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2021 Vol.37 Issue.5,Published 2021-10-26

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441 Some Ordering Properties of Coherent Systems under Different Random Environments
LING Xiaoliang; GAO Yu; LI Ping

In reliability engineering, components of the coherent system may be dependent since they operate in a common random environment. This paper uses multivariate distortion function to describe the dependence among lifetimes of components and the structure function of the coherent system. Some sufficient conditions are given to compare two coherent systems under different random environments in the sense of usual stochastic order, failure rate order, reversed failure rate order and likelihood ratio order.

2021 Vol. 37 (5): 441-448 [Abstract] ( 110 ) [HTML 1KB] [ PDF 672KB] ( 127 )
449 Optimization Analysis of M/M/1 Queue with Two Types of Priority Customers
ZHANG Yitong; XU Xiuli

This paper consideres an M/M/1 queue model with preemptive and non-preemptive priorities. Customers are divided into three priority levels. The first category of customers enjoy the priority of preemptive, the second category of customers enjoy the priority of non-preemptive, and the third category of customers have no priority. When the first kind of customers arrive, they will interrupt the second or third kind of customers who are receiving the service immediately; When
the second type of customers arrive, if there are only the third type of customers in the system, the customer must wait for the current service to be completed before accepting the service. Customers of the same type follow the queuing rule of FCFS. A multi-dimensional vector Markov process is constructed here using the supplementary variable method, and the probability generating functions of three types of joint distribution of queue length are obtained. Then, each category of customer's average queue length and the probabilities in service are obtained. The influence of the change of service rate on the average queue length of all kinds of customers in the system is studied using MATLAB software. Finally, different cost functions are constructed for optimization analysis.

2021 Vol. 37 (5): 449-460 [Abstract] ( 12 ) [HTML 1KB] [ PDF 1061KB] ( 72 )
461 Study on Optimal Discount Coefficient of Multi-Event Catastrophe Bonds under Risk Feedback Conditions
SUN Zhentao; YAO Dingjun; CHENG Gongpin

Due to the correlation between the trigger events of traditional multi-event catastrophe bonds, investors are more likely to suffer losses and their returns are more volatile. In order to reduce the impact of this disadvantage, taking earthquake catastrophe bond as an example, this paper reconstructs the payment function of multi-events catastrophe bond considering risk feedback. Under the objective of maximizing the hedging efficiency of insurance companies, we obtain the optimal discount coefficient of the cash flow payment after triggering all risk events by Monte Carlo simulation. The results show that under the
new payment function, the expected return of investors increases and the volatility of return decreases, which makes the bond more attractive to the market. Finally, we focuses on the analysis of the impact of catastrophe triggering parameters on the optimal discount coefficient.

2021 Vol. 37 (5): 461-477 [Abstract] ( 5 ) [HTML 1KB] [ PDF 1598KB] ( 60 )
478 The Empirical Bayes Estimators of the Probability Parameter of the Beta-Negative Binomial Model under Zhang's Loss Function
ZHOU Mingqin; ZHANG Yingying; SUN Ya; SUN Ji; RONG Tengzhong; LI Manman

For the probability parameter of the beta-negative binomial model, we recommend and analytically calculate the Bayes estimator under Zhang's loss function which penalizes gross overestimation and gross underestimation equally. This estimator minimizes the posterior expected Zhang's loss (PEZL). We also calculate the usual Bayes estimator under the squared error loss function. Moreover, we obtain the PEZLs evaluated at the two Bayes estimators. After that, we show two theorems about the estimators of the hyperparameters of the beta-negative binomial model by the moment method and the maximum likelihood estimation (MLE) method, respectively. Hence, the empirical Bayes estimator of the probability parameter under Zhang's loss function is obtained with the hyperparameters estimated from the two theorems. In the numerical simulations, we have illustrated three things. Firstly, we have exemplified the two inequalities of the Bayes estimators and the PEZLs. Secondly, we have illustrated that the moment estimators and the maximum likelihood estimators (MLEs) are consistent estimators of the hyperparameters. Thirdly, we have calculated the goodness-of-fit of the beta-negative binomial model to the simulated data.
Finally, we consider four cases to fit a real insurance claims data by utilizing the beta-negative binomial model.

2021 Vol. 37 (5): 478-494 [Abstract] ( 13 ) [HTML 1KB] [ PDF 604KB] ( 52 )
495 Research on Estimation Methods for the Upper-Truncated Geometric Mixture Distribution
ZHUANG Wei

Geometric model is a widely used discrete probabilistic model. The uppper-truncated geometric model which is derived from geometric model also has important applications in real life. In this paper we will specify the upper-truncated geometric mixed model, study its non-parametric and parameter estimation methods. The maximum conditional likelihood estimation of unknown parameters is proposed based on gaussian distribution, beta distribution and gamma distribution. Meanwhile, combined with simulation and an example, the applications of the upper-truncated geometric model is illustrated.

2021 Vol. 37 (5): 495-506 [Abstract] ( 10 ) [HTML 1KB] [ PDF 725KB] ( 48 )
507 Precise Rates in the Generalized Law of the Iterated Logarithm in Multidimensional Euclidean Space
XU Mingzhou; CHENG Kun

Let \{X,X_n,n\ge1\} be a sequence of i.i.d. random vectors with \ep X=(0,0,\cdots,0)_{m\times1} and \cov(X,X)=\Sigma, and set S_n=\tsm_{i=1}^nX_i, n\ge1. For every d>0 and a_n=o((\ln\ln n)^{-d}), we obtain the precise rates in the generalized law of the iterated logarithm for a kind of weighted infiniteseries of \pr(|S_n|\ge(\varepsilon+a_n)\sigma\sqrt{n}(\ln\ln n)^d).

2021 Vol. 37 (5): 507-514 [Abstract] ( 6 ) [HTML 1KB] [ PDF 511KB] ( 53 )
515 Deviations for Weak Record Numbers in Simple Random Walks
LI Yuqiang; YAO Qiang

Record numbers are important statistics in random walk models. Their deviation principles are unknown as far as we know. In this article, we provide the asymptotic probabilities of large and moderate deviations for the number of weak records in one-dimensional symmetric simple random walks.

2021 Vol. 37 (5): 515-522 [Abstract] ( 15 ) [HTML 1KB] [ PDF 480KB] ( 52 )
523 The Estimation of Time Varying Volatility Based on the Long Stock Return Series with Its Application
WANG Jiangyan; LIN Jinguan; CHEN Xulan

The price fluctuations in the stock market and the changes in the rate of return brought by it have attracted the attention of experts. In this context, this paper focuses on the changes in the volatility of long-term asset return series, and being used in analysis of the Shanghai Composite Index. Since the most commonly used GARCH model is available when the observation period is short, and the volatility for long-term asset return series tends to have long memory, this paper
proposes an improved time varying GARCH model. In order to fit the change of volatility well, we decompose the variance of volatility into a conditional part and an unconditional part. Through reasonable model transformation, the conditional variance follows the GARCH process, while the unconditional variance, which is changing smoothly over time, is estimated by the nonparametric method (B-spline estimation). The simulation research shows that the model proposed in this paper can better capture the change of volatility in a long run. In order to verify the proposed estimation method, the daily return series of the Shanghai Composite Index are taken for the empirical analysis. In the end, we found that: (i) The nonparametric estimation method proposed in this paper performs well. (ii) The variation of the unconditional variance has a strong correlation with the economic recession; (iii) An apparent variation in the time-varying GARCH model can be explained by the variation of the non-stationary component.

2021 Vol. 37 (5): 523-543 [Abstract] ( 7 ) [HTML 1KB] [ PDF 2182KB] ( 65 )
544 Summary of Growth of Mathematical Stochastics
CHEN Mu-Fa

This paper summarizes the growth of mathematical stochastics. It includes two periods of early history, the syllabus of ``College Mathematics'' for non-mathematics majors facing the new century, the teaching materials and research institutions of ``mathematical stochastics'', the mark of grow up of mathematical stochastics, etc.

2021 Vol. 37 (5): 544-550 [Abstract] ( 26 ) [HTML 1KB] [ PDF 12543KB] ( 72 )
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