Evaluating Probabilistic Forecasts with Bayesian Signal Detection Models

We propose the use of signal detection theory (SDT) to evaluate the performance of both probabilistic forecasting systems and individual forecasters. The main advantage of SDT is that it provides a principled way to distinguish the response from system diagnosticity, which is defined as the ability to distinguish events that occur from those that do not. There are two challenges in applying SDT to probabilistic forecasts. First, the SDT model must handle judged probabilities rather than the conventional binary decisions. Second, the model must be able to operate in the presence of sparse data generated within the context of human forecasting systems. Our approach is to specify a model of how individual forecasts are generated from underlying representations and use Bayesian inference to estimate the underlying latent parameters. Given our estimate of the underlying representations, features of the classic SDT model, such as the receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC), follow immediately. We show how our approach allows ROC curves and AUCs to be applied to individuals within a group of forecasters, estimated as a function of time, and extended to measure differences in forecastability across different domains. Among the advantages of this method is that it depends only on the ordinal properties of the probabilistic forecasts. We conclude with a brief discussion of how this approach might facilitate decision making.     

Uniform Convergence of Compactly Supported Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of non stationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.     

Protection of privacy in efficient application of randomized response techniques

In estimating the proportion of people bearing a sensitive attribute A, say, in a given community, following Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, certain randomized response (RR) techniques are available for application. These are intended to ensure efficient and unbiased estimation protecting a respondent’s privacy when it touches a person’s socially stigmatizing feature like rash driving, tax evasion, induced abortion, testing HIV positive, etc. Lanke (Int Stat Rev 44:197–203, 1976), Leysieffer and Warner (J Am Stat Assoc 71:649–656, 1976), Anderson (Int Stat Rev 44:213–217, 1976, Scand J Stat 4:11–19, 1977) and Nayak (Commun Stat Theor Method 23:3303–3321, 1994) among others have discussed how maintenance of efficiency is in conflict with protection of privacy. In their RR-related activities the sample selection is traditionally by simple random sampling (SRS) with replacement (WR). In this paper, an extension of an essential similarity in case of general unequal probability sample selection even without replacement is reported. Large scale surveys overwhelmingly employ complex designs other than SRSWR. So extension of RR techniques to complex designs is essential and hence this paper principally refers to them. New jeopardy measures to protect revelation of secrecy presented here are needed as modifications of those in the literature covering SRSWR alone. Observing that multiple responses are feasible in addressing such a dichotomous situation especially with Kuk’s (Biometrika 77:436–438, 1990) and Christofides’ (Metrika 57:195–200, 2003) RR devices, an average of the response-specific jeopardizing measures is proposed. This measure which is device dependent, could be regarded as a technical characteristic of the device and it should be made known to the participants before they agree to use the randomization device. The views expressed are the authors’, not of the organizations they work for. Prof Chaudhuri’s research is partially supported by CSIR Grant No. 21(0539)/02/EMR-II.     

Buffon's needle and the estimation of migration distances

A procedure is suggested for estimating migration distances from data on the proportion of migrants crossing regional boundaries. The method makes use of Buffon's needle, a problem in geometrical probability from the eighteenth century that was originally used as an empirical means of estimating π. The procedure is described for various scenarios that differ in their assumptions about region shape, the spatial distribution of population, and the distribution of migration distances. An application to migration distances in the United States is given, and additional attention is given to the estimation of intraregional migration distances.     

An additive subdistribution hazard model for competing risks data

The cumulative incidence function plays an important role in assessing its treatment and covariate effects with competing risks data. In this article, we consider an additive hazard model allowing the time-varying covariate effects for the subdistribution and propose the weighted estimating equation under the covariate-dependent censoring by fitting the Cox-type hazard model for the censoring distribution. When there exists some association between the censoring time and the covariates, the proposed coefficients’ estimations are unbiased and the large-sample properties are established. The finite-sample properties of the proposed estimators are examined in the simulation study. The proposed Cox-weighted method is applied to a competing risks dataset from a Hodgkin's disease study.     

R–estimation of normed bivariate density functions

Normed bivariate density funtions were introduced by HOEFFDING (1940/41). In the present paper estimators for normed bivariate ranks and on a FOURIER series expansion in LEGENDRE polynomials. The estimation of normed bivarate density functions under positive dependence is also described     

Parameter Estimation for the Discrete Stable Family

The discrete stable family constitutes an interesting two-parameter model of distributions on the non-negative integers with a Paretian tail. The practical use of the discrete stable distribution is inhibited by the lack of an explicit expression for its probability function. Moreover, the distribution does not possess moments of any order. Therefore, the usual tools—such as the maximum-likelihood method or even the moment method—are not feasible for parameter estimation. However, the probability generating function of the discrete stable distribution is available in a simple form. Hence, we initially explore the application of some existing estimation procedures based on the empirical probability generating function. Subsequently, we propose a new estimation method by minimizing a suitable weighted L ²-distance between the empirical and the theoretical probability generating functions. In addition, we provide a goodness-of-fit statistic based on the same distance.     

Probability density function of quotient of order statistics from the pareto,power and weibull distributions

This paper deals with the probability density functions of quotient of order statistics. We use the Mellin transform technique, to find the distribution of the quotient Z= X/Xwhere X.,X(i < j) are the ith and jth order statistics from the Pareto, Power and Weibull distributions     

Updating finite markov chains by using techniques of group matrix inversion

Several jackknife methods for the proportional hazards model are proposed. Instead of deleting observations in the calculation of the pseudovalues, we delete the conditional probabilities from the partial likelihood function. The parameter estimators and variance estimators for both the linear and weighted linear jackknife methods are strongly consistent. A limitted simulation study is conducted.     

Lottomania and other anomalies in the market for lotto

This research is concerned with the determination of the demand for “lotto” in Israel. While an important focus of our research is upon the effects on the demand for lotto of ticket pricing and jackpot announcements, we also investigate several empirical phenomena that are apparently inconsistent with expected utility theory. These include an effect we call “lottomania” which is induced by rollover, and “prize fatigue” when the jackpot does not increase. Another aberration from expected utility theory is that the underlying odds of winning have no measurable effect on sales.