Many economic and econometric applications require the integration of functions lacking a closed form antiderivative, which is therefore a task that can only be solved by numerical methods. We propose a new family of probability densities that can be used as substitutes and have the property of closed form integrability. This is especially advantageous in cases where either the complexity of a problem makes numerical function evaluations very costly, or fast information extraction is required for time-varying environments. Our approach allows generally for nonparametric maximum likelihood density estimation and may thus find a variety of applications, two of which are illustrated briefly:

Estimation of Value at Risk based on approximations to the density of stock returns.

Recovering risk neutral densities for the valuation of options from the option price - strike price relation

We propose a new family of density function that posses both flexibility and closed form expressions for moments and anti-derivatives, making them particularly appealing for applications. We illustrate its usefulness by applying our new family to obtain density forecasts of U.S. inflation. Our methods generate forecasts that improve on standard methods based on AR-ARCH models relying on normal or Student's t-distributional assumptions.