The birth and death of firms govern firm population dynamics, and understanding these processes can guide urban planning and policy. Longitudinal data with full entry and exit records allow direct analysis of birth and death rates and how they depend on external factors and firm-level properties. However, real-world data are often incomplete, with missing records of extinct firms or exit dates. In such cases, it is unclear if and how we can extract information about the birth and death processes. By modeling how these processes shape firms’ age distributions and survival fractions, we show how one can gain insights even from incomplete data. While age distributions are insufficient for inferring both processes, survival fractions reveal how death rates depend on firm age and sector size. Applying our approach to 14 major sectors in Singapore, we find that death rates decline with age and rise with sector size, with a multiplicative interaction between both effects. Assuming sigmoidal dependence on both factors, we infer sector-specific death models that accurately reproduce the data and enable reconstruction of key system features, e.g., sector size trajectories and birth-death rate correlations.