In this paper, a new compounding distribution, named the Weibull–Poisson distribution is introduced. The shape of failure rate function of the new compounding distribution is flexible, it can be decreasing, increasing, upside-down bathtub-shaped or unimodal. A comprehensive mathematical treatment of the proposed distribution and expressions of its density, cumulative distribution function, survival function, failure rate function, the kth raw moment and quantiles are provided. Maximum likelihood method using EM algorithm is developed for parameter estimation. Asymptotic properties of the maximum likelihood estimates are discussed, and intensive simulation studies are conducted for evaluating the performance of parameter estimation. The use of the proposed distribution is illustrated with examples.

Precisely forecasting the price of crude oil is challenging due to its fundamental properties of nonlinearity, volatility, and stochasticity. This paper introduces a novel hybrid model, namely, the KV-MFSCBA-G model, within the decomposition–integration paradigm. It combines the mixed-frequency convolutional neural network–bidirectional long short-term memory network-attention mechanism (MFCBA) and generalized autoregressive conditional heteroskedasticity (GARCH) models. The MFCBA and GARCH models are employed to respectively forecast the low-frequency and high-frequency components decomposed through variational mode decomposition optimized by Kullback–Leibler divergence (KL-VMD). The classification of these components is performed using the fuzzy entropy (FE) algorithm. Therefore, this model can fully exploit the advantages of deep learning networks in fitting nonlinearities and traditional econometric models in capturing volatilities. Furthermore, the intelligent optimization algorithm and the low-frequency economic variable are introduced to improve forecasting performance. Specifically, the sparrow search algorithm (SSA) is employed to determine the optimal parameter combination of the MFCBA model, which is incorporated with monthly global economic conditions (GECON) data. The empirical findings of West Texas Intermediate (WTI) and Brent crude oil indicate that the proposed approach outperforms other models in evaluation indicators and statistical tests and has good robustness. This model can assist investors and market regulators in making decisions.

Considering the complexity pattern of the gold price, this paper adopts the decomposition‐reconstruction‐forecast‐mergence scheme to perform the international gold price forecast. The original gold price data are decomposed into 12 intrinsic mode functions and a residual by the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method, and then the 13 sequences are reconstructed into a high‐frequency subsequence (IMFH), a low‐frequency subsequence (IMFL), and the residual (Res). According to the different characteristics of the subsequences, the IMFL and Res are forecasted by the support vector regression (SVR) model. Besides, in order to further improve the prediction accuracy of IMFH, we have developed a novel hybrid method based on the support vector regression (SVR) model and the grey wolf optimizer (GWO) algorithm with SVR for predicting the IMFH of gold prices, i.e., the CEEMDAN‐GWO‐SVR model. This hybrid model combines the methodology of complex systems with machine learning techniques, making it more appropriate for analyzing relationships such as high‐frequency dependences and solving complex nonlinear problems. Finally, the final result is obtained by combining the forecasting results of the three subsequences. The empirical results show that the proposed model demonstrates the highest prediction ability among all of the investigated models in a comparison of prediction errors with other individual models.

This article intensively studies the stock market volatility spillover effects between China and the countries along the Belt and Road (B&R) based on the covered selection of Morgan Stanley Capital International Inc (MSCI) index by using multiplicative error model to measure stock market volatility with daily price range. The results show that during the whole sample period, there are bilateral linkages of volatility between the stock markets of China and all of B&R countries. Most of B&R and China’s markets are sensitive to positive news but the asymmetry is trivial. Financial crisis intensified the volatility spillover effects across countries while the markets’ volatilities tend to be influenced by the negative shocks from foreign markets. The B&R markets as risk absorbers exhibit significant sensitivities to the negative news from Chinese market during the crisis period.