In the measurement of actual random phenomenon, the observed data often result in a loss or are distorted due to the existence of a definite dynamic range of measurement instruments. In this study, a trial on the stochastic signal processing for the incomplete data with loss or distortion is newly proposed. Concretely, by regarding the observed data within a finite dynamic measurement range as a random variable with a amplitude saturation, a new unified expression of the probability distribution function matched to this amplitude limitation is derived in a series expansion form. Next, as an application of the above probability expression, a state estimation method based on the above incomplete observations is theoretically proposed through an establishment of wide sence digital filter under the actual situation of existence of the additional noise. Finally, the validity and the effectiveness of the proposed method are experimentally confirmed.