Mean Time to Failure of Weighted k-out-of-n: G Systems

The purpose of this article is to develop a Monte-Carlo simulation algorithm for computing mean time to failure (MTTF) of weighted-k-out-of-n:G and linear consecutive-weighted-k-out-of-n:G systems. Our algorithm is based on the use of appropriately defined stochastic process which represents the total weight of the system at time t. These stochastic processes are explicitly defined and used along with the ordered component lifetimes to simulate MTTF of the systems with weighted components.     

关于“概率论与数理统计”教学的几点新体会

提出改变传统的以教师为中心的教学结构,建立一种既能发挥教师的指导作用,又能充分体现学生主体作用的新型教学结构;采用中英文结合的教学模式,在以母语教学为主体的教学实践中贯穿双语教学,使学生能用英语进行数学方面的交流,并充分利用网络优势,扩大学生的知识面.     

A Starting Point for Navigating the World of Daily Fantasy Basketball

Fantasy sports, particularly the daily variety in which new lineups are selected each day, are a rapidly growing industry. The two largest companies in the daily fantasy business, DraftKings and Fanduel, have been valued as high as $2 billion. This research focuses on the development of a complete system for daily fantasy basketball, including both the prediction of player performance and the construction of a team. First, a Bayesian random effects model is used to predict an aggregate measure of daily NBA player performance. The predictions are then used to construct teams under the constraints of the game, typically related to a fictional salary cap and player positions. Permutation based and K-nearest neighbors approaches are compared in terms of the identification of “successful” teams—those who would be competitive more often than not based on historical data. We demonstrate the efficacy of our system by comparing our predictions to those from a well-known analytics website, and by simulating daily competitions over the course of the 2015–2016 season. Our results show an expected profit of approximately $9,000 on an initial $500 investment using the K-nearest neighbors approach, a 36% increase relative to using the permutation-based approach alone. Supplementary materials for this article are available online.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

Systemic risk statistics with scenario analysis

In this paper, we introduce two new classes of risk statistics, named convex and positively homogeneous systemic risk statistics, respectively. Structural decomposition results and representation results for them are provided. These new risk statistics can be considered as a kind of systemic risk extension of risk statistics introduced by Kou, Peng, and Heyde, and also empirical versions of system risk measures introduced by Cehn, Iyengar, and Mollemi and Kromer, Overbeck, and Zich. Finally, some examples are also given.     

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.     

On tests of symmetry,marginal homogeneity and quasi-symmetry in two-way contingency tables based on minimum φ-divergence estimator with constraints

The restricted minimum φ-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum φ-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221–237], is employed to obtain estimates of the cell frequencies of an I×I contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated φ-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function φ₂ for estimation and the best function φ₁ for testing.     

Csiszar’s ϕ-divergences for testing the order in a Markov chain

x ₁, ..., x _n+r can be treated as the sample values of a Markov chain of order r or less (chain in which the dependence extends over r+1 consecutive variables only), and consider the problem of testing the hypothesis H ₀ that a chain of order r− 1 will be sufficient on the basis of the tools given by the Statistical Information Theory: _ϕ-Divergences. More precisely, if p _{a 1 ....., a r}: _{a r +1} denotes the transition probability for a r ^th order Markov chain, the hypothesis to be tested is H ₀:p _{a 1 ....., a r}: _{a r +1} = p _{a 2 ....., a r}: _{a r +1}, a _i ∈{1, ..., s}, i = 1, ..., r + 1 The tests given in this paper, for the first time, will have as a particular case the likelihood ratio test and the test based on the chi-squared statistic. Received: August 3, 1998; revised version: November 25, 1999     

Order statistics from non-identical right-truncated Lomax random variables with applications

In this paper, we derive some recurrence relations for the single and the product moments of order statistics from n independent and non-identically distributed Lomax and right-truncated Lomax random variables. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and product moments of all order statistics in a simple recursive manner. The results for order statistics from the multiple-outlier model (with a slippage of p observations) are deduced as special cases. We then apply these results by examining the robustness of censored BLUE's to the presence of multiple outliers. Received: November 30, 1998; revised version: March 8, 2000     

Generalised K-S confidence regions: some exact results

One-sided confidence regions for continuous cumulative distribution function are constructed using empirical cumulative distribution functions and the generalized Kolmogorov-Smirnov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. Significance levels necessary for implementation are given. Some other K-S type distances useful in constructing a confidence region with nonconstant width are also included.