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I am confused about "parametric" and "non-parametric": Our topic is nonparametric estimators for probability of default. So first of all, we consider the generalized linear models, as an example we have probit and logit:

$\pi (x)=G(\beta_0 +\sum \beta_i x_i)$

These generalized linear models are parametric models right? They are parametric, since we have parameters beta, which have to be estimated.

Next we consider semiparametric credit scoring, the generalized partial linear model:

$E(Y|X,T)=G(\beta ' X + m(T))$

m(.) is e.g. a Kernel, a smooth function

So the parametric term is again the beta ' x and the non-parametric is the kernel, right?

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  • $\begingroup$ Is this homework? If so, it should be tagged as such. $\endgroup$ – chl Feb 9 '13 at 20:16
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In statistical inference, the terms parametric, semi-parametric, and non-parametric refer to the assumptions necessary for inference to be valid. For instance, the T-test is a non-parametric test for differences in means. This is because the asymptotic sampling distribution of the test-statistic has a normal distribution by the central limit theorem (so there are mild regularity constraints on our probability model for observed data, such as having finite variance).

A parametric test requires assumptions about the probability model for inference to be correct. As you mentioned, many of the GLMs use the mean-variance relationship determined by the link function (and it's relationship to data generating mechanisms for observed data) to improve inference efficiency. Correct assumptions about the probability model for the observed data is a sufficient condition to get more efficient inference relative to non-parametric approaches. When robust standard errors are used (a special case of the GEE), this becomes a semi-parametric approach. The GEE requires only that the mean model is correct to have correct inference (the mean-variance relationship could have been misspecified, but asymptotically, the 95% confidence intervals for the sampling distribution of the test statistic will be correct).

The model you've written up there is an example of a semi-parametric modeling approach. The $m(T)$ can be estimated iteratively using the EM algorithm, or depending on the link function $G(\cdot)$ can be factored into a partial likelihood like with the Cox proportional hazards model.

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  • $\begingroup$ Nonparametric tests also require assumptions about the probability model: the distinction between parametric and non-parametric does not lie there! $\endgroup$ – whuber Feb 9 '13 at 23:58
  • $\begingroup$ I realized my answer may need refining. Can you give an example of the assumptions, aside from very general regularity conditions, behind non-parametric tests? $\endgroup$ – AdamO Feb 10 '13 at 1:46
  • $\begingroup$ @AdamO ok, but what is about the first equation? Is it semiparametric or nonparametric? (the $π(x)=G(β0+∑βixi)$) $\endgroup$ – user20690 Feb 10 '13 at 10:22
  • $\begingroup$ @itsmeagain Can you please register your account. If you are the OP, a moderator could merge your accounts. $\endgroup$ – chl Feb 10 '13 at 10:35
  • $\begingroup$ I don't know what you mean by "regularity conditions," Adam, because I don't think they apply. Distinction concerns the topology of the space of possible distributions. When it is a finite dimensional manifold (so that locally one can use a finite-dimensional real vector to parameterize its elements), the situation is parametric. Otherwise it is non-parametric. When the non-parametric space can be decomposed as a product of a finite-dimensional manifold and another space, we can call it "semi-parametric." $\endgroup$ – whuber Feb 10 '13 at 16:35
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So the parametric term is again the beta ' x and the non-parametric is the kernel, right?

Correct. It's called "nonparametric" because, rather than trying to fit the trend for $T$ using fixed terms like a polynomial or a spline (or cutting it into bands), you're letting the data determine the fit for you. In practice, you do still have some parameters which control how the trend is to be fitted: the bandwidth of the kernel, in this case. A large bandwidth means the fitted curve will be fairly smooth, so it won't be heavily influenced by random noise. The tradeoff is that it may oversmooth the underlying trend, meaning it could miss points where it changes slope or curvature. Conversely, a small bandwidth is more likely to pick up such features, but will also be more strongly affected by random noise.

In principle, any continuous variable that you fit in the parametric part can be replaced by a nonparametric fit. For example, you could fit a trend for age, or income, the same way as you do for $T$. This lets you capture nonlinear behaviour in a data-driven way, ie you don't have to specify what sort of behaviour in advance. You could also handle this problem by cutting the variable into bands and modelling it as a nominal variable, but this is a statistically less efficient way of doing things.

These models are also called generalized additive models, a term which I think is more common than semiparametric modelling.

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