5
$\begingroup$

I'm encountering the following PDF of continuous scalar real $X$ with semi-infinite support $]0,+\infty[$:

$$ f_X(x) = C ~ x^{-\alpha} ~_1F_1\left ( a,b;-\frac{d}{x^\beta} \right ),~~~~~~\beta>0;~\alpha>1;~a,b>2;~d>0 $$

where $_1F_1(\cdot,\cdot;\cdot)$ is a confluent hypergeometric function (sometimes denoted $\Phi(\cdot,\cdot;\cdot)$) and $C$ is the usual normalization constant.

Is there a particular name for this (family of) distribution(s)? Or has it been encountered or analyzed somewhere before, perhaps without a name attached?

$\endgroup$

closed as too broad by whuber Jul 16 at 22:37

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Never seen that; where did you encounter it? I like your choosing (?) to write an open interval via the symmetric square bracket; I myself also an advocate for that notation, learned from Bourbaki I guess. $\endgroup$ – Megadeth Jul 15 at 14:55
  • $\begingroup$ Maybe 'encounter' was not very precise. I am analyzing/solving a problem (in applied reliability) and got to this result. For some special cases of values for $a$ and $b$, it reduces to known distributions and simple PDFs of chi- and gamma type, so I have some confidence that this PDF is correct, or at least meaningful. BTW, the brackets are a legacy from my old (pre-18) school days :-) $\endgroup$ – Lucozade Jul 15 at 15:20
  • $\begingroup$ You are going to have problems finding a name, because most of these aren't valid density functions: they attain negative values. As $a$ increases, these functions have more and more simple zeros on the positive $x$ axis, alternating between positive and negative values between those zeros. $\endgroup$ – whuber Jul 16 at 13:47
  • 1
    $\begingroup$ $ C ~ x^{\nu-1} ~_1F_1\left (\alpha,\beta;-x \right )$ is the density of the confluent hypergeometric function kind one distribution (which is a ratio of a Gamma distribution over an independent Beta distribution, I think). It seems to me that your distribution is a transformation of this distribution. $\endgroup$ – Stéphane Laurent Jul 16 at 14:21
  • $\begingroup$ @whuber: thanks for raising this point. The listed coefficients are not arbitrary but are themselves expressions that involve other parameters, with further restrictions, so there is no issue regarding positivity. I did not want to clutter my query with nonpertinent details, instead focus on functions involved rather than parameter value ranges. $\endgroup$ – Lucozade Jul 16 at 20:50