through those zones of the Minkowski surface with high curvatures, namely the "corners". At corners with high curvatures an alternative computational algorithm is introduced in which a family of ellipsoidal inner and outer bounds of the Minkowski's sum are used to construct a "narrow strip" for the actual slice of Minkowski sum. This strip can narrow persistently for a few more number of constructing bounds to precisely coincide on the actual slice of Minkowski sum. This algorithm is also applicable to the cases of slice of Minkowski sum of ellipsoids with high aspect ratio. In line with the main goal, some ellipsoidal inner and outer bounds of the Minkowski sum are reviewed, including the so called "Kurzhanski's" bounds. Also some ellipsoidal approximations are suggested for the Minkowski sum, which they can be used, as well as the inner and outer bounds, in calculation of the suggested parametric approximation of the slice of Minkowski sum.

Noisy data is always encountered in real applications, such as bioinformatics, neuroimage and remote sensing. Existing methods mainly consider linear or generalized linear errors-in-variables regression, while relatively little attention is paid for the multivariate response case, and how to evaluate the estimation performance under perturbed covariates is still an open question. In this paper, we consider the information-theoretic limitations of estimating a low-rank matrix in the multi-response errors-in-variables regression model. By application of the information theory and statistical techniques on concentration inequalities, the minimax lower bound is provided in terms of the squared Frobenius loss, which recaptures the rate provided under the clean covariate assumption in previous literatures. Hence our result further indicates that though under the more realistic errors-in-variables situation, no more samples are required so as to achieve a rate-optimal estimation.

We compare the well known Rayleigh-Ritz variational method (RRVM) with a recently proposed approach based on supersymmetric quantum mechanics and the Gram-Schmidt orthogonalization method (SSQMGS). We apply both procedures to a particular class of double-well harmonic oscillators that had been conveniently chosen for the application of the latter approach. The RRVM eigenvalues converge smoothly from above providing much more accurate results with less computational effort. Present results show that the unproved SSQMGS upper bounds do not hold.

A rigorous perturbation analysis of the singular value decomposition of a real matrix of full column rank is presented. It is shown that the SVD perturbation problem is well posed only in case of distinct singular values. The analysis involves the solution of coupled systems of linear equations and produces asymptotic (local) componentwise perturbation bounds of the entries of the orthogonal matrices participating in the decomposition of the given matrix and of its singular values. Local bounds are derived for the sensitivity of the singular subspaces measured by the angles between the unperturbed and perturbed subspaces. An iterative scheme is described to find global bounds on the respective perturbations. The analysis implements the same methodology used previously to determine componentwise perturbation bounds of the Schur form and the QR decomposition of a matrix.

In this study we introduce the new classes $\mathcal{M}_{\alpha}(\sin)$ and $\mathcal{M}_{\alpha}(\cos)$ of $\alpha$-convex functions associated with sine and cosine functions. Also, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of Zalcman functional for the class $\mathcal{M}_{\alpha}(\cos)$ for the case $n=3$, showing that the Zalcman conjecture holds for this value. Moreover, the problem of the Fekete-Szeg\H{o} functional estimate for these classes is studied.