A Viable Alternative to Resorting to Statistical Tables

It is shown in this article that, given the moments of a distribution, any percentage point can be accurately determined from an approximation of the corresponding density function in terms of the product of an appropriate baseline density and a polynomial adjustment. This approach, which is based on a moment-matching technique, is not only conceptually simple but easy to implement. As illustrated by several applications, the percentiles so obtained are in excellent agreement with the tabulated values. Whereas statistical tables, if at all available or accessible, can hardly ever cover all the potentially useful combinations of the parameters associated with a random quantity of interest, the proposed methodology has no such limitation.     

Accuracy Analysis of the Percentile Method for Estimating Non Normal Manufacturing Quality

We consider asymmetric kernel estimates based on grouped data. We propose an iterated scheme for constructing such an estimator and apply an iterated smoothed bootstrap approach for bandwidth selection. We compare our approach with competing methods in estimating actuarial loss models using both simulations and data studies. The simulation results show that with this new method, the estimated density from grouped data matches the true density more closely than with competing approaches.     

Influence diagnostics for the structural sharpe model under normal/independent distributions

A method is proposed in this paper to assess the local influence of minor perturbations for the Sharpe model when the normal distribution is replaced by normal/independent (NI) distributions. The family of NI distributions is an attractive class of symmetric heavy-tailed densities that includes as special cases the normal, t-Student, slash, and the contaminated normal distributions. Since the returns of the market are not observable, the statistical analysis is carried out in the context of an errors-in-variables model. An influence analysis for detecting influential observations (atypical returns) is developed to investigate the sensitivity of the maximum likelihood estimators. Diagnostic measures are obtained based on the conditional expectation of the complete-data log-likelihood function. The results are illustrated by using a set of shares of companies traded in the Chilean stock market.     

Discrete distributions with maximum expected value of the maximum

We give an upper bound for the expected value of the largest order statistic of a simple random sample of size n from a discrete distribution on N points. We also characterize the distributions that attain such bound. In the particular case n=2, we obtain a characterization of the discrete uniform distribution. © 1998 Elsevier Science B.V. All rights reserved.     

Distribution of Exposure Concentrations and Doses for Constituents of Environmental Tobacco Smoke

The ultimate goal of the research reported in this series of three articles is to derive distributions of doses of selected environmental tobacco smoke (ETS)-related chemicals for nonsmoking workers. This analysis uses data from the 16-City Study collected with personal monitors over the course of one workday in workplaces where smoking occurred. In this article, we describe distributions of ETS chemical concentrations and the characteristics of those distributions (e.g., whether the distribution was log normal for a given constituent) for the workplace exposure. Next, we present population parameters relevant for estimating dose distributions and the methods used for estimating those dose distributions. Finally, we derive distributions of doses of selected ETS-related constituents obtained in the workplace for people in smoking work environments. Estimating dose distributions provided information beyond the usual point estimate of dose and showed that the preponderance of individuals exposed to ETS in the workplace were exposed at the low end of the dose distribution curve. The results of this analysis include estimations of hourly maxima and time-weighted average (TWA) doses of nicotine from workplace exposures to ETS (extrapolated from 1 day to 1 week) and doses derived from modeled lung burdens of ultraviolet-absorbing particulate matter (UVPM) and solanesol resulting from workplace exposures to ETS (extrapolated from 1 day to 1 year).     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

Coupon collector's problem and generalized Pareto distributions

Let N_n={1,2,…,n}. We sample with replacement from the set N_n assuming that each element has probability 1/n of being drawn. Let M_n be the waiting time determined by certain stoping rules in the coupon collector's problem. We investigate models for the asymptotic behavior of the excesses of M_n over the high thresholds.     

On some mixture models based on the Birnbaum-Saunders distribution and associated inference

In this paper, we consider three different mixture models based on the Birnbaum-Saunders (BS) distribution, viz., (1) mixture of two different BS distributions, (2) mixture of a BS distribution and a length-biased version of another BS distribution, and (3) mixture of a BS distribution and its length-biased version. For all these models, we study their characteristics including the shape of their density and hazard rate functions. For the maximum likelihood estimation of the model parameters, we use the EM algorithm. For the purpose of illustration, we analyze two data sets related to enzyme and depressive condition problems. In the case of the enzyme data, it is shown that Model 1 provides the best fit, while for the depressive condition data, it is shown all three models fit well with Model 3 providing the best fit.     

Methods to Distinguish Between Polynomial and Exponential Tails

Two methods to distinguish between polynomial and exponential tails are introduced. The methods are based on the properties of the residual coefficient of variation for the exponential and non‐exponential distributions. A graphical method, called a CV‐plot, shows departures from exponentiality in the tails. The plot is applied to the daily log‐returns of exchange rates of US dollar and Japanese yen. New statistics are introduced for testing the exponentiality of tails using multiple thresholds. They give better control of the significance level than previous tests. The powers of the new tests are compared with those of some others for various sample sizes.     

Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions

In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.