Randomized response (RR) techniques are commonly used to estimate the proportion of a sensitive attribute in a finite population while the preserving respondents' privacy. In this paper, we develop a novel two-stage randomized response model to jointly estimate the population proportion $\eta$ of a sensitive characteristic and the probability $T$ that a respondent provides a true answer under direct questioning. First the proposed design first implements direct questioning and only applies a randomization device to non-respondents, thus reducing unnecessary randomization and improving data quality. The randomization mechanism incorporates three additional questions within a two-stage probabilistic framework, thus offering enhanced privacy protection and improved statistical efficiency. Closed-form expressions for the proposed estimators are derived, and their statistical properties, including unbiasedness and variance, are obtained under simple random sampling. Analytical efficiency comparisons with existing models are established through mean-squared error criteria. The methodology is further extended to stratified random sampling, and the corresponding estimators and variance expressions are derived. Monte Carlo simulations are conducted to assess the finite-sample performance and to support the theoretical findings. The results demonstrate that the proposed estimators outperform competing approaches in both efficiency and privacy protection. The proposed framework provides a mathematically rigorous and practically effective tool for inference on sensitive characteristics in survey sampling.

Accurately forecasting gold returns is critical for investors, policymakers, and risk managers, yet it remains challenging due to the coexistence of deterministic cycles, stochastic volatility, and nonlinear dependencies. This study introduced a novel hybrid model, local mean decomposition (LMD), stochastic product function (SPF), autoregressive integrated moving average (ARIMA), and long short-term memory (LSTM) (LMD-SPF-ARIMA-LSTM), that integrates signal decomposition, statistical diagnostics, and machine learning to address these complexities. LMD first decomposes the return series into product functions (PFs). An SPF then applies formal tests (augmented Dickey-Fuller (ADF) for stationarity, Brock-Dechert-Scheinkman (BDS) for linearity, and correlation filtering) to classify PFs, allocating them to ARIMA for linear-stationary components and to LSTM for nonlinear-nonstationary ones. Using daily gold return data from June 2020 to May 2025, the proposed framework achieved substantially improves over benchmarks, reducing mean absolute error (MAE) by more than 55% compared to ARIMA, lowering root mean squared error (RMSE) by 57% relative to LSTM, and attaining 85.31% directional accuracy-over three percentage points higher than the best competing hybrid. Unlike previous LMD-ARIMA-LSTM approaches that treat all decomposed components uniformly, our method tailors the modeling strategy to the statistical properties of each PF, reducing redundancy, lowering computational cost, and enhancing generalization. These results not only demonstrate the methodological significance of combining decomposition with statistically informed model selection but also provide practical value by delivering more reliable and interpretable forecasts for financial decision-making in volatile markets.