On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables

The distribution of the ratio of two independent normal random variables X and Y is heavy tailed and has no moments. The shape of its density can be unimodal, bimodal, symmetric, asymmetric, and/or even similar to a normal distribution close to its mode. To our knowledge, conditions for a reasonable normal approximation to the distribution of Z = X/Y have been presented in scientific literature only through simulations and empirical results. A proof of the existence of a proposed normal approximation to the distribution of Z, in an interval I centered at β = E(X) /E(Y), is given here for the case where both X and Y are independent, have positive means, and their coefficients of variation fulfill some conditions. In addition, a graphical informative way of assessing the closeness of the distribution of a particular ratio X/Y to the proposed normal approximation is suggested by means of a receiver operating characteristic (ROC) curve.     

Multinomial Selection Problem: A Study of BEM and AVC Algorithms

The two well-known and widely used multinomial selection procedures Bechhofor, Elmaghraby, and Morse (BEM) and all vector comparison (AVC) are critically compared in applications related to simulation optimization problems.

Two configurations of population probability distributions in which the best system has the greatest probability p _i of yielding the largest value of the performance measure and has or does not have the largest expected performance measure were studied.

The numbers achieved by our simulations clearly show that none of the studied procedures outperform the other in all situations. The user must take into consideration the complexity of the simulations and the performance measure probability distribution properties when deciding which procedure to employ.

An important discovery was that the AVC does not work in populations in which the best system has the greatest probability p _i of yielding the largest value of the performance measure but does not have the largest expected performance measure.     

An Alternative to Transfer Function Forecasting Based on Subspace Methods

In the time series literature, recent interest has focused on the so-called subspace methods. These techniques use canonical correlations and linear regressions to estimate the system matrices of an ARMAX model expressed in state space form. In this article, we use subspace methods to forecast two series with the help of some exogenous variables related to them. We compare the results with those obtained using traditional transfer function models and find that the forecasts obtained with both methods are similar. This result is very encouraging because, in contrast to transfer function models, subspace methods can be considered as almost automatic.     

Sampling hyperparameters in hierarchical models: Improving on Gibbs for high-dimensional latent fields and large datasets

ABSTRACT

A common Bayesian hierarchical model is where high-dimensional observed data depend on high-dimensional latent variables that, in turn, depend on relatively few hyperparameters. When the full conditional distribution over latent variables has a known form, general MCMC sampling need only be performed on the low-dimensional marginal posterior distribution over hyperparameters. This improves on popular Gibbs sampling that computes over the full space. Sampling the marginal posterior over hyperparameters exhibits good scaling of compute cost with data size, particularly when that distribution depends on a low-dimensional sufficient statistic.     

Nonparametric product partition models for multiple change-points analysis

ABSTRACT

We propose an extension of parametric product partition models. We name our proposal nonparametric product partition models because we associate a random measure instead of a parametric kernel to each set within a random partition. Our methodology does not impose any specific form on the marginal distribution of the observations, allowing us to detect shifts of behaviour even when dealing with heavy-tailed or skewed distributions. We propose a suitable loss function and find the partition of the data having minimum expected loss. We then apply our nonparametric procedure to multiple change-point analysis and compare it with PPMs and with other methodologies that have recently appeared in the literature. Also, in the context of missing data, we exploit the product partition structure in order to estimate the distribution function of each missing value, allowing us to detect change points using the loss function mentioned above. Finally, we present applications to financial as well as genetic data.     

A review of classic non-asymptotic methods for comparing two proportions by means of independent sampLES

The 2 × 2 tables used to present the data in an experiment for comparing two proportions by means of two observations of two independent binomial distributions may appear simple but are not. The debate about the best method to use is unending, and has divided statisticians into practically irreconcilable groups. In this article, all the available non-asymptotic tests are reviewed (except the Bayesian methodology). The author states which is the optimal (for each group), referring to the tables and programs that exist for them, and contrast the arguments used by supporters of each of the options. They also sort the tangle of solutions into "families", based on the methodology used and/or prior assumptions, and point out the most frequent methodological mistakes committed when comparing the different families.     

Asymptotic Behavior of the Hazard Rate Kernel Estimator Under Truncated and Censored Data

In this article, we give the asymptotic mean integrated squared error and the mean squared error for the kernel estimator of the hazard rate from truncated and censored data. Martingale techniques and combinatory calculus are used to obtain these results. A probability bound and the optimal bandwidth choice are also given.     

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

When estimating loss distributions in insurance, large and small losses are usually split because it is difficult to find a simple parametric model that fits all claim sizes. This approach involves determining the threshold level between large and small losses. In this article, a unified approach to the estimation of loss distributions is presented. We propose an estimator obtained by transforming the data set with a modification of the Champernowne cdf and then estimating the density of the transformed data by use of the classical kernel density estimator. We investigate the asymptotic bias and variance of the proposed estimator. In a simulation study, the proposed method shows a good performance. We also present two applications dealing with claims costs in insurance.