Springer Science and Business Media LLC

The receiver operating characteristic (ROC) curve is a popular graphical tool for describing the accuracy of a diagnostic test. Based on the idea of estimating the ROC curve as a distribution function, we propose a new kernel smoothing estimator of the ROC curve which is invariant under nondecreasing data transformations. We prove that the estimator has better asymptotic mean squared error properties than some other estimators involving kernel smoothing and we present an easy method of bandwidth selection. By simulation studies, we show that for the limited sample sizes, our proposed estimator is competitive with some other nonparametric estimators of the ROC curve. We also give an example of applying the estimator to a real data set.

The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, $${\beta\in\Re^r}$$ may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing is known about the mean vector θ).

The asymptotic null distribution of the Lagrange Multiplier stationarity test, and analytical expressions for the limiting characteristic functions, are derived for a number of time series models which include a piecewise continuous deterministic trend. Some case studies illustrate the performance of the method.

Many experimental situations with controllable, independent design variables have an associated null hypothesis 0 of zero dependence of observations on the design variables. The necessity of randomization of the design variables for asymptotic discriminability between 0 and its complement is considered in terms of likelihood ratios. Applications are made to stochastic processes and comparative experiments.

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.

Nearest neighbour methods traditionally used to estimate density of a sessile biological population treat individuals as points. The present paper suggests distance-based density estimators which treat individuals as circles with variable areas. Distribution of distance between a sample point and thek-th (k = 1, 2, 3, …) nearest circle is derived. Maximum likelihood estimator of density is obtained from a random sample of point tok-th order distances. Assuming a skewed distribution for the circle radius, moment estimators of density and mean circle area are derived.

Trong bài báo này, chúng tôi nâng cao nghiên cứu về các lớp lão hóa và so sánh ngẫu nhiên của thời gian sống còn tại thời điểm ngẫu nhiên (RLRT) và thời gian không hoạt động tại thời điểm ngẫu nhiên (ITRT). Chúng tôi cung cấp một số thuộc tính bảo toàn mới của các lớp lão hóa tổng quát (cụ thể là s-IFR, s-DFR) và thứ tự ngẫu nhiên tổng quát (s-FR) cho RLRT và ITRT, trong đó s là một số nguyên không âm. Một ứng dụng trong lý thuyết độ tin cậy cũng được khảo sát. Các kết quả này củng cố một số kết quả có sẵn trong tài liệu và dự kiến sẽ hữu ích trong lý thuyết độ tin cậy, khoa học pháp y, kinh tế lượng, lý thuyết hàng đợi và khoa học bảo hiểm.