The paper deals with state estimation problem of nonlinear non-Gaussian discrete dynamic systems for improvement of accuracy and consistency. An efficient new algorithm called the adaptive Gaussian-sum square-root cubature Kalman filter(AGSSCKF) with a split-merge scheme is proposed. It is developed based on the squared-root extension of newly introduced cubature Kalman filter(SCKF) and is built within a Gaussian-sum framework. Based on the condition that the probability density functions of process noises and initial state are denoted by a Gaussian sum using optimization method, a bank of SCKF are used as the sub-filters to estimate state of system with the corresponding weights respectively, which is adaptively updated. The new algorithm consists of an adaptive splitting and merging procedure according to a proposed split-decision model based on the nonlinearity degree of measurement. The results of two simulation scenarios(one-dimensional state estimation and bearings-only tracking) show that the proposed filter demonstrates comparable performance to the particle filter with significantly reduced computational cost. An efficient new algorithm called the adaptive Gaussian-sum square-root cubature Kalman filter (AGSSCKF) with a split-merge scheme is proposed. Based on the condition that the probability density functions of process noises and initial states are denoted by a Gaussian sum using optimization method, a bank of SCKF are used as the sub-filters to estimate state of system with the corresponding weights updated, which is adaptively updated. The new algorithm consists of an adaptive splitting and merging procedure according to a proposed split-decision model based on the nonlinearity degree of measurement. The results of two simulation scenarios (one-dimensional state estimation and bearings-only tracking) show that the proposed filter demonstrates comparable performance to the particle filter with significantly reduced computational cost.