Acta Mathematicae Applicatae Sinica, English Series

There are already a lot of models to fit a set of stationary time series, such as AR, MA, and ARMA models. For the non-stationary data, an ARIMA or seasonal ARIMA models can be used to fit the given data. Moreover, there are also many statistical softwares that can be used to build a stationary or non-stationary time series model for a given set of time series data, such as SAS, SPLUS, etc. However, some statistical softwares wouldn’t work well for small samples with or without missing data, especially for small time series data with seasonal trend. A nonparametric smoothing technique to build a forecasting model for a given small seasonal time series data is carried out in this paper. And then, both the method provided in this paper and that in SAS package are applied to the modeling of international airline passengers data respectively, the comparisons between the two methods are done afterwards. The results of the comparison show us the method provided in this paper has superiority over SAS’s method.

In this paper, we will show that under some smallness conditions, the planar diffusion wave $$\bar v\left( {\frac{{x_1 }} {{\sqrt {1 + t} }}} \right)$$ is stable for a quasilinear wave equation with nonlinear damping: v tt − Δf(v) + v t + g(v t ) = 0, x = (x 1, x 2, ⋯, x n ) ∈ ℝ n , where $$\bar v\left( {\frac{{x_1 }} {{\sqrt {1 + t} }}} \right)$$ is the unique similar solution to the one dimensional nonlinear heat equation: $v_t - f(v)_{x_1 x_1 } = 0,f'(v) > 0$ , v(±∞, t) = v ±, v + ≠ v −. We also obtain the L ∞ time decay rate which reads $\left\| {v - \bar v} \right\|_{L^\infty } = O(1)(1 + t) - \tfrac{r} {4} $ , where r = min{3, n}. To get the main result, the energy method and a new inequality have been used.

The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.

In this paper, we consider a two-dimensional perturbed risk model with stochastic premiums and certain dependence between the two marginal surplus processes. We obtain the Lundberg-type upper bound for the infinite-time ruin probability by martingale approach, discuss how the dependence affects the obtained upper bound and give some numerical examples to illustrate our results. For the heavy-tailed claims case, we derive an explicit asymptotic estimation for the finite-time ruin probability.

A thorough investigation of the system $$\frac{{d^2 y(x)}}{{dx^2 }} + p(x)y(x) = 0$$ with periodic impulse coefficients $$\begin{gathered} p(x) = \left\{ {\begin{array}{*{20}c} {1, 0 \leqslant x< x_0 (2\pi > x_0 > 0)} \\ { - \eta , x_0 \leqslant x< 2\pi (\eta > 0)} \\ \end{array} } \right. \hfill \\ p(x) = p(x + 2\pi ), ---\infty< x< \infty \hfill \\ \end{gathered} $$ is given, and the method can be applied to one with other periodic impulse coefficients.

In 1960, J. B. Rosen gave a famous Gradient Projection Method in [1]. But the convergence of the algorithm has not been proved for a long time. Many authors paid much attention to this problem, such as X.S. Zhang proved in [2] (1984) that the limit point of {x k} which is generated by Rosen's algorithm is a K-T piont for a 3-dimensional caes, if {x k} is convergent. D. Z. Du proved in [3] (1986) that Rosen's algorithm is convergent for 4-dimensional. In [4] (1986), the author of this paper gave a general proof of the convergence of Rosen's Gradient Projection Method for ann-dimensional case. As Rosen's method requires exact line search, we know that exact line search is very difficult on computer. In this paper a line search method of discrete steps are presented and the convergence of the algorithm is proved.

Chúng tôi trình bày một phân tích về sự hội tụ của một phương pháp ngẫu nhiên để mô hình hóa các chất lỏng phức tạp sử dụng các trường cấu hình Brownian (BCF) cho dòng cắt. Phân tích này xem xét cấu trúc đặc biệt của các phương trình vi phân ngẫu nhiên cho dòng cắt. Chúng tôi thiết lập tốc độ hội tụ tối ưu. Chúng tôi cũng phân tích bản chất của sai số bằng cách cung cấp các độ lớn bậc cao nhất của nó.

This paper considers the repeated likelihood ratio test for the variance of normal distribution with un-known mean. The large deviations for Type I, and Type II error probabilities have been developed.

This paper is devoted to the goodness-of-fit test for the general autoregressive models in time series. By averaging for the weighted residuals, we construct a score type test which is asymptotically standard chi-squared under the null and has some desirable power properties under the alternatives. Specifically, the test is sensitive to alternatives and can detect the alternatives approaching, along a direction, the null at a rate that is arbitrarily close to n −1/2. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of alternatives. The performance of the tests is evaluated through simulation studies.