Why is the variance of the Parzen density estimator infinite?

[crbazevedo]

Hello everyone,

I'm new to this forum and I'm glad to have found such a high quality resource where we can have such valuable guidance and discussions.

I've read somewhere that the variance of p(x) = {\frac{1}{n}}\sum_{i=1}^{n}\delta(x-x_i) \forall x \in \Re[\tex], <br /> <br /> in which D_n = \left\{x_1, \cdots, x_n\right\}[\tex] independent realizations of a continuous random variable X[\tex], and \delta[\tex] is the dirac delta function, is infinite, irrespective to n[\tex] and D_n[\tex]. &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; My questions are straightforward: &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; 1) Why is that (my guess that the integral involved in the variance calculation does not converge)?; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; 2) What does it mean to have infinite variance in practical terms?; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; 3) Any practical examples where a distribution with infinite variance would be useful?; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Please note that I don&amp;amp;amp;amp;amp;#039;t have a strong background in statistics.&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Any help will be much appreciated.&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Cheers,&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Carlos

 

[bpet]

I've read somewhere that the variance of p(x) = {\frac{1}{n}}\sum_{i=1}^{n}\delta(x-x_i) \forall x \in \Re[\tex], <br /> <br /> in which D_n = \left\{x_1, \cdots, x_n\right\}[\tex] independent realizations of a continuous random variable X[\tex], and \delta[\tex] is the dirac delta function, is infinite, irrespective to n[\tex] and D_n[\tex]. &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt;

&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Assuming you mean to find the variance of &amp;amp;amp;amp;lt;i&amp;amp;amp;amp;gt;the random variable with density&amp;amp;amp;amp;lt;/i&amp;amp;amp;amp;gt; p(x), this would be infinite only if the variance of X is infinite (for example if X has the Cauchy or Pareto distribution).