Example 1: One might say that the utility function in a given model is increasing and concave in consumption. But it only becomes parametric once one says that u(c)=ln(c) or u(c)=c1-A/1-A. At this point only parameters such as A remain to be specified or estimated.
Example 2: In an econometric model one often imposes assumptions such as that the the relationship being estimated is linear, thence to do a linear regression. These are parametric assumptions. One might also make some estimates of the 'regression function' (the relationship) without such parametric assumptions. This field is called nonparametric estimation.
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