In this paper we develop a new closure theory for moment approximations in kinetic gas theory and derive hyperbolic moment equations for 13 fluid variables including stress and heat flux. Classical equations have either restricted hyperbolicity regions like Grad's moment equations or fail to include higher moments in a practical way like the entropy maximization approach. The new closure is based on Pearson-Type-IV distributions which reduce to Maxwellians in equilibrium, but allow anisotropies and skewness in non-equilibrium. The closure relations are essentially explicit and easy to evaluate. Hyperbolicity is shown numerically for a large range of values. Numerical solutions of Riemann problems demonstrate the capability of the new equations to handle strong non-equilibrium.