This paper investigates the potential of movable antennas (MAs) to enhance physical layer security within a multiple-input multiple-output multiple-antenna eavesdropper (MIMOME) system. We consider a practical scenario where the transmitter operates with imperfect eavesdropper channel state information (ECSI), knowing only the instantaneous line-of-sight (LoS) component and the statistical properties of non-line-of-sight (NLoS) component. To rigorously quantify secrecy performance under the ECSI uncertainty, we adopt the ergodic secrecy rate (ESR) as the metric. Since deriving an exact analytical expression for the ESR is intractable, we leverage random matrix theory to derive a deterministic equivalent. This avoids heavy Monte Carlo simulations and also provides explicit insights into the effects of channel spatial statistics on secrecy performance. Building upon the deterministic equivalent, we formulate a joint maximization problem for the transmit precoding matrix and the antenna positions at the legitimate transmitter. To tackle the non-convexity of this optimization problem, we develop a comprehensive alternating optimization framework. Specifically, the precoding matrix is optimized via a majorization-minimization (MM) algorithm, where the gradient is computed by solving an implicit fixed-point equation. For the antenna position optimization, the complexity of the objective function prevents the construction of standard MM surrogate. To this end, we further propose a novel AMSGrad-based surrogate function that relies solely on gradient information. We provide a rigorous theoretical proof that guarantees the convergence of this proposed algorithm despite relaxing the strict majorization conditions.