The article presents a performance-based comparative analysis of popular deep neural network (DNN) models such as 1-dimensional convolutional neural network (1D-CNN) and long short-term memory (LSTM) for position estimation of shape memory alloy (SMA)-based wire actuator. These DNN models utilize the self-sensing property (SSP) for position estimation of the SMA actuator. The phase-dependent electrical resistivity of SMA wire acts as SSP, where the electrical resistivity in the form of SMA wire resistance acts as inputs to the proposed models for precise estimation of the current position of the SMA actuator. For effective position control of the SMA actuator, accurate position sensor feedback is required, utilizing SSP results in the elimination of this external sensor. This will improve the overall system in terms of compactness and reduced interface complexity. Coming to DNN models, 1D-CNN has been meagerly explored in the current literature landscape for self-sensing estimation of SMA actuators. These 1D-CNN models are becoming quite popular for time series prediction for various applications and are emerging as an alternative to widely used LSTM models. In this paper, a novel implementation of a 1D-CNN model for SMA actuator position estimation has been done. A comparative analysis between 1D-CNN and LSTM has been done for prediction capability and inference speed based on performance measures such as Mean Square Error (MSE), Mean Absolute Error (MAE), symmetric Mean Absolute Percentage Error (sMAPE), data distribution, and average inference speed. The proposed comparative results show that 1D-CNN has a matching performance with the LSTM model with respect to prediction capability, however, 1D-CNN offers faster inference speed. The analysis of the proposed work can be useful for choosing a suitable DNN model for deployment on low computing platforms such as microcontrollers for SMA actuator-based real-time applications where time latency is a critical parameter.