Bayesian prediction bounds for the pareto lifetime model

Suppose that the length of time in years for which a business operates until failure has a Pareto distribution. Let x₁ ≤ x₂ x₃ ≤…≤z_k denote the survival lifetimes of the first k of a random sample of n businesses. Bayesian predictions are to be made on the ordered failure times of t h e remaining (n-k) businesses, using the conditional probability density function. Examples are given to illustrate our results.     

A new rank test for the K-Sample problem

The treatment sum of squares in the one-way analysis of variance can be expressed in two different ways: as a sum of comparisons between each treatment and the remaining treatments combined, or as a sum of comparisons between the treatments two at a time. When comparisons between treatments are made with the Wilcoxon rank sum statistic, these two expressions lead to two different tests; namely, that of Kruskal and Wallis and one which is essentially the same as that proposed by Crouse (1961,1966). The latter statistic is known to be asymptotically distributed as a chi-squared variable when the numbers of replicates are large. Here it is shown to be asymptotically normal when the replicates are few but the number of treatments is large. For all combinations of numbers of replicates and treatments its empirical distribution is well approximated by a beta distribution     

Predicting Insurance Losses Under Cross-Classification: A Comparison of Alternative Approaches

Various mathematical and statistical models for estimation of automobile insurance pricing are reviewed. The methods are compared on their predictive ability based on two sets of automobile insurance data for two different states collected over two different periods. The issue of model complexity versus data availability is resolved through a comparison of the accuracy of prediction. The models reviewed range from the use of simple cell means to various multiplicative-additive schemes to the empirical-Bayes approach. The empirical-Bayes approach, with prediction based on both model-based and individual cell estimates, seems to yield the best forecast.     

A Bootstrap Stationarity Test for Predictive Regression Invalidity

In order for predictive regression tests to deliver asymptotically valid inference, account has to be taken of the degree of persistence of the predictors under test. There is also a maintained assumption that any predictability in the variable of interest is purely attributable to the predictors under test. Violation of this assumption by the omission of relevant persistent predictors renders the predictive regression invalid, and potentially also spurious, as both the finite sample and asymptotic size of the predictability tests can be significantly inflated. In response, we propose a predictive regression invalidity test based on a stationarity testing approach. To allow for an unknown degree of persistence in the putative predictors, and for heteroscedasticity in the data, we implement our proposed test using a fixed regressor wild bootstrap procedure. We demonstrate the asymptotic validity of the proposed bootstrap test by proving that the limit distribution of the bootstrap statistic, conditional on the data, is the same as the limit null distribution of the statistic computed on the original data, conditional on the predictor. This corrects a long-standing error in the bootstrap literature whereby it is incorrectly argued that for strongly persistent regressors and test statistics akin to ours the validity of the fixed regressor bootstrap obtains through equivalence to an unconditional limit distribution. Our bootstrap results are therefore of interest in their own right and are likely to have applications beyond the present context. An illustration is given by reexamining the results relating to U.S. stock returns data in Campbell and Yogo (2006 Campbell, J. Y. and Yogo, M. (2006), “Efficient Tests of Stock Return Predictability,” Journal of Financial Economics, 81, 27–60.[Crossref], [Web of Science ®] , [Google Scholar]). Supplementary materials for this article are available online.     

Serial unblocked designs balanced for correlation between neighbours,II

The theory in Part I contained an error that was inferred from the output of a program, written in SAS by Eric P. Smith and David D. Morris. The program produces random BUS designs in accordance with the algorithm of Part I. The theory is here corrected by using a combinatorial argument that involves elementary number theory. The algorithm needs no change but its interpretation is now adjusted.     

An exact test for nbue class of survival functions

Let F(x) be a life distribution. An exact test is given for testing H₀ F is exponential, versusH₁Fε NBUE (NWUE); along with a table of critical values for n=5(l)80, and n=80(5)65. An asymptotic test is made available for large values of n, where the standardized normal table can be used for testing.     

On the minimum expected quantity for the validity of the chi-squared test in 2 2 2 tables

A 2 2 2 contingency table can often be analysed in an exact fashion by using Fisher's exact test and in an approximate fashion by using the chi-squared test with Yates' continuity correction, and it is traditionally held that the approximation is valid when the minimum expected quantity E is E S 5. Unfortunately, little research has been carried out into this belief, other than that it is necessary to establish a bound E>E*, that the condition E S 5 may not be the most appropriate (Martín Andrés et al., 1992) and that E* is not a constant, but usually increasing with the growth of the sample size (Martín Andrés & Herranz Tejedor, 1997). In this paper, the authors conduct a theoretical experimental study from which they ascertain that E* value (which is very variable and frequently quite a lot greater than 5) is strongly related to the magnitude of the skewness of the underlying hypergeometric distribution, and that bounding the skewness is equivalent to bounding E (which is the best control procedure). The study enables estimating the expression for the above-mentioned E* (which in turn depends on the number of tails in the test, the alpha error used, the total sample size, and the minimum marginal imbalance) to be estimated. Also the authors show that E* increases generally with the sample size and with the marginal imbalance, although it does reach a maximum. Some general and very conservative validity conditions are E S 35.53 (one-tailed test) and E S 7.45 (two-tailed test) for alpha nominal errors in 1% h f h 10%. The traditional condition E S 5 is only valid when the samples are small and one of the marginals is very balanced; alternatively, the condition E S 5.5 is valid for small samples or a very balanced marginal. Finally, it is proved that the chi-squared test is always valid in tables where both marginals are balanced, and that the maximum skewness permitted is related to the maximum value of the bound E*, to its value for tables with at least one balanced marginal and to the minimum value that those marginals must have (in non-balanced tables) for the chi-squared test to be valid.     

Estimation in an empirical bayes model for longitudinal and cross‐sectionally clustered binary data

Some studies generate data that can be grouped into clusters in more than one way. Consider for instance a smoking prevention study in which responses on smoking status are collected over several years in a cohort of students from a number of different schools. This yields longitudinal data, also cross‐sectionaliy clustered in schools. The authors present a model for analyzing binary data of this type, combining generalized estimating equations and estimation of random effects to address the longitudinal and cross‐sectional dependence, respectively. The estimation procedure for this model is discussed, as are the results of a simulation study used to investigate the properties of its estimates. An illustration using data from a smoking prevention trial is given.     

Estimating the number of clusters

Hartigan (1975) defines the number q of clusters in a d ‐variate statistical population as the number of connected components of the set {f > c}, where f denotes the underlying density function on R^d and c is a given constant. Some usual cluster algorithms treat q as an input which must be given in advance. The authors propose a method for estimating this parameter which is based on the computation of the number of connected components of an estimate of {f > c}. This set estimator is constructed as a union of balls with centres at an appropriate subsample which is selected via a nonparametric density estimator of f. The asymptotic behaviour of the proposed method is analyzed. A simulation study and an example with real data are also included.