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?


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

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  • $\begingroup$ 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