Parameters are by definition the characteristics of a theoretical distribution. If you can't assume a theoretical distribution, it means that you have no parameter to estimate. It does not prevent you from computing sample statistics; however, you cannot pretend that these sample statistics are estimators of parameters from a putative theoretical distribution.

I agree entirely with the argument, that the estimated functional should be determined by the probability distribution. For regularly varying pdf  $F$ , say at infinity, we assume existence of the limit   $\lim \frac{1 - F(ax)}{1 - F(x)} $  as  $x \to  \infty$,  which is then necessarily of the form   $ a^{\alpha}$, Then  $ \alpha$ would be  the parameter of my concern. The mentioned limit theorems for extreme values allow us to build an estimator of $\alpha$. And this is an opposite meaning if compared to the cited verse.

In Denis' terms, I just wanted to stress that every theoretical pd is putative:) In particular existence of an estimated  parameter of the distribution is assumed. And there is no proof of existence of the parameter  based on finite samples (obviously, unless the size of the sample and of the population are equal).

Unfortunately, no one limit theorem can provide complete answer, becouse we cannot pass to limit basing on FINITE SAMPLE from any distribution. Rarely it is possible to find the sample size, starting from which it is possible to guarantee, we are close enough to limit. Our goal is to give definitions of corresponding notion and analogues of theorems which allows us to avoide such difficulties.

I do not agree  with  Domsta  that  "we  can  never  estimate  the  expectation  and  any other functional on the probabilities".  Of course, we  cannot  estimate  mean of distribution. However,  we  can estimate CDF by i ts empirical counterpart. The situation is easy: empirical cdf is a sum of BOUNDED (by 1) random variables, and we can estimate variance by a constant. To have precision of estimator i a finite sample we can apply Chebyshev inequality (for example). We have similar situation while estimating characteristic function of any distribution. It is, again, bounded by 1. So, what is possible to estimate? Any continuous (in weak distance) functional of a distribution (more precisely, uniformly continuous functional). However, it is (or seems to be) impossible for non continuous (in weak metric) functional. Our goal here is  an attempt to change some popular non-continuous functionals by others, which are continuous and "close" (in intuitive sense) to discontinuous initial versions.

@ Lev Klebanov

Dear sir, The discussion enters a dangerous field of "one-said-this and the-other-this and I-have-meant-this.." which can be end-less. The best way is just to summarize, possibly without less important remark. Now is my turn, so let me stress that I have opposed and still oppose just one claim from the project (on impossibility of estimating the degree of heavy tails). I am still accepting the meaning, that the tails can be estimated via standard statistical algorithms (e.g. with the use of limit theorems for extreme values). The sentence that it is impossible to estimate the mean value is given as an example of a  a possible consequence of  accepting the impossibility to estimate the parameters $\alpha$ and $\lambda$. In brief, the message was intented to mean isthe following:

If it is impossible to estimate the parameters of tails, then it is also impossible to estimate the mean of any distribution; and let me add: even if the random variable is bounded.

Of cource, you are fully right that we have tools like Chebshev inequality etc. BUT, even for 0-1 valued rv,  the estimating of the mean (ie. of the probability of 1) from iid sequences is not possible in other way  than to produce sentences about likelyhood of a given interval. And probably there is no-one who can disagree that the likelyhood is equal  1  in only one case, when the interval equals [0;1].  In view of the above, my retoric question is whether the same is true when estimating the parameters of heavy tails. It is retoric for me, since in my opinion the same impossibility is related to this case as for the 0-1 valued probabilities.  Summarizing, let me state my opinion:

for the parameters of tails we can also build confidence intervals as well as for the mean value of a 0-1 valued random variable.

Best regards,

Joachim Domsta

Our discussion seems to be a liitle bit strange. I think, we are talking about different things. Why? We can understand the phrases "It is possible to estimate", or "It is possible to calculate" in different ways. One way is, say, "to calculate precisely", or "to estimate with probability 1". If sense is such, so, I agree, it is impossible to estimate mean even if the random variable is bounded". However, I am talking about different sense. We can estimate mean (which is a value of a functional of distribution, which is continuous in weak topology on a set of random variables, bounded by a fixed constant) not precisely, but by giving boundaries for it. The boundaries provide us with an approximation for mean, wich is true with some guaranteed probability. This probability can be close to one if the number of observations (summands) is large enough. What does it mean "large enough"? It means, that for any \epsilon>0 there exists N such that for n>N (and so on,.. Of course, everybody knows thiis!). But is it possible to find this N for all random variables? If we have a class of random variables bounded by one known constant, then it is possible to find this N independently of the choice of random variable from our class. Do we have such situation for the class of all random variables with finite (but unknown) mean value? Obviously, no! This N will depend on each random variable in the class. The supremum of such N's over this class will be infinite. Therefore, we have no idea do we have enogh observations or not. In this sense, the problem of mean estimation has no solution in the class of all random variables with  finite mean. The same situation is true for tail parameters. However, the situation in this last problem is just worse. Here we need to calculate the parameters of the tail of individual tail, and then to find some conditions of "uniformity" over sample size. The parameters of individual distribution are definite by limit process. So, we have to find our N (the position for good tail approximation ) for individual distribution. Generally (in essentially non-parametric situation) we have no good boundaries for this N. How can we propose the conditions for supremum of all such N's to be finite? Yes, in some simple situations, it is possible. But the conditions appears to be expressed in very strong boundaries for the tail. So, we have to change something here. What is the reason for such situation? Just the fact, that corresponding functional (parameter of the tail) in not weakly continuous, and the set of corresponding distributions is not weakly compact. What would I do? I would like to change the difinition of tail parameters, to have weakly continuos functional. Do I have any ideas in this direction? I hope, yes. The problem seems to be similar to that which had been considered in the paper Klebanov L.B., Rachev S.T., Szekely G.J. Pre-limit Theorems and Their Applications, Acta Applicandae Mathematicae, v.58, 1999, 159-174.  You may find it on this site. However, the method of the paper do not provide us with complete solution. So, my coauthors and I are still thinking about suitable solution.

I hope, I will clear enough. If no, so, I am ready for further discussions.

Many thanks for your comments and this discussion!

@ Lev Klebanov

Dear sir, thank you vey much for the full explanation, which is consistent in main points with my point of view, and really, in some places  I have been talking about a little bit different  objects. The only difference now is  that I am still not sure that the situation for tail parameters is substantially different if compared, say, to the estimation of the mean value. Therefore, exploiting your readines for further discussion, in order to expalin my point of view in more detail, I am preparing a proposal of an algorithm of statistical estimation of the tail parameters. Possibly, again, our understandings meat at common places again.

Also, thank you for the discussion and for the references in your last and previous posts.

With best regards, Joachim Domsta

PS. Possibly I will be ready for further answers not earlier than in circa 10 days; But I will try be sooner back, anyway. JD

@Joachim Domsta 

Thank you so much! I am eager to see your project. Best wishes, Lev,

@ Lev Klebanov

The ten days have extended themselves  twice in duration:)  Sorry for this delay, caused by difficulties in formulation of my algorithm. Finaly it obtained an acceptable  form of a 5 pages long article (attached), which definitely can be smoother and shorter. But this could extend the waiting time by "another circa 10 days". Thus I have chosen to release it already now in a  rough form. Thus let me be forgiven some (certainly existing) errors. Hopefully, all these are "removable".

Let me state in this brief letter, that the presented algorithm of building a test for the  exponent of the tail of the p.d. confirms the expected and predicted by you features of such a test, simultaneously confirming the possibility of determining the value of the exponent in a rigorous ststistical test. The parallel to the estimation of the mean value is not exposed sufficiently, yet it can be seen from the article.

Best regards