[math-fun] Inverse function of error function
If x=erf(y), then say y=erfinv(x). SMP jocks: What is the Maclaurin series of erfinv(x)? Obviously the erfinv function is useful for statistics. It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
(sqrt(pi) x)/2 + 1/24 pi^(3/2) x^3 + 7/960 pi^(5/2) x^5 + O(x^7) http://www.wolframalpha.com/input/?i=taylor+series+of+InverseErf On Tue, Jul 8, 2014 at 10:24 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If x=erf(y), then say y=erfinv(x).
SMP jocks: What is the Maclaurin series of erfinv(x)?
Obviously the erfinv function is useful for statistics. It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I doubt erfinv(z) is entire, since as x -> +-oo, erf'(x) -> 0. In fact erfinv'(x) -> oo as |x| -> 1/2. This means its power series about 0 can't have a radius of convergence > 1/2. --Dan P.S. This is related to a simple -- some would say obvious -- insight I hadn't thought of before today: If you want to pick a real number at random from a distribution given by a certain pdf equal to f(x), and you can invert its cdf F(x) = Integral_{-oo,x} f(t) dt to get Finv(x), then you just need to pick a number u at random from U[0,1]. Then Finv(u) is your random pick from the distribution in question. (Of course you can also just start with the cdf F(x).) On Jul 8, 2014, at 10:31 PM, Mike Stay <metaweta@gmail.com> wrote:
(sqrt(pi) x)/2 + 1/24 pi^(3/2) x^3 + 7/960 pi^(5/2) x^5 + O(x^7)
http://www.wolframalpha.com/input/?i=taylor+series+of+InverseErf
On Tue, Jul 8, 2014 at 10:24 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If x=erf(y), then say y=erfinv(x).
SMP jocks: What is the Maclaurin series of erfinv(x)?
Obviously the erfinv function is useful for statistics. It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Coincidentally, I have been wondering about just how closely the (scaled) Gaussian curve approaches the (discrete) binomial coefficients; this involves inverting the standard investigation in (say) Feller vol I, chap. VII (2.11) p.170 to inspect horizontal rather than vertical distance between the curves. With y = exp(-x^2/2) / sqrt(2\pi) the unit Gaussian function, easily h := 1/sqrt(n/4); y := binomial(n, k)/(2^n*h); x := sign(k - n/2)*sqrt(-2*log(y*sqrt(2*Pi))); k' := x/h + n/2; now experimentally it appears that |k - k'| < 1/2 , except near endpoints k < O(sqrt(n) etc. So what I want is some strict bound on |k - k'| --- maybe well-known? Incidentally, I have a note in my copy of Feller that his analysis may be incomplete, with a reference to Y. S. Chow & H. Teicher "Probability Theory" Springer (1978), sect 2.3 Lemma 2 . Can anybody comment? Fred Lunnon On 7/9/14, Mike Stay <metaweta@gmail.com> wrote:
(sqrt(pi) x)/2 + 1/24 pi^(3/2) x^3 + 7/960 pi^(5/2) x^5 + O(x^7)
http://www.wolframalpha.com/input/?i=taylor+series+of+InverseErf
On Tue, Jul 8, 2014 at 10:24 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If x=erf(y), then say y=erfinv(x).
SMP jocks: What is the Maclaurin series of erfinv(x)?
Obviously the erfinv function is useful for statistics. It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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On 7/9/14, Warren D Smith <warren.wds@gmail.com> wrote:
If x=erf(y), then say y=erfinv(x).
SMP jocks: What is the Maclaurin series of erfinv(x)?
Obviously the erfinv function is useful for statistics.
It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
--Sorry, unfortunately, that reasoning was not valid. The same reasoning would lead to the "conclusion" that the inverse function of exp(y), i.e. ln(x), exists everywhere and is "entire" -- but actually, ln is a multivalued function and has a singularity at 0. The flaw in my wrong reasoning was that something infinitely far away in the y plane, may not be in the x plane. So... attempting to correct myself, I think the Maclaurin series for erfinv(x) ought to have radius of convergence equal to lim(y-->infinity) |erf(y)| = 1. Less nice. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
participants (4)
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Dan Asimov -
Fred Lunnon -
Mike Stay -
Warren D Smith