Talk:Quantile

message: I think there might be a slight mistake in the equvivalent characterization of the p- and q-quantil. Should the second line not be
P(X>x)\leq p or somthing simlair?
:No, I do not believe that the definition should be P(X>x)\leq p, because it is my opinion that it is almost always done as P(X<x)\leq p.  I believe that the author added the second equivalent to account for the actual method used in estimating the quantiles, where some times you round up and some times you round down.  Along those lines, I believe that the two equivalent functions could just as easily be expressed as P(X\le x)\cong p\mbox{ and }P(X\ge x)\cong 1-p.  I hope I didn't go on and on as much as I did last time (⇩⇩⇩See below⇩⇩⇩!!!) sorry about that I don't know what I was thinking. NormDor 13:38, 22 June 2006 (UTC)
thread_title: Equivalent Characterisation

message: Two images from Yrithinnd's list of Commons images might be of interest here?
*:Image:Quantile graph.png
*:Image:Normalverteilung.png
--DO11.10 00:19, 5 May 2007 (UTC)
thread_title: Two images

message: I came across the word tertile, but found no article for it, so I made a stub. The word is infrequent in use, but I ask that someone with a better command of the statistical lingo than mine refine the article. --Shingra 08:29, 23 July 2007 (UTC)
replies: text: The tertile stub has since been rewritten and transwikified: tertile. --Shingra (talk) 11:24, 23 November 2007 (UTC)
thread_title:  Tertile 

message: I came across this
http://mathforum.org/library/drmath/view/60969.html
which i believe have a much simplier explanation.
Should we put it in the external links at least?  —Preceding unsigned comment added by 189.33.225.219 (talk) 05:41, 25 December 2007 (UTC)
thread_title:  Dr.Math 

message: I had added the "Quantiles of a sample" section, distinct from the "Quantiles of a population" section, because I have used the ideas of the former for several years, and had assumed that they were commonplace.  However, since the time of adding that section, I have not seen any published works that are similar to my approach.  Thus, I must conclude that my approach is actually original research — and I have removed it.  Quantling (talk) 20:39, 10 August 2009 (UTC)
thread_title:  Estimating the quantiles of a population 

message: I added a section / stub on approximate quantiles from a stream because these methods are becoming very popular. The section should be expanded a bit to summarize the method and explain the pro and cons, but at least, the issue is visible.  — Preceding unsigned comment added by Jdfekete (talk • contribs) 07:24, 20 June 2020 (UTC)
thread_title:  Approximate quantiles from a stream 

message: I have removed the following
* Monte Carlo simulations show that method R-5 is the preferred method for continuous data. ref name="schoonjans"/ref
as this is a letter in a medical journal and it looks as it the writers are expressing personal opinions having looked a particular case rather than widely held view among statisticians.--Rumping (talk) 22:59, 13 December 2011 (UTC)
thread_title: Preferred method 

message: I've added Quartiles to the list of Specialized quantiles. Is there a reason it wasn't there in the first place, while less common quantiles like duo-deciles are present? --Adam Matan (talk) 11:35, 8 September 2013 (UTC)
thread_title: Quartiles

message: Dear Leegrc,
The recent edit on estimating sample quantiles was undone by you. According to the Wikipedia rules, fully referenced new information should not be reverted, especially not when based on unreferenced arguments.
Your argument that the quantile at the kth value depends on "the context and purpose" suggest that the matter is somehow subjective. But this is mathematics. The quantiles are defined by the CDF, and the CDF is defined by the non-exceedance probabilities. The probability itself is not subjective, of course, but is defined by Kolmogorov's axioms. It follows that the quantile point values for the order ranks cannot depend on anything related to their use.  
On the contrary, they are known exactly, please read the reference http://dx.doi.org/10.1155/2014/326579
Best regards,
RTELAM
RATLAM (talk) 08:33, 12 September 2015 (UTC)
thread_title: Week of September 6, 2015

message: : Thank you for your note.  This is how I think about it.  Yes, I agree that when a population is fully described, its quantiles are fully characterized by the CDF, except for ties such as the median of an even number of elements, where there is some ambiguity.  However, when one has a sample drawn from an unknown population, one does not know the CDF of the population.  What one then chooses for the estimate of the quantile can depend upon the assumptions and purposes.  For example, if it is known that the sample is drawn from some normal distribution with known variance but unknown mean, that can be used to interpolate when  (from the big table in the article) is not an integer.  On the other hand, if it is known that the sample is drawn from a uniform distribution with unknown endpoints, that can be used to interpolate in a different way.  R and the other packages support multiple approaches, not simply in hopes that people will accidentally choose the wrong option, but because there are circumstances where each is applicable.  Even with so many options, the list is not and cannot be exhaustive.
:Your critique of these thoughts would be appreciated.  Thank you Leegrc (talk) 12:14, 14 September 2015 (UTC)
replies: text: The article you have been citing (Makkonen, L., Pajari, M. (2014) Defining sample quantiles by the true rank probability. Journal of Probability and Statistics, ID 326579. http://dx.doi.org/10.1155/2014/326579) has zero citations according to Google Scholar.  It fails the notability criterion of Wikipedia and I have removed those citations.  Leegrc (talk) 15:20, 14 September 2015 (UTC)
thread_title: Week of September 13, 2015

message: I disagree with the last statement (8). As discussed, the probability of not exceeding the -th smallest value in the sample of  is . Because CDF is defined by the non-exceedance probability, R-6 gives the best empirical estimate of the quantiles. “Efficiency” and “bias” are related to the expected value. Nothing can be said about them based on a single artificial sample.
RATLAM (talk) 09:15, 14 October 2015 (UTC)
replies: text: Let me try to recap, with an even simpler experiment.  We choose 3 values independently from the probability distribution that is uniform over some interval , in order to infer the point at which the CDF of this uniform distribution equals 75%, which occurs at  exactly.  We agree that the expected value of the largest of the three, , will be .  We agree that this is not the same as saying that 50% of the time  will be less than or equal to .  We agree that this is not the same as saying that the expected value  minimizes the quadratic loss formula.  We agree that these three objectives (, , and minimum square deviation around ) are different ways to home in on , which is the absolutely true point at which the CDF of the uniform distribution equals 75%.  Although you might characterize it as a distorted use of the words, we agree that it is common usage for people to characterize an approach based upon any of these three objectives as "quantile estimation from a sample".  Do I have that right?  Leegrc (talk) 13:05, 14 October 2015 (UTC)

text: The goal is to find  because 75% of future draws will not exceed that value.  However, almost surely any formula for making an estimate from three sampled values will not give exactly .  That is, any formula for making an estimate from three sampled values will almost surely fail to give us a 75% nonexceedance probability for future draws.  We could give up here, or we could relax our goal somewhat.  One option for getting near the true goal of achieving exactly 75% nonexceedance regardless of the sample of three values is the use of R-6, where we achieve the goal that on average the estimator gives 75% nonexceedance.  That is, R-6 is almost surely failing to deliver 75% nonexceedance for any particular sample of three values, but it is guaranteeing 75% nonexceedance on average.  Who are we to say that R-6 is better at approximating the true every-sample goal than an approach that guarantees 75% nonexceedance on median, by which I mean that the median value of the estimator is exactly , the 75% nonexceedance value?  Leegrc (talk) 13:50, 16 October 2015 (UTC)
thread_title: Week of October 11, 2015

message: This long-lasting discussion between two parties is related to a more than century long disputation about plotting positions. So far, it seems to be closed without agreement. I wonder whether the following contribution could help in finding some kind of convergence. 
In methods R-4 … R-9, a probability estimate of the form = is associated to each observed, order-ranked value .  and  are constants specific to each method. The different values of  and  form the essential difference between the methods.  is an estimate of , the true probability of  not exceeding . We can concentrate on the properties of  pairs where  is , because all other  pairs predicted by the methods are obtained either by interpolation or extrapolation.
By a loose definition, a consistent estimator of a parameter is a sequence which converges to the correct value of the parameter when the number of samples “grows to infinity”. Leegrc states that all methods R-4 … R-9 are consistent estimators of . I disagree. It is true and nice to know that when  increases indefinitely, all such estimates   converge in probability to , but this does not mean that the methods are consistent estimators because  is the size of one sample, not the number of samples.
Instead of  I would like to use notation    where  = 4,…,9 refers to the index of  methods R-4 … R-9 and  to the index of the sample. In order statistics, a sample consists of a set of order-ranked values . Increasing the number of samples (=) means increasing the number of such sets, each of size , not increasing . Given a probability , the sequence provided by any of the methods above is   which does not converge with increasing . It follows that none of the methods is a consistent estimator of  as such. 
We can construct a consistent estimator  e.g. by calculating the mean of successive values, i.e.  =   which is a consistent estimator of parameter   when = . In the same way,  = can be regarded as an approximately consistent estimator of  when  is = . When we estimate two different parameters, the estimators are different and the existence of a consistent estimator does not tell which method is better.  
The previous discussion has dealt with the question: If   is close to , does it matter how we define “close to”? Is it e.g. OK to minimize the sum of absolute values of deviations  as in R-8 or should we minimize the sum of squared deviations as in R-6?  When the loss functions are minimized, the resulting  and  are different. The original question can now be reformulated: Does it matter whether we are close to  or close to  when  ≠ . The answer should be obvious but let’s continue. It follows that  = =. Any estimator  applied to  samples gives  =  which both converge to the same value, say . If  is the parameter to be estimated and  is a consistent estimator of , then  =  = . This is impossible because  ≠ . Consequently, either R-6 or R-8 is wrong or they both are. 
Since both LEEGRC and RATLAM approve R-6, there is no need to promote it here anymore. All other methods applying plotting positions other than  contradict R-6 in the same way as R-8. If we prefer estimates of  which are unbiased and consistent for  probability values, we choose R-6 from the Table. It is very difficult to find a situation in which a biased estimate, given by any other method and converging to a parameter other than  should be preferred. Therefore, the article could be improved by adding a warning and, in the long run, by introducing new methods, e.g. R-6 modified in such a way that the interpolation is carried out on probability paper, and by moving the methods other than R-6 to a historical review with explanation.
BERKOM 18:59, 18 December 2015 (UTC)
thread_title:  Week of December 15, 2015 

message: :BERKOM (talk • contribs) writes:
::"It is very difficult to find a situation in which a biased estimate, given by any other method and converging to a parameter other than  should be preferred. Therefore …"
:Fortunately, all listed methods R-4 through R-9, not just R-6, converge to the same value as  goes to infinity; they are all consistent.  Whether one cares about bias on the -axis for specific values of  vs. minimizing absolute deviation, etc. is another matter, over which people are free to disagree, hence the plethora of equally important options.  No "warning" or markings as "historical" would be appropriate.  𝕃eegrc (talk) 14:19, 21 July 2016 (UTC)
thread_title:  Week of July 17, 2016 

message: It is true that when  increases indefinitely, all methods R-4 ... R-9 converge to the same value. More generally, when  is an arbitrary positive real number, the probability  associated to each observed order-ranked value  also produces an estimator which converges to the same value when  increases indefinitely. It is obvious that such a convergence  is not enough to justify any of the methods in any relevant case, i.e. when  is definite.
BERKOM 14:16, 22 November 2016 (UTC)
replies: text: Agreed that consistency is only one of several criteria one could desire for an estimator.  Agreed that since all of R-4, …, R-9 are consistent, consistency does not help us choose any of R-4, …, R-9 as better than the rest.  If the cumulative probability distribution function is continuous then R-6 will be best in the situation where one first samples  random real values, then randomly selects  of them, then asks what is the probability that the unselected value will fall into any of the  intervals defined by the  selected values, and then seeks to provide quantile estimates that give those interval probabilities in an unbiased fashion.  If that is not your situation or your optimization criterion then you may prefer, e.g., R-8.  𝕃eegrc (talk) 20:04, 22 November 2016 (UTC)
thread_title:  Week of November 20, 2016 

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thread_title:  External links modified 

message: The central message in the "Notes" column was obscured by extra sentences
on handling values outside the range of the sample data (for example
"When p < 1 / (N+1), use x1. When p ≥ N / (N + 1), use xN.")
Besides being a distraction, there was also a question of accuracy.  There is
no general agreement on how to extrapolate outside the range of data
elements — different packages have made different choices.  For example, MS Excel's
PERCENTILE.EXC refuses to guess and returns @NA.  Linear extrapolation
from the two nearest end points is also reasonable given that linear
interpolation is being used for all the other segments.  The article previously
described flattening the curve and assuming the underlying distribution
is constant at the end-points (contrary to Hyndman & Fan's property
P5 which posits "that for a continuous distribution, we expect there to
be a positive probability for values beyond the range of the data").
To table make table clearer and to improve accuracy, those notes were moved to a bullet-point comment underneath the table.  Additionally, the notes no longer take a position on how or whether to extrapolate beyond the range of the sample data.
Rdhettinger (talk) 17:59, 31 May 2019 (UTC)
thread_title: Simplify table of interpolation methods by moving the extrapolation notes below the table 

message: I just made a preliminary edit where I made it clear that the word percentile (and probably also quantile) has two distinct meanings. It's either the scalar as this article talks about, or one of the intervals that the scalar values limit. In my commit comment I wrote:
"The word percentile (and, I guess, any quantile) can also mean the interval that the scalar percentiles (or, more generally, the scalar quantiles) limit. They're just different meanings used in different contexts. I guess this should really be stated more prominently, as the interval meaning is in fact being used in scientific papers, and the existence of two separate but equally valid meanings depending on context seems to be a source of confusion."
I mention it here as well, as I believe this is in fact an important distinction to make the readers aware of, to avoid confusion. Words can have different meanings depending on context, and that is fine, as long as it's clearly defined and clearly understood.
I do believe, as mentioned, that we should state this distinction more prominently than under the "discussion", as it is quite relevant. I just don't know how. Maybe someone else has an idea of how to do it? It should of course also be stated more precisely. My edit was just meant to at least not write that it is wrong to use the interval meaning of percentile or quantile. It is in fact correct in the proper context and with a universally agreed upon meaning.
--Jhertel (talk) 12:05, 17 August 2020 (UTC)
thread_title:  Two distinct meanings of quantile: scalar and interval 

message: The current definition reads that "q-quantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes". Which does not sound that sane. Take, for example, the normal distribution. Not only is its value set infinite, it is not even bounded. And even if we were to take some bounded interval from it, that interval would still be an infinite set, because it is continuous. Should the word "finite" be removed from this definition? 188.242.96.147 (talk) 13:57, 8 October 2023 (UTC)
thread_title:  "Finite" set of values