There is an increasing interest in accurately predicting natural human arm motions for areas like human-robot interaction, wearable robots, and ergonomic simulations. This paper studies the problem of predicting natural fingertip and joint trajectories in human arm reaching movements. Compared to the widely-used minimum jerk model, the 5-parameter logistic model can represent natural fingertip trajectories more accurately. Based on 3520 human arm motions recorded by a motion capture system, regression learning is used to predict the five parameters representing the fingertip trajectory for a given target point. Then, the elbow swivel angle is predicted using regression learning to resolve the kinematic redundancy of the human arm at discrete fingertip positions. Finally, discrete joint angles are solved based on the predicted elbow swivel angles and then fitted to a continuous 5-parameter logistic function to obtain the joint trajectory. This method is verified using 48 test motions, and the results show that this method can generate accurate human arm motions.