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https://www.researchgate.net/publication/222105754_Sample_Quantiles_in_Statistical_Packages

We believe there is a similar need
to adopt a standard sample quantile definition, and we propose 
that $\hat{Q}_8(p)$ is the best choice.

The above paper says this. It is not clear based on what it is the best. Could anybody help explain?

Since this is an old paper. Is this recommendation still the best according to the current knowledge?

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  • $\begingroup$ The article does give some arguments: it says $\hat Q_8$ has the stated general desirable properties except P3 (similarly to some of the other estimators) as well as the specific property that it is approximately "median-unbiased". The other estimators also have specific properties and intuitions, and it is not really possible to conclude that just one is best. But its main argument is that it confusing for students to have multiple definitions and choosing one as the default would be a good idea. $\endgroup$
    – Henry
    Commented Jul 31, 2021 at 1:17
  • $\begingroup$ ... My problem with that is that if students are taught there is "one definition of quantile", many of them would forget $\hat Q_1$ (or perhaps $\hat Q_2$) is closer to the real meaning of quantile for a discrete distribution, and that what is being discussed here is for estimating the population quantile based on a finite sample from a continuous distribution. This already happens with the $\frac1{n-1}$ method for sample standard deviations $\endgroup$
    – Henry
    Commented Jul 31, 2021 at 1:23
  • $\begingroup$ @Henry. In teaching, the first part of my discussion about sample quantiles is to mention that the exact point within the interval given by the textbook definition differs from one software package and one textbook author to another. [Perhaps the worst part is that methods for textbook answers do not always agree with textbook examples, as grad students making answer keys may use a variety of computer programs.] $\endgroup$
    – BruceET
    Commented Jul 31, 2021 at 2:15

1 Answer 1

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R lists several types of slightly different definitions of sample quantiles. Each is claimed to be "better" for some specific applications. Textbook "definitions," sometimes giving intervals for "a quantile," rather than specific points for "the quantile," are often not sufficiently specific to distinguish among "types."

Quantiles are often used for very large samples, and for them the differences among types are often relatively small and can usually be ignored.

It is untidy that there is no consensus as to "the best" definition. Your linked article is not the first complaint about the untidiness, nor I suspect, the last.

Example: In R type=7 is default:

set.seed(1234); x = rnorm(200, 100, 15)  
quantile(x,.75); quantile(x,.75, type=8)
     75% 
108.2989   # default type 7 
     75% 
108.3642   # link's type 8

sort(x)[149:151]
[1] 107.7064 108.2500 108.4458

For comparison, quantile 0.75 of $\mathsf{Norm}(\mu=100,\sigma=15)$ is $110.1123.$

qnorm(.75, 100, 15)
[1] 110.1173
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