[WP34s] A set of new quantile functions
05-25-2014, 09:40 PM (This post was last modified: 05-28-2014 01:23 PM by Dieter.)
Post: #1
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
[WP34s] A set of new quantile functions
Some 34s quantile functions for the statistical distributions do not seem to work as fast and reliably as they could. I recently suggested an update for Student's t quantile, based on a new estimate and the Halley method for fast convergence.

I now would like to propose a similar solution for the Chi² quantile. Here things are a bit more complex: a regular Halley iteration can be set up easily, as shown in another earlier thread, but there are issues with large degrees of freedom. Although the first Chi² estimate is only approx. 11% low in the worst case, this can mean that the CDF of this can be off by several orders of magnitude. This leads to very slow convergence in the first 2...10 iterations which essentially apply the same correction until the estimate is sufficiently accurate and rapid convergence starts.

So I chose a different approach: instead of cdf(x) – p, better solve ln(cdf(x)/p) = 0. This requires a bit of calculus, but it is not too hard to do.

The Student case was already discussed earlier, so the code below is just a slight update. Please note that the incomplete Beta function recently was updated and now requires a new order of its three arguments in X, Y and Z. The code below assumes the previous order.

Compared to the current version included in the 34s, the Normal quantile was streamlined and shortened. It also takes advantage of the new Normal estimate.

The result below is part of a set of programs for the Normal, Student and Chi² quantile. They all rely on a short routine that provides an estimate for the Normal quantile. Compared to the routine previously used in the 34s, only some constants were changed to provide better accuracy. More important, "GNQ" now accepts any probability between 0 and 1 (exclusively) and it returns a correctly signed result. As an additional benefit, min(p, 1–p) is returned in Y which can be handy here and there.

Finally, the following set of programs is what I got until now. This code is to be run in double precision mode, four stack levels are sufficient. As usual, the number of degrees of freedom is assumed in register J. Otherwise the programs only use R00 and R01 and flags 00 and 01. I could not find any severe errors, but please do your own thorough tests and do not consider this perfect. As usual, any error reports and suggestions are welcome.

Code:
 *LBL'NQF' // Normal quantile function XEQ'GNQ' CF 00 x<0? SF 00     // save sign in flag 00 x[<->] Y STO 00    // store min(p, 1-p) # 002 STO 01    // not more than 2 iterations are required RCL Z ABS FILL INC X RSD 03 x[!=]1? GTO 000 DEC 01    // estimate < 0.005 requires just one iteration DROP SDL 016 x>1? GTO 000 DROP      // estimate < 5E-17 is already exact GTO 004 *LBL 000 DROP *LBL 001 FILL x<1?      // any threshold between 0.254 and 1.28 should be ok GTO 002 [PHI][sub-u](x) RCL- 00 GTO 003   // large x: evaluate upper cdf(x) - p *LBL 002 x[^2]     // small x: evaluate (0.5-p) minus integral from 0 to x # 1/2     // by means of the incomplete Gamma function [times] # 1/2 I[GAMMA][sub-p] RCL[times] L +/- # 1/2 RCL- 00 + *LBL 003 x[<->] Y [phi](x) / ENTER[^]  // use extrapolation scheme as shown in Abramowitz & Stegun p. 954 x[^3] RCL T x[^2] STO+ X INC X [times] # 006 / # 1/2 RCL[times] T RCL[times] Z RCL[times] Z + + + DSE 01 GTO 001 *LBL 004 FS?C 00   // adjust sign +/-       // and exit END *LBL'TQF' // Student's t quantile function ENTER[^] +/- INC X MIN CF 00 x[!=]? L SF 00     // save sign in flag 00 STO 00    // store min(p, 1-p) # 006 STO 01    // do at most 6 iterations (usually 3 or 4) # 012 RCL J +/- y[^x] RCL 00 x>? Y    // threshold for tail estimate is p = 12^(-dof) GTO 000 RCL J    // tail estimate for p close to 0 (or 1) STO+ X [times] # [pi] RCL L # 1/2 x[^2] DEC X + / [sqrt] [times] RCL J [^x][sqrt]y RCL J [sqrt] x[<->] Y / GTO 001    // start iteration *LBL 000 XEQ'GNQ'   // estimate for all other p, *not* close to 0 or 1 x[^2]      // t is a tranformation of the Normal z-quantile # eE RCL[times] J 1/x INC X [times] RCL/ J e[^x]-1 RCL[times] J [sqrt] *LBL 001 FILL # 1/2     // use different methods for evaluating cdf(t) - p x>? Y     // for t close or not close to the center GTO 002 DROP t[sub-u](x) RCL- 00 GTO 003 *LBL 002 DROP x[^2]     // for t close to the center ENTER[^]  // evaluate cdf(t) - p by means of the incomplete Beta function RCL+ J / # 1/2 RCL[times] J RCL L I[beta]   // IMPORTANT: newer 34s firmware requires different order of arguments! See below. RCL[times] L +/- # 1/2 RCL- 00 + *LBL 003 RCL T     // use Halley method for fast convergence t[sub-p](x) / ENTER[^] RCL[times] T RCL J INC X [times] RCL T x[^2] RCL+ J STO+ X / DEC X / - CNVG? 00  // if the last two approximations agree in 14 digits SKIP 003  // the latest result can be considered converged DSE 01 GTO 001 ERR 20    // quit with error if no convergence within 6 iterations FS?C 00 +/-       // else set sign and exit END *LBL'CQF' // Chi^2 quantile function x=0? RTN       // catch case p = 0 STO 00 # 006 STO 01    // do at most 6 iterations (usually 3...4) # 019 SDR 001   // threshold for lower tail estimate is 1/pi * 1.9^-n RCL J x=1? DEC X     // handle threshold for 1 dof separately (1/pi is fine) +/- y[^x] # [pi] / RCL 00 x<? Y GTO 000   // jump to lower tail estimate XEQ'GNQ'  // else get correctly signed Normal quantile # 222     // and do a Wilson-Hilferty transformation SDR 003 RCL/ J STO Z [sqrt] [times] INC X RCL- Y x[^3] RCL[times] J   // Wilson-Hilferty estimate # eE           // is adjusted for p close to 1 RCL[times] J   // i.e. for Chi^2 > e*dof + 8 # 008 + x[<->] Y x<? Y          // Chi^2 not large? GTO 001        // then start iteration # 1/2          // otherwise adjust estimate [times] LN # 1/2 RCL[times] J DEC X [times] +/- RCL 00 +/- LN1+x + # 1/2 RCL[times] J LN[GAMMA] + STO+ X +/- GTO 001 *LBL 000   // lower tail estimate RCL[times] J # 1/2 [times] LN # 1/2 RCL[times] J LN[GAMMA] + STO+ X RCL/ J e[^x] STO+ X *LBL 001   // iteration starts here FILL       // uses Halley method for f(x) = ln(cdf(chi^2) / p) x[>=]? J   // with different methods for low and high Chi^2 GTO 002 [chi][^2]  // low Chi^2 ENTER[^] RCL/ 00 LN GTO 003    // provide lower cdf in Y and ln(cdf/p) in X *LBL 002 # 001 ENTER[^] RCL- 00 RCL Z [chi][^2][sub-u] STO- Z    // Z := 1-Z, i.e. upper cdf => lower cdf - RCL/ 00   // provide lower cdf in Y and ln1+x((1-p) - upper_cdf) / p) in X LN1+x     // the latter is mathematically equivalent to ln(lower_cdf/p) *LBL 003 x[<->] Y  // start Halley approximation RCL Z [chi][^2][sub-p] x[<->] Y / STO Z / RCL J DEC X DEC X RCL- T RCL/ T [<->] ZXYT  // all this can be done with four stack levels ;-) STO+ X - # 004 / RCL[times] Y +/- INC X / - CNVG? 00  // if the last two approximations agree in 14 digits SKIP 003  // the latest result can be considered converged DSE 01 GTO 001 ERR 20    // quit with error if no convergence within 6 iterations END       // else exit **LBL'GNQ'  // Provides a guess for the Normal quantile ENTER[^]    // input: 0 < p < 1 +/-         // output: X = signed quantile, Y = min(p, 1-p) INC X       // max. absolute and relative error < 0.006 MIN CF 01 x[!=]? L SF 01       // save sign in flag 01 # 002 SDR 001     // threshold is 0.2  (was 0.23 before) x[<->] Y x[>=]? Y    // use different estimates for p < or > 0.2 GTO 000 FILL        // tail estimate for p < 0.2 or > 0.8 LN STO+ X +/- ENTER[^] DEC X # [pi] [times] STO+ X [sqrt] RCL[times] T LN STO+ X +/- [sqrt] x[<->] Y # 132 STO+ X SDR 003     // 0.264  (was 0.254 in earlier versions) x[<->] Y / + GTO 001 *LBL 000 ENTER[^]    // central estimate for 0.2 < p < 0.8 +/- # 1/2 + # [pi] STO+ X [sqrt] [times] ENTER[^] x[^3] # 005       // was 6 in earlier versions / + *LBL 001 FS?C 01   // adjust sign +/-       // negative for p<0.5, positive for p>0.5 END

Edit: there was an update in the incomplete Beta function. If your 34s displays this function as "Iβ" you are running the older version and the original code above works fine. If the function is named Ix instead, this is the newer version with a different order of the three arguments. In this case use the following code between LBL 02 and 03 in "TQF":

Code:
... *LBL 002 RCL[times] J  // X was 1/2, so this is n/2 x[<->] Y x[^2] ENTER[^] RCL+ J /             // x = t²/(t²+n) # 1/2 x[<->] Y      // stack now is  x  1/2  n/2  t I[sub-x] # 1/2 [times] +/- # 1/2 RCL- 00 + *LBL 003 ...

And finally: Pauli, what do you think ?-)

Dieter
05-26-2014, 07:40 AM (This post was last modified: 05-26-2014 07:42 AM by walter b.)
Post: #2
 walter b On Vacation Posts: 1,957 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
AFAICS, Pauli uploaded some modifications following your suggestions last night. Will be in next build of WP 34S and WP 31S.

d:-)
05-26-2014, 08:06 AM
Post: #3
 Paul Dale Senior Member Posts: 1,726 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
These new distribution functions aren't in the image. I'll get to them once things calm down a bit here.

The changes were to the L.R. command.

- Pauli
05-26-2014, 10:45 AM
Post: #4
 Paul Dale Senior Member Posts: 1,726 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
A new build will depend on Marcus. The L.R. changes are committed.

The distribution changes aren't going to happen immediately. I've more important things to worry about than calculators at present.

- Pauli
05-26-2014, 06:43 PM
Post: #5
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-26-2014 09:36 AM)fhub Wrote:  That's a bit strange in my opinion:
First you complain about the illogical order of arguments for IBeta a few weeks ago, then Pauli has changed that (and it was good so), and now you provide a new set of code for distribution functions with exactly the same old (illogical) order of arguments?

Yes. The essential point was not the order of the Iβ arguments, but the fact that the function worked different from what the manual said. This has been corrected. Iβ now works "as advertized", which indeed is more logical.

But there is another point. The beauty of the German language has gifted us with the term "Die normative Kraft des Faktischen". What do you think, how many 34s are currently using the latest firmware? Five percent? Ten? That's why I assumed the previous Iβ implementation. All others may add a simple stack shuffle command.

Quote:It would really be nice to have a (quite) bugfree WP34s version finally

That's why I humbly proposed these three routines. The previous versions sometime threw errors or required inacceptable execution times.

What about you? Your TVM solver has made it into the 34s and added some essential functionality. Great. But what about another project? While I worked on the quantile functions I discovered both the Iβ bug as well as the L.R. problem. Both have been corrected – and the 34s became a bit better. This is a community effort – "ask not what the 34s can do for you – ask what you can do for the 34s project". :-)

Quote:(and I don't care whether any high-level function is accurate to 37 or 'only' 36 digits internally ).

It's more like 30...34. There are limitations in the algorithm and the functions it uses as well as in the hardware. But I think 30 or 32 digits are good enough. If you want all 34 in DP mode, having these functions coded in C is the only way. ;-)

Dieter
05-26-2014, 06:52 PM
Post: #6
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-26-2014 10:45 AM)Paul Dale Wrote:  The distribution changes aren't going to happen immediately. I've more important things to worry about than calculators at present.

Take your time. And most important: the suggested code is just a humble proposal. I do not say it is perfect and I strongly suggest you (and the rest of the community) do some thorough tests before this is included. Maybe there is a better approach, a faster solver, a better χ² estimate than good old Wilson-Hilferty, or whatever. I only can say that up to now everything seems to work fine here on my hardware 34s – within the limiations of the respective CDFs and Iβ resp. IΓ. Maybe these could get updated as well...?-)

Dieter
05-28-2014, 01:29 PM
Post: #7
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-25-2014 09:40 PM)Dieter Wrote:  Finally, the following set of programs is what I got until now.

Addendum: I updated the original post and added a modified piece of code for use with the newest 34s firmware with the modified incomplete Beta function. I also noticed that this function now appears as Ix, which I think is a good idea.

Dieter
05-28-2014, 02:19 PM
Post: #8
 walter b On Vacation Posts: 1,957 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-28-2014 01:29 PM)Dieter Wrote:  I updated the original post and added a modified piece of code for use with the newest 34s firmware with the modified incomplete Beta function. I also noticed that this function now appears as Ix, which I think is a good idea.

Thanks.

d:-)
05-29-2014, 12:43 PM (This post was last modified: 05-29-2014 02:12 PM by Dieter.)
Post: #9
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-26-2014 10:45 AM)Paul Dale Wrote:  The distribution changes aren't going to happen immediately.

That's fine since another update may be required as well.

I did some further tests with the recently suggested Student quantile function and found that close to zero the CDF of the calculated quantile often significantly differed from the original probability. Some tests with 42-digit precision showed that the quantile was fine (33 or 34 correct digits), while there are some issues with the WP34s t(x) function.

Near t=0 the Student CDF should be roughly 0,5 + r·t where r is a factor between 1/pi and 1/sqrt(2·pi), i.e. somewhere between 0,3 and 0,4. So the double precision results for the Student CDF should look like this:

Code:
    t          CDF(t) -----------------------------------------------------------------    0,1         0,53...    0,01        0,503...    0,001       0,5003...    1E-5        0,50000 3...    1E-10       0,50000 00000 3...    1E-20       0,50000 00000 00000 00000 3...    1E-33       0,50000 00000 00000 00000 00000 00000 0003 or ...4 =< 1E-34       0,50000 00000 00000 00000 00000 00000 0000 -----------------------------------------------------------------

But actually there is a much larger t below which a plain 0,5 is returned:

Code:
    t          WP34s t(x) function for 10 dof -----------------------------------------------------------------    1E-16       0,50000 00000 00000 03891 08383 96603 1050    1E-17       0,50000 00000 00000 00000 00000 00000 0000 -----------------------------------------------------------------

For further investigations I used a little program that provides a 34-digit t-value of the same magnitude as entered (pi/3 * input):

Code:
LBL A  pi  x  3  / ENTER  x² ENTER RCL+J  /  1/2 ENTER RCLxJ x<> Z  Ix     // with recent firmware  1/2  x RCL+L  // = exact CDF for t close to zero x<> Y t(x)   // = CDF as given by the current 34s function  -     // return absolute error RTN 10 STO J  // set 10 degrees of freedom 0,1    [A]   1,1 E-33 0,01   [A]  -5,81 E-32 0,001  [A]  -4,085 E-31 1E-5   [A]  -4,2508 E-29 1E-10  [A]   2,8046 E-24 1E-15  [A]  -6,2697 E-19 1E-16  [A]   1,8365 E-18 5E-17  [A]   2,0374 E-17 1E-17  [A]   4,0747 E-18 1E-18  [A]   4,0747 E-19 1E-19  [A]   4,0747 E-20 ... 1E-29  [A]   4,0747 E-30 1E-30  [A]   4,075 E-31 1E-31  [A]   4,07 E-32 1E-32  [A]   4,1 E-33 1E-33  [A]   4 E-34 1E-34  [A]   0

Since the true result is approx. 0,5 the returned error can be read as "units in the xth digit". So for t near 5 E–17 the error is 2 units in the 17th digit. In other words, accuracy degrades down to single precision level. I assume this happens due to digit cancellation when the argument x = n/(t²+n) for the incomplete Beta function is evaluated, since for very small t this rounds to 1. That's why I suggest that for small t the Student CDF should be evaluated using 1–x = t²/(t²+n), as shown in the routine above.

BTW the recently posted Student quantile function is not affected by all this as it already uses the suggested method for |t| < 0,5.

EDIT:

I looked at the current Student CDF code in t.wp34s, and indeed it works as assumed. Here is a fix that uses a different method for small t less than 1:

Code:
                        ... cdf_t_return::          x<1?                         JMP cdf_t_small                         RCL J                         SWAP                          x[^2]                         RCL+ J                         /                         Num 1/2                         RCL[times] J                         Num 1/2                         x[<->] Z                         I[sub-x]                         Num 1/2                         [times]                         RTN cdf_t_small::           x[^2]                         ENTER[^]                         RCL+ J                         /                         Num 1/2                         ENTER[^]                         RCL[times] J                         x[<->] Z                         I[sub-x]                         Num 1/2                         [times]                         +/-                         RCL+ L                         RTN cdf_t_invert::          GSB cdf_t_return                         +/-                           INC X                         RTN

Dieter
05-30-2014, 06:13 AM
Post: #10
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-25-2014 09:40 PM)Dieter Wrote:  I now would like to propose a similar solution for the Chi² quantile. Here things are a bit more complex: a regular Halley iteration can be set up easily, as shown in another earlier thread, but there are issues with large degrees of freedom. Although the first Chi² estimate is only approx. 11% low in the worst case, this can mean that the CDF of this can be off by several orders of magnitude. This leads to very slow convergence in the first 2...10 iterations which essentially apply the same correction until the estimate is sufficiently accurate and rapid convergence starts.

Recent tests showed that a similar behaviour may also occur with the Student quantile function. Up to now I only noticed this in extreme cases out of the SP working range, and here six or seven iterations seem to be sufficient. So the number of allowed iterations initially stored in R01 should be set to 8 for the moment. I will take a closer look at this and see if an approach similar to the Chi² case will eliminate this.

Dieter
06-01-2014, 01:18 PM
Post: #11
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(05-30-2014 06:13 AM)Dieter Wrote:  Recent tests showed that a similar behaviour may also occur with the Student quantile function.
(...)
I will take a closer look at this and see if an approach similar to the Chi² case will eliminate this.

It does. Here is an improved version of the Student quantile function that uses the same approach as the Chi² quantile: it solves ln(cdf/p) = 0. It does not require more than four stack levels. With eight levels the whole thing might get shortened a bit.

The 34s code is rather complex and hard to understand, so here it is in a more readable form:

Code:
 imax = 5 q = min(p, 1-p) t = guess_t(q, n) For i = 1 To imax          u = t / (n + t * t)     r = (n + 1) * u     pdf = tpdf(t, n)             If t >= 0.325 Then  ' here the cdf is < 0.4 for n = 1...\infty        cdf = tcdf(t, n) ' upper tail Student CDF        s = pdf / cdf        r = r - s        a = Ln(cdf / q) / s     Else        cdf = 0.5 * ((1 - 2*q) - betai(t*u, 1/2, n/2))        a = cdf / pdf     End If          dt = a / (a * r / 2 - 1)     t = t - dt Next return t * sign(p-0.5)

The 34s code is the same as before, only the iteration loop at LBL 01 is replaced with this:

Code:
... *LBL 001 FILL # 1/2 x>? Y           // 0.325 would be perfect, but 0.5 or even 1 will do either GTO 002 SIGN RCL+ J          // n+1 [times] x[<->] Y x[^2] RCL+ J /               // r := f"/f' = (n+1)t²/(n+t²) x[<->] Y t[sub-u](x)     // cdf = upper tail CDF RCL L t[sub-p](x) RCL/ Y          // pdf/cdf   cdf  r  t x[<->] Y RCL/ 00 LN              // ln(cdf/p)  pdf/cdf  r  t x[<->] Y /               // -f/f' = ln(cdf/p) * cdf/pdf x[<->] Y        // r  ln(cdf/p)*cdf/pdf  t  t RCL- L          // r - pdf/cdf # 1/2 [times]         // (r - pdf/cdf)/2 = f"/f'/2 RCL[times] Y GTO 003 *LBL 002 RCL[times] J    // X was 1/2, so this is n/2 x[<->] Y x[^2] ENTER[^] RCL+ J /               // x = t²/(t²+n) # 1/2 x[<->] Y        // stack now is  x  1/2  n/2  t I[sub-x] +/- RCL 00 STO+ X +/- INC X + # 1/2 [times]       // = f(x) = upper tail cdf(x) - p x[<->] Y t[sub-p](x) /             // = -f(x)/f'(x) ENTER RCL[times] T  // evaluate -f/f' * f"/f'/2 RCL J         // = -f/f' * (n+1)t/2(t²+n) INC X [times] RCL T x[^2] RCL+ J STO+ X / *LBL 003 DEC X / - CNVG? 00 SKIP 003 DSE 01 GTO 001 ERR 20 FS?C 00 +/- RTN

As before, 5 iterations are initially stored in R01. If the new code is too long and/or uses too much memory, the iteration limit in R01 may be removed. More than 5 iterations are only required in a limited domain for very small p and high dof. Up to now I have not found any cases within the SP working range. So if some more iterations are OK in these cases, the original code may be used as well. The worst I've seen so far are 12 loops (at 5000 dof, p=10^-5400). However, the new code does it in three. ;-)

Dieter
06-02-2014, 08:32 PM
Post: #12
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
It's me again. Here is the third reply to my own post. #-)

(06-01-2014 01:18 PM)Dieter Wrote:
(05-30-2014 06:13 AM)Dieter Wrote:  Recent tests showed that a similar behaviour may also occur with the Student quantile function.
(...)
I will take a closer look at this and see if an approach similar to the Chi² case will eliminate this.

It does. Here is an improved version of the Student quantile function that uses the same approach as the Chi² quantile: it solves ln(cdf/p) = 0.
...
If the new code is too long and/or uses too much memory, the iteration limit in R01 may be removed.

Final update: there is another solution. The original (shorter) TQF version can be used with a slight modification.

The program uses two different estimates. One for the central region and another for p close to 0 or 1. At least one of these seems to be good enough to ensure reasonably fast convergence. But in some cases, especially for very high dof values, choosing the one or the other is quite tricky since there is just a narrow corridor of probabilities where both approximations work equally well. So the crucial point is the decision whether to use the central or the tail estimate.

Up to now this was done by checking whether the input is greater or less than 12–n. This mostly works well, but it is not sufficiently exact for a few critical cases. After some tests I can suggest a different threshold that seems to work fine: simply replace the constant 12 with the term sqrt(n) + 7,5.

So these lines near the beginning of TQF...

Code:
... # 012 RCL J +/- y[^x] ...

...are replaced with these:

Code:
... RCL J [sqrt] # 075 SDR 001 + RCL J +/- y[^x] ...

And that's it.

So there are essentially two ways of solving the problem of slow convergence in some extreme cases:

1. The safe method: use a robust algorithm. This is the approach of the second version I posted, i.e. solving ln(cdf/p) = 0. However, this requires more code and memory.

2. Since the initial estimates are good enough, the threshold for switching between the two may be fine-tuned. With the suggested fix the original shorter version of TQF should work fine. I could not find a case where more than four iterations were required.

Dieter
06-02-2014, 10:06 PM
Post: #13
 Paul Dale Senior Member Posts: 1,726 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(06-02-2014 08:32 PM)Dieter Wrote:  It's me again. Here is the third reply to my own post. #-)

I'm reading these, just not acting at present.

- Pauli
06-03-2014, 08:35 PM (This post was last modified: 06-03-2014 08:35 PM by Dieter.)
Post: #14
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: [WP34s] A set of new quantile functions
(06-02-2014 10:06 PM)Paul Dale Wrote:  I'm reading these, just not acting at present.

Thank you for your feedback. Today I found an even better way of defining the threshold that works far more accurately than required. Should this be used anyway? Yes, I think so. Why? Because we can. ;-)

So if the updated Student quantile should eventually make it into the 34s, please let me know. But again: take your time.

Dieter
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