Time series can contain both linear and nonlinear components, and linear and nonlinear artificial neural networks (L&NL-ANNs) have been proposed to forecast them. Although L&NL-ANNs can produce well forecasting results, these neural networks have a handicap in point of view statistical science like other neural networks. The forecasts obtained from L&NL-ANNs may change depending on the time series samples, but this variation is neglected in the literature. The objective of this study has overcome this handicap and producing a method which can give point forecasts, confidence intervals and some weights significance hypothesis tests besides the proposed method performs linearity and nonlinearity hypothesis tests. The proposed method is compared with other conventional methods using a Monte Carlo simulation study and real-world time series data which are Nikkei 225, Dow Jones and Istanbul Stock Exchange time series datasets as well as Australian beer consumption time series. The performance of the proposed method is evaluated using the application and simulation results and found to perform well overall with respect to other methods. It is shown that bootstrapped L&NL-ANN produced the smallest mean and variance of forecast errors for results obtained from different random initial parameters.