[R] quantile from quantile table calculation without original data

[David Winsemius]

On 3/5/21 1:14 AM, PIKAL Petr wrote:
> Dear all
>
> I have table of quantiles, probably from lognormal distribution
>
>   dput(temp)
> temp <- structure(list(size = c(1.6, 0.9466, 0.8062, 0.6477, 0.5069,
> 0.3781, 0.3047, 0.2681, 0.1907), percent = c(0.01, 0.05, 0.1,
> 0.25, 0.5, 0.75, 0.9, 0.95, 0.99)), .Names = c("size", "percent"
> ), row.names = c(NA, -9L), class = "data.frame")
>
> and I need to calculate quantile for size 0.1
>
> plot(temp$size, temp$percent, pch=19, xlim=c(0,2))
> ss <- approxfun(temp$size, temp$percent)
> points((0:100)/50, ss((0:100)/50))
> abline(v=.1)
>
> If I had original data it would be quite easy with ecdf/quantile function but without it I am lost what function I could use for such task.

The quantiles are in reverse order so tryoing to match the data to 
quantiles from candidate parameters requires subtracting them from unity:

 > temp$size
[1] 1.6000 0.9466 0.8062 0.6477 0.5069 0.3781 0.3047 0.2681 0.1907
 > qlnorm(1-temp$percent, -.5)
[1] 6.21116124 3.14198142 2.18485959 1.19063854 0.60653066 0.30897659 
0.16837670 0.11708517 0.05922877
 > qlnorm(1-temp$percent, -.9)
[1] 4.16346589 2.10613313 1.46455518 0.79810888 0.40656966 0.20711321 
0.11286628 0.07848454 0.03970223
 > qlnorm(1-temp$percent, -2)
[1] 1.38589740 0.70107082 0.48750807 0.26566737 0.13533528 0.06894200 
0.03756992 0.02612523 0.01321572
 > qlnorm(1-temp$percent, -1.6)
[1] 2.06751597 1.04587476 0.72727658 0.39632914 0.20189652 0.10284937 
0.05604773 0.03897427 0.01971554
 > qlnorm(1-temp$percent, -1.6, .5)
[1] 0.64608380 0.45951983 0.38319004 0.28287360 0.20189652 0.14410042 
0.10637595 0.08870608 0.06309120
 > qlnorm(1-temp$percent, -1, .5)
[1] 1.1772414 0.8372997 0.6982178 0.5154293 0.3678794 0.2625681 
0.1938296 0.1616330 0.1149597
 > qlnorm(1-temp$percent, -1, .4)
[1] 0.9328967 0.7103066 0.6142340 0.4818106 0.3678794 0.2808889 
0.2203318 0.1905308 0.1450700
 > qlnorm(1-temp$percent, -0.5, .4)
[1] 1.5380866 1.1710976 1.0127006 0.7943715 0.6065307 0.4631076 
0.3632657 0.3141322 0.2391799
 > qlnorm(1-temp$percent, -0.55, .4)
[1] 1.4630732 1.1139825 0.9633106 0.7556295 0.5769498 0.4405216 
0.3455491 0.2988118 0.2275150
 > qlnorm(1-temp$percent, -0.55, .35)
[1] 1.3024170 1.0260318 0.9035201 0.7305712 0.5769498 0.4556313 
0.3684158 0.3244257 0.2555795
 > qlnorm(1-temp$percent, -0.55, .45)
[1] 1.6435467 1.2094723 1.0270578 0.7815473 0.5769498 0.4259129 
0.3241016 0.2752201 0.2025322
 > qlnorm(1-temp$percent, -0.53, .45)
[1] 1.6767486 1.2339052 1.0478057 0.7973356 0.5886050 0.4345169 
0.3306489 0.2807799 0.2066236
 > qlnorm(1-temp$percent, -0.57, .45)
[1] 1.6110023 1.1855231 1.0067207 0.7660716 0.5655254 0.4174793 
0.3176840 0.2697704 0.1985218

Seems like it might be an acceptable fit. modulo the underlying data 
gathering situation which really should be considered.

You can fiddle with that result. My statistical hat (not of PhD level 
certification) says that the middle quantiles in this sequence probably 
have the lowest sampling error for a lognormal, but I'm rather unsure 
about that. A counter-argument might be that since there is a hard lower 
bound of 0 for the 0-th quantile that you should be more worried about 
matching the 0.1907 value to the 0.01 order statistic, since 99% of the 
data is know to be above it. Seems like efforts at matching the 0.50 
quantile to 0.5069 for the logmean parameter and matching the 0.01 
quantile 0.1907 for estimation of  the variance estimate might be 
preferred to worrying too much about the 1.6 value which would be in the 
right tail (and far away from your region of extrapolation.)


Further trial and error:


 > qlnorm(1-temp$percent, -0.58, .47)
[1] 1.6709353 1.2129813 1.0225804 0.7687497 0.5598984 0.4077870 
0.3065638 0.2584427 0.1876112
 > qlnorm(1-temp$percent, -0.65, .47)
[1] 1.5579697 1.1309763 0.9534476 0.7167775 0.5220458 0.3802181 
0.2858382 0.2409704 0.1749275
 > qlnorm(1-temp$percent, -0.65, .5)
[1] 1.6705851 1.1881849 0.9908182 0.7314290 0.5220458 0.3726018 
0.2750573 0.2293682 0.1631355
 > qlnorm(1-temp$percent, -0.65, .4)
[1] 1.3238434 1.0079731 0.8716395 0.6837218 0.5220458 0.3986004 
0.3126657 0.2703761 0.2058641
 > qlnorm(1-temp$percent, -0.68, .4)
[1] 1.2847179 0.9781830 0.8458786 0.6635148 0.5066170 0.3868200 
0.3034251 0.2623852 0.1997799
 > qlnorm(1-temp$percent, -0.65, .39)
[1] 1.2934016 0.9915290 0.8605402 0.6791257 0.5220458 0.4012980 
0.3166985 0.2748601 0.2107093
 >
 > qlnorm(1-temp$percent, -0.65, .42)
[1] 1.3868932 1.0416839 0.8942693 0.6930076 0.5220458 0.3932595 
0.3047536 0.2616262 0.1965053
 > qlnorm(1-temp$percent, -0.68, .42)
[1] 1.3459043 1.0108975 0.8678396 0.6725261 0.5066170 0.3816369 
0.2957468 0.2538940 0.1906976


(I did make an effort at searching for quantile matching as a method for 
distribution fitting, but came up empty.)


-- 

David.

>
> Please, give me some hint where to look.
>
>
> Best regards
>
> Petr
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