### 26 December 2023, Volume 39 Issue 6

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• MO Xiaoyun, QIN Guohua, OU Hui
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 791-801. https://doi.org/10.3969/j.issn.1001-4268.2023.06.001
The actuarial calculation of annuities is closely related to the interest rate model. In standard annuities, the interest rate for each period is a fixed constant. In practice, the interest rate for each period can be a variable or even a random variable. These random
variables constitute a stochastic process of interest rate. In many cases, the stochastic process of interest rate is a Markov process. This article studies the actuarial calculation of annuities under the Markov stochastic interest rate model. It is proved that if the interest rate process is a time-homogeneous Markov chain, then the discounting process is also a time-homogeneous Markov chain, and they have the same' initial distribution and the same' one-step transition probability matrix. With the help of the interest rate discounting process, the expectation and variance of the present value of annuities under the Markov stochastic interest rate model are calculated. This article introduced annuity polynomials, operators, and annuity operator polynomials. It makes the expressions of expectation and variance for annuities very concise, and easy to program and calculate.
• DUAN Xiaogang
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 802. https://doi.org/10.3969/j.issn.1001-4268.2023.06.002
Simple random sampling, both with and without replacement, are fundamental in traditional survey sampling. Based on an idea of imaginary census'', we provide in this note a new way for calculating the sample variance of simple sample average, as well as understanding the intrinsic relationship between simple random sampling with and without replacement. The key concept is an imaginary census'' matrix, which records the exact sampling trajectory of each draw without replacement, until all population units were sampled out. The random matrix possesses a nice probabilistic symmetry, and each of its column summed to be a fixed number. The new framework, in a sense, is a fusion of two existing classic techniques in traditional survey sampling. One depends on the random vector of 0\,--\,1 valued random variables indicating which population units were sampled, and the other is the symmetrization technique. Our method appears valuable for understanding several important sampling strategies, even to the branch of survey sampling itself. For illustration, we present two examples pertain to unequal probability sampling with replacement and adaptive cluster sampling, with a focus on understanding these sampling strategies from the perspective of simple random sampling.
• LIN Bingqing, ZHUANG Zefan
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 813-831. https://doi.org/10.3969/j.issn.1001-4268.2023.06.003
With the wide use of new generation sequencing technology, single-cell RNA data has gradually become the mainstream object of research. However, it is costly to obtain single-cell RNA data directly from organisms. Therefore, how to obtain these data simply and quickly is an important problem. In order to meet the needs of comparative experiments, the simulation method of single-cell RNA data usually needs not only the statistics of the simulation data are close to the original data, but also the gene and cell samples that can retain the original data in the simulation data. Here, we introduce a data-based simulation method. On the basis of retaining the gene and cell samples of the original data, we can simulate the single-cell RNA data at low cost and ensure that the simulation results are similar to the original data in most characteristics. Through a large number of numerical experiments, it is proved that the proposed method is superior to other simulation methods in terms of distribution of gene expression.
• LIANG Wenjuan, WANG Xiaofei, ZHOU Zonghao
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 832-848. https://doi.org/10.3969/j.issn.1001-4268.2023.06.004
This paper mainly studies the accurate reliability inference of Wiener process with measurement errors. A test statistic is proposed to judge whether there is measurement error among the population. By constructing some pivotal quantities, the exact confidence intervals of diffusion parameters, the generalized confidence intervals of other model parameters, and the quantiles of life and reliability function are given. Furthermore, the generalized prediction interval of future degradation level is derived. In order to evaluate the performance of the proposed inference method, some simulation results are studied. Finally, two examples are given to illustrate the applicability of the proposed procedure.
• CAI Jingheng, WANG Ruoning
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 849-858. https://doi.org/10.3969/j.issn.1001-4268.2023.06.005
This paper mainly proposes Bayesian methods to analyze the accelerated failure time models. In this model, the distribution of the error terms is unknown and approximated with a P\'{o}lya tree distribution. This paper employs the Bayesian Lasso and Markov chain Monte Carlo methods for parameter estimation and variable selection. Simulation studies demonstrate that the proposed methods can identify the important factors and provide accurate estimates. Finally, the proposed model is applied to identify the risk factors of survival times of the Type II diabetic patients.
• WANG Yeshunying, MENG Hui, LIAO Pu
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 859-878. https://doi.org/10.3969/j.issn.1001-4268.2023.06.006
We investigate the equilibrium reinsurance strategy in an infinite reinsurance space for an ambiguity-averse insurer (AAI) under a continuous-time framework. We assume that the surplus process of the AAI follows the Cram\'{e}r-Lundberg (C-L) model perturbed by standard Brownian motion, and the insurer invests his surplus in a risk-free asset. We present the equilibrium reinsurance strategy and its corresponding value function by solving extended Hamilton-Jacobi-Bellman (HJB) system equations, and we find that the AAI's equilibrium reinsurance strategy to maximize the time-inconsistent penalty-dependent mean-variance
reward function is a combination of quota-share with excess of loss reinsurance or its dual form. Detailed numerical analyses are presented to illustrate the various effects of insurer aversion to various uncertainties and other parameters on the equilibrium reinsurance strategy and its corresponding value function.
• SUN Qi, ZHANG Mei
CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS. 2023, 39(6): 879-896. https://doi.org/10.3969/j.issn.1001-4268.2023.06.007
For a supercritical branching processes with immigration \{Z_n\} with offspring distribution \{p_i,i\ge 0\}, it is known that under suitable conditions on the offspring and immigration distributions, $Z_n/m^n$ converges almost surely to a finite and strictly positive limit, where $m$ is the offspring mean. In certain situation p_0>0, we study the limiting properties of the probabilities