Suppose we already know the pdf of the spacings between gamma order statistics, where can we find applications that use this pdf?
Thanks so much.
@ Mohamed I. Riffi
Dear sir, can you please explain what you call the pdf of the spacings between gamma order statistics. And what is the form of the knowledge under discussion. First impression is that there is no problem if the question concerns the joint distribution of the diferences between the consecutive order statistics. Indeed, e.g. for the uniform distribution of the observable on [0,1] the joint distribution of
(X_1-X_0=1, X_2 - X_1, .. . , X_{n-1} - X_{n-2}, X_{n} - X_{n-1}, 1=X_{n+1} - X_n)
equals the probability distribution of the sequence
(Z_j := Y_j/S_{n+1}: j= 1,2,..., n+1), where Y_j, j=1,2,. ., n+1, form iid with exponential pd, and S_{n+1} is the sum of all Y-s.
Now, if you know the pdf of the observable X , for simplicity, let it be F(x), x\ge 0, strictly increasing and continuous, F(0) = 0, then the differences between the consecutive order statistics modified by F, i.e. the sequence of differences of U_j := F(X_j) - F(X_{j-1} ) j=1,2,..., n+1 (suitably defined by F(X_0) = 0, F(X_{n+1}) = 1 ) is the same as above defined sequence of Z-s.
Thus, the problem may appear when someone needs further details, e.g. the distribution of the non-modified spacings X_j - X_{j-1}. This requires much more work, and probably is performable for particular cases, only. With respect to the above: is your probability distribution of spacings for gamma order statistics given explicitly?
Further answers on applications, obviously depend on what you keep in your dossier. Independently, it would be interesting, also, how your pd. is obtained.
Regards
@ Mohamed I. Riffi
I am really sorry, I didn't notice the reference to your article presented in the project. Thus, the above answer treet, please, as an instantenous response given to the question caused by its formulation "as it is". Since all my additional questions are explained in the paper, treat them please as non-existing (if possible:).
Sincerely, Joachim Domsta
@ Joachim Domst
Thanks so much for your answers.
To make my question clear, I could express the pdf of X_(i+m) - X_(i) as a finite mixture of Gamma distributions. Now, I am trying to use this in some areas of statistics like queuing theory, for example.
Best Regards'
Mohamed I. Riffi
Dear Mahamed I. Riffi,
I like your result. It is an application of the class of Takacs distributions, whose two-sided Laplace transforms are expressed by rational functions with real poles. It is interesting that the class is closed wrt to linear combinations and to ordering (like taking the maximum or minimum). This explaines why you could obtain the form of the distributions of spacing as a combination of gamma distributions. Note, please, that the combinations with negative coefficients are possible, as well as with all positive, provided they sum up to one.The exact formulas are interesting for itself. I agree, that further application of this result is possible within reliability of composed systems (the queuing systems included). Unfortunately my experience in this is not enough to suggest some more specified phenomena. Let me suggest, however: why shouldn't you look rather for a problem and then solve it with the use of the knowledge and experience you have got during the research? There are many systems with traveling objects, with many service-points, with service time not necessarily of the gamma distribution but of other Takacs type (like e.g. maximum of couple of exponential pd-s), which reach a reseacher only if someone needs the particular analysis. Thus better source would be to meet an "applied scientist" who has problem with such systems and try to solve the real question(s). Even your problem of spacing can be probably extended to Takacs class distribution functions. This seems to be an ansolved problem, too.
At the end, let me ask whether it is possible to obtain your result via the basic proposition of my first answer (i.e. by modifying the order statistics by F ).
Thanks in advance.
Sincerely yours,
Joachim Domsta
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