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Since calculating the amount of fractal in the ARFIMA time series and increasing its accuracy and bring it closer to reality is very important, this article intends to investigate the possibility of modifying this computational formula by changing the focus criterion and using simulation. In the present paper, by analysing and simulating the fractal parameter for time series ARFIMA model and redefining and reviewing the Fractal mathematical, a fractal calculus and dimension in comparison with Euclidean norms introduced. In this regard, first, a new criterion about fractal or Hausdorff component for measuring the forms of fractal time series introduced, then the effects and functional inquiries using simulation data searched, and some mathematical proofs through simulation of data achieved. The findings showed that, the deviation of the new estimator from the simulated initial value is less, and closer to reality as this new criterion introduced by changing the focus criterion and replacing the mean with the median due to less sensitivity to out-dated data. The new criterion is better for determining the fractal parameter and identifying its degree of effectiveness. Finally, the findings empirically indicated that the proposed criterion is more efficient and better than the others for calculating fractal dimensions.