Statistical Inference for a General Family of Modified Exponentiated Distributions

In this paper, a modified exponentiated family of distributions is introduced. The new model was built from a continuous parent cumulative distribution function and depends on a shape parameter. Its most relevant characteristics have been obtained: the probability density function, quantile function, moments, stochastic ordering, Poisson mixture with our proposal as the mixing distribution, order statistics, tail behavior and estimates of parameters. We highlight the particular model based on the classical exponential distribution, which is an alternative to the exponentiated exponential, gamma and Weibull. A simulation study and a real application are presented. It is shown that the proposed family of distributions is of interest to applied areas, such as economics, reliability and finances.

Kurtosis of von Neumann entanglement entropy

In this work, we study the statistical behavior of entanglement in quantum bipartite systems under the Hilbert-Schmidt ensemble as assessed by the standard measure - the von Neumann entropy. Expressions of the first three exact cumulants of von Neumann entropy are known in the literature. The main contribution of the present work is the exact formula of the corresponding fourth cumulant that controls the tail behavior of the distribution. As a key ingredient in deriving the result, we make use of what we refer to as unsimplifiable summation bases leading to a complete cancellation. In addition to providing further evidence of the conjectured Gaussian limit of the von Neumann entropy, the obtained formula also provides an improved finite-size approximation to the distribution.

TAIL BEHAVIOR OF STOPPED LÉVY PROCESSES WITH MARKOV MODULATION

This article concerns the tail probabilities of a light-tailed Markov-modulated Lévy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an application to the stationary distribution of wealth in a simple economic model in which agents with constant absolute risk aversion are subject to random mortality and income fluctuation.

A UNIFORM BOUND ON THE OPERATOR NORM OF SUB-GAUSSIAN RANDOM MATRICES AND ITS APPLICATIONS

For an $N \times T$ random matrix $X(\beta )$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta )$ that may depend on a possibly infinite-dimensional parameter $\beta \in \mathbf {B}$ , we obtain a uniform bound on its operator norm of the form $\mathbb {E} \sup _{\beta \in \mathbf {B}} ||X(\beta )|| \leq CK \left (\sqrt {\max (N,T)} + \gamma _2(\mathbf {B},d_{\mathbf {B}})\right )$ , where C is an absolute constant, K controls the tail behavior of (the increments of) $x_{it}(\cdot )$ , and $\gamma _2(\mathbf {B},d_{\mathbf {B}})$ is Talagrand’s functional, a measure of multiscale complexity of the metric space $(\mathbf {B},d_{\mathbf {B}})$ . We illustrate how this result may be used for estimation that seeks to minimize the operator norm of moment conditions as well as for estimation of the maximal number of factors with functional data.

Assessment of temporal change in the tails of probability distribution of daily precipitation over India due to climatic shift in 1970s

Daily precipitation extremes are crucial in the hydrological design of major water control structures and are expected to show a changing tendency over time due to climate change. The magnitude and frequency of extreme precipitation can be assessed by studying the upper tail behavior of probability distributions of daily precipitation. Depending on the tail behavior, the distributions can be classified into two categories: heavy-tailed and light-tailed distributions. Heavier tails indicate more frequent occurrences of extreme precipitation events. In this paper, we have analyzed the temporal change in the tail behavior of daily precipitation over India from pre- to post-1970 time periods as per the global climatic shift. A modified Probability Ratio Mean Square Error norm is used to identify the best-fit distribution to the tails of daily precipitation among four theoretical distributions (e.g., Pareto-type II, Lognormal, Weibull, and Gamma distributions). The results indicate that the Lognormal distribution, which is a heavy-tailed distribution, fits the tails of daily precipitation for the majority of the grids. It is inferred from the study that there is an increase in the heaviness of tails of daily precipitation data over India from pre- to post-1970 time periods.