Empirical distribution function

<h2 id="definition">Definition</h2>
Let (X1, …, Xn) be <a href="/facts/Iid/othIRaWt">independent, identically distributed</a> real random variables with the common <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> F(t). Then the empirical distribution function is defined as<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

F
                ^
              
            
          
          
            n
          
        
        (
        t
        )
        =
        
          
            
              
                
                  number of elements in the sample
                
              
              ≤
              t
            
            n
          
        
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          
            1
          
          
            
              X
              
                i
              
            
            ≤
            t
          
        
        ,
      
    
    {\displaystyle {\widehat {F}}_{n}(t)={\frac {{\mbox{number of elements in the sample}}\leq t}{n}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t},}

where 
 
 
 
 
 
 1
 
 
 A
 
 
 
 
 {\displaystyle \mathbf {1} _{A}}
 
 is the <a href="/facts/Indicator_function/QEuh04NM">indicator</a> of <a href="/facts/Event_(probability_theory)/aJ0l4oQb">event</a> A. For a fixed t, the indicator 
 
 
 
 
 
 1
 
 
 
 X
 
 i
 
 
 ≤
 t
 
 
 
 
 {\displaystyle \mathbf {1} _{X_{i}\leq t}}
 
 is a <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli random variable</a> with parameter p = F(t); hence 
 
 
 
 n
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 
 
 {\displaystyle n{\widehat {F}}_{n}(t)}
 
 is a <a href="/facts/Binomial_distribution/UMoFMjDj">binomial random variable</a> with <a href="/facts/Mean/swcEd4Pg">mean</a> nF(t) and <a href="/facts/Variance/ULBJKXD1">variance</a> nF(t)(1 − F(t)). This implies that 
 
 
 
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 
 
 {\displaystyle {\widehat {F}}_{n}(t)}
 
 is an <a href="/facts/Bias_of_an_estimator/oxIvEgmd">unbiased</a> estimator for F(t).
However, in some textbooks, the definition is given as

F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 =
 
 
 1
 
 n
 +
 1
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 n
 
 
 
 
 1
 
 
 
 X
 
 i
 
 
 ≤
 t
 
 
 
 
 {\displaystyle {\widehat {F}}_{n}(t)={\frac {1}{n+1}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t}}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
<h2 id="asymptotic-properties">Asymptotic properties</h2>
Since the ratio (n + 1)/n approaches 1 as n goes to infinity, the asymptotic properties of the two definitions that are given above are the same.
By the <a href="/facts/Strong_law_of_large_numbers/X3Bjcy3v">strong law of large numbers</a>, the estimator 
 
 
 
 
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 
 
 
 {\displaystyle \scriptstyle {\widehat {F}}_{n}(t)}
 
 converges to F(t) as n → ∞ <a href="/facts/Almost_sure_convergence/pWdatFlY">almost surely</a>, for every value of t:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

F
                ^
              
            
          
          
            n
          
        
        (
        t
        )
         
        
          
            →
            
              a.s.
            
          
        
         
        F
        (
        t
        )
        ;
      
    
    {\displaystyle {\widehat {F}}_{n}(t)\ {\xrightarrow {\text{a.s.}}}\ F(t);}

thus the estimator 
 
 
 
 
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 
 
 
 {\displaystyle \scriptstyle {\widehat {F}}_{n}(t)}
 
 is <a href="/facts/Consistent_estimator/HAKDWYEM">consistent</a>. This expression asserts the pointwise convergence of the empirical distribution function to the true cumulative distribution function. There is a stronger result, called the <a href="/facts/Glivenko%E2%80%93Cantelli_theorem/o5E7Tjda">Glivenko–Cantelli theorem</a>, which states that the convergence in fact happens uniformly over t:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

‖
        
          
            
              
                F
                ^
              
            
          
          
            n
          
        
        −
        F
        
          ‖
          
            ∞
          
        
        ≡
        
          sup
          
            t
            ∈
            
              R
            
          
        
        
          
            |
          
        
        
          
            
              
                F
                ^
              
            
          
          
            n
          
        
        (
        t
        )
        −
        F
        (
        t
        )
        
          
            |
          
        
         
        
          →
          
        
         
        0.
      
    
    {\displaystyle \|{\widehat {F}}_{n}-F\|_{\infty }\equiv \sup _{t\in \mathbb {R} }{\big |}{\widehat {F}}_{n}(t)-F(t){\big |}\ \xrightarrow {} \ 0.}

The sup-norm in this expression is called the <a href="/facts/Kolmogorov%E2%80%93Smirnov_test/Kr5MqpfH">Kolmogorov–Smirnov statistic</a> for testing the goodness-of-fit between the empirical distribution 
 
 
 
 
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 
 
 
 {\displaystyle \scriptstyle {\widehat {F}}_{n}(t)}
 
 and the assumed true cumulative distribution function F. Other <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm functions</a> may be reasonably used here instead of the sup-norm. For example, the <a href="/facts/Lp_norm/gbad0Rv1">L2-norm</a> gives rise to the <a href="/facts/Cram%C3%A9r%E2%80%93von_Mises_criterion/IdtN5xQV">Cramér–von Mises statistic</a>.
The asymptotic distribution can be further characterized in several different ways. First, the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a> states that pointwise, 
 
 
 
 
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 (
 t
 )
 
 
 
 {\displaystyle \scriptstyle {\widehat {F}}_{n}(t)}
 
 has asymptotically normal distribution with the standard 
 
 
 
 
 
 n
 
 
 
 
 {\displaystyle {\sqrt {n}}}
 
 rate of convergence:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

n
          
        
        
          
            (
          
        
        
          
            
              
                F
                ^
              
            
          
          
            n
          
        
        (
        t
        )
        −
        F
        (
        t
        )
        
          
            )
          
        
         
         
        
          
            →
            
              d
            
          
        
         
         
        
          
            N
          
        
        
          
            (
          
        
        0
        ,
        F
        (
        t
        )
        
          
            (
          
        
        1
        −
        F
        (
        t
        )
        
          
            )
          
        
        
          
            )
          
        
        .
      
    
    {\displaystyle {\sqrt {n}}{\big (}{\widehat {F}}_{n}(t)-F(t){\big )}\ \ {\xrightarrow {d}}\ \ {\mathcal {N}}{\Big (}0,F(t){\big (}1-F(t){\big )}{\Big )}.}

This result is extended by the <a href="/facts/Donsker%E2%80%99s_theorem/7PdQrWrk">Donsker’s theorem</a>, which asserts that the <a href="/facts/Empirical_process/HUQFdWLB">empirical process</a> 
 
 
 
 
 
 
 n
 
 
 (
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 −
 F
 )
 
 
 
 {\displaystyle \scriptstyle {\sqrt {n}}({\widehat {F}}_{n}-F)}
 
, viewed as a function indexed by 
 
 
 
 
 t
 ∈
 
 R
 
 
 
 
 {\displaystyle \scriptstyle t\in \mathbb {R} }
 
, <a href="/facts/Convergence_in_distribution/pWdatFlY">converges in distribution</a> in the <a href="/facts/Skorokhod_space/R9Jz9QLh">Skorokhod space</a> 
 
 
 
 
 D
 [
 −
 ∞
 ,
 +
 ∞
 ]
 
 
 
 {\displaystyle \scriptstyle D[-\infty ,+\infty ]}
 
 to the mean-zero <a href="/facts/Gaussian_process/MrBq7kYW">Gaussian process</a> 
 
 
 
 
 
 G
 
 F
 
 
 =
 B
 ∘
 F
 
 
 
 {\displaystyle \scriptstyle G_{F}=B\circ F}
 
, where B is the standard <a href="/facts/Brownian_bridge/vnStxc6h">Brownian bridge</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> The covariance structure of this Gaussian process is

E
        ⁡
        [
        
        
          G
          
            F
          
        
        (
        
          t
          
            1
          
        
        )
        
          G
          
            F
          
        
        (
        
          t
          
            2
          
        
        )
        
        ]
        =
        F
        (
        
          t
          
            1
          
        
        ∧
        
          t
          
            2
          
        
        )
        −
        F
        (
        
          t
          
            1
          
        
        )
        F
        (
        
          t
          
            2
          
        
        )
        .
      
    
    {\displaystyle \operatorname {E} [\,G_{F}(t_{1})G_{F}(t_{2})\,]=F(t_{1}\wedge t_{2})-F(t_{1})F(t_{2}).}

The uniform rate of convergence in Donsker’s theorem can be quantified by the result known as the <a href="/facts/Hungarian_embedding/nJVNTdfC">Hungarian embedding</a>:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

lim sup
 
 n
 →
 ∞
 
 
 
 
 
 n
 
 
 
 ln
 
 2
 
 
 ⁡
 n
 
 
 
 
 
 ‖
 
 
 
 
 n
 
 
 (
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 −
 F
 )
 −
 
 G
 
 F
 ,
 n
 
 
 
 
 
 ‖
 
 
 
 ∞
 
 
 <
 ∞
 ,
 
 
 a.s.
 
 
 
 {\displaystyle \limsup _{n\to \infty }{\frac {\sqrt {n}}{\ln ^{2}n}}{\big \|}{\sqrt {n}}({\widehat {F}}_{n}-F)-G_{F,n}{\big \|}_{\infty }<\infty ,\quad {\text{a.s.}}}

Alternatively, the rate of convergence of 
 
 
 
 
 
 
 n
 
 
 (
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 −
 F
 )
 
 
 
 {\displaystyle \scriptstyle {\sqrt {n}}({\widehat {F}}_{n}-F)}
 
 can also be quantified in terms of the asymptotic behavior of the sup-norm of this expression. Number of results exist in this venue, for example the <a href="/facts/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality/CTeMebhH">Dvoretzky–Kiefer–Wolfowitz inequality</a> provides bound on the tail probabilities of 
 
 
 
 
 
 
 n
 
 
 ‖
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 −
 F
 
 ‖
 
 ∞
 
 
 
 
 
 {\displaystyle \scriptstyle {\sqrt {n}}\|{\widehat {F}}_{n}-F\|_{\infty }}
 
:<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

Pr
        
        
          
            (
          
        
        
          
            n
          
        
        ‖
        
          
            
              
                F
                ^
              
            
          
          
            n
          
        
        −
        F
        
          ‖
          
            ∞
          
        
        >
        z
        
          
            )
          
        
        ≤
        2
        
          e
          
            −
            2
            
              z
              
                2
              
            
          
        
        .
      
    
    {\displaystyle \Pr \!{\Big (}{\sqrt {n}}\|{\widehat {F}}_{n}-F\|_{\infty }>z{\Big )}\leq 2e^{-2z^{2}}.}

In fact, Kolmogorov has shown that if the cumulative distribution function F is continuous, then the expression 
 
 
 
 
 
 
 n
 
 
 ‖
 
 
 
 
 F
 ^
 
 
 
 
 n
 
 
 −
 F
 
 ‖
 
 ∞
 
 
 
 
 
 {\displaystyle \scriptstyle {\sqrt {n}}\|{\widehat {F}}_{n}-F\|_{\infty }}
 
 converges in distribution to 
 
 
 
 
 ‖
 B
 
 ‖
 
 ∞
 
 
 
 
 
 {\displaystyle \scriptstyle \|B\|_{\infty }}
 
, which has the <a href="/facts/Kolmogorov_distribution/Kr5MqpfH">Kolmogorov distribution</a> that does not depend on the form of F.
Another result, which follows from the <a href="/facts/Law_of_the_iterated_logarithm/Yqds7Bco">law of the iterated logarithm</a>, is that <a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

lim sup
          
            n
            →
            ∞
          
        
        
          
            
              
                
                  n
                
              
              ‖
              
                
                  
                    
                      F
                      ^
                    
                  
                
                
                  n
                
              
              −
              F
              
                ‖
                
                  ∞
                
              
            
            
              2
              ln
              ⁡
              ln
              ⁡
              n
            
          
        
        ≤
        
          
            1
            2
          
        
        ,
        
        
          a.s.
        
      
    
    {\displaystyle \limsup _{n\to \infty }{\frac {{\sqrt {n}}\|{\widehat {F}}_{n}-F\|_{\infty }}{\sqrt {2\ln \ln n}}}\leq {\frac {1}{2}},\quad {\text{a.s.}}}

and

lim inf
          
            n
            →
            ∞
          
        
        
          
            2
            n
            ln
            ⁡
            ln
            ⁡
            n
          
        
        ‖
        
          
            
              
                F
                ^
              
            
          
          
            n
          
        
        −
        F
        
          ‖
          
            ∞
          
        
        =
        
          
            π
            2
          
        
        ,
        
        
          a.s.
        
      
    
    {\displaystyle \liminf _{n\to \infty }{\sqrt {2n\ln \ln n}}\|{\widehat {F}}_{n}-F\|_{\infty }={\frac {\pi }{2}},\quad {\text{a.s.}}}

<h2 id="confidence-intervals">Confidence intervals</h2>

As per <a href="/facts/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality/CTeMebhH">Dvoretzky–Kiefer–Wolfowitz inequality</a> the interval that contains the true CDF, 
 
 
 
 F
 (
 x
 )
 
 
 {\displaystyle F(x)}
 
, with probability 
 
 
 
 1
 −
 α
 
 
 {\displaystyle 1-\alpha }
 
 is specified as

F
          
            n
          
        
        (
        x
        )
        −
        ε
        ≤
        F
        (
        x
        )
        ≤
        
          F
          
            n
          
        
        (
        x
        )
        +
        ε
        
        
           where 
        
        ε
        =
        
          
            
              
                ln
                ⁡
                
                  
                    2
                    α
                  
                
              
              
                2
                n
              
            
          
        
        .
      
    
    {\displaystyle F_{n}(x)-\varepsilon \leq F(x)\leq F_{n}(x)+\varepsilon \;{\text{ where }}\varepsilon ={\sqrt {\frac {\ln {\frac {2}{\alpha }}}{2n}}}.}

As per the above bounds, we can plot the Empirical CDF, CDF and confidence intervals for different distributions by using any one of the statistical implementations.

<h2 id="statistical-implementation">Statistical implementation</h2>
A non-exhaustive list of software implementations of Empirical Distribution function includes:

<ul><li>In <a href="https://www.rdocumentation.org/packages/stats/versions/3.6.1/topics/ecdf">R software</a>, we compute an empirical cumulative distribution function, with several methods for plotting, printing and computing with such an “ecdf” object.</li>
<li>In <a href="https://www.mathworks.com/help/stats/cdfplot.html">MATLAB</a> we can use Empirical cumulative distribution function (cdf) plot</li>
<li><a href="https://www.jmp.com/support/help/14-2/cdf-plot.shtml">jmp from SAS</a>, the CDF plot creates a plot of the empirical cumulative distribution function.</li>
<li><a href="https://blog.minitab.com/blog/data-analysis-and-quality-improvement-and-stuff/the-empirical-cdf-part-2-software-vs-etch-a-sketch">Minitab</a>, create an Empirical CDF</li>
<li><a href="http://www.mathwave.com/articles/distribution-fitting-graphs.html">Mathwave</a>, we can fit probability distribution to our data</li>
<li><a href="https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/ecdfplot.htm">Dataplot</a>, we can plot Empirical CDF plot</li>
<li><a href="https://scipy.github.io/devdocs/reference/generated/scipy.stats.ecdf.html">Scipy</a>, we can use scipy.stats.ecdf</li>
<li><a href="https://www.statsmodels.org/dev/generated/statsmodels.distributions.empirical_distribution.ECDF.html#statsmodels.distributions.empirical_distribution.ECDF">Statsmodels</a>, we can use statsmodels.distributions.empirical_distribution.ECDF</li>
<li><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ecdf.html">Matplotlib</a>, using the matplotlib.pyplot.ecdf function (new in version 3.8.0)<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li>
<li><a href="https://seaborn.pydata.org/generated/seaborn.ecdfplot.html">Seaborn</a>, using the seaborn.ecdfplot function</li>
<li><a href="https://plotly.com/python/ecdf-plots/">Plotly</a>, using the plotly.express.ecdf function</li>
<li><a href="https://www.statisticshowto.com/empirical-distribution-function/">Excel</a>, we can plot Empirical CDF plot</li>
<li><a href="/facts/ArviZ/gIUKyL1L">ArviZ</a>, using the <a href="https://python.arviz.org/en/latest/api/generated/arviz.plot_ecdf.html#arviz.plot_ecdf">az.plot_ecdf</a> function</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/C%C3%A0dl%C3%A0g/R9Jz9QLh">Càdlàg</a> functions</li>
<li><a href="/facts/Count_data/vPP2VEyk">Count data</a></li>
<li><a href="/facts/Distribution_fitting/bm097384">Distribution fitting</a></li>
<li><a href="/facts/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality/CTeMebhH">Dvoretzky–Kiefer–Wolfowitz inequality</a></li>
<li><a href="/facts/Empirical_probability/WyHXAJ43">Empirical probability</a></li>
<li><a href="/facts/Empirical_process/HUQFdWLB">Empirical process</a></li>
<li><a href="/facts/Quantile/oWr1MC3s">Estimating quantiles from a sample</a></li>
<li><a href="/facts/Frequency_(statistics)/FtFNmhDS">Frequency (statistics)</a></li>
<li><a href="/facts/Empirical_likelihood/3jThGNe8">Empirical likelihood</a></li>
<li><a href="/facts/Kaplan%E2%80%93Meier_estimator/0yEyJzcV">Kaplan–Meier estimator</a> for censored processes</li>
<li><a href="/facts/Survival_function/jkdCmDKq">Survival function</a></li>
<li><a href="/facts/Q%E2%80%93Q_plot/h2jKBjYm">Q–Q plot</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Galen_R._Shorack/S0BQvl7j">Shorack, G.R.</a>; <a href="/facts/Jon_A._Wellner/D9B6Dgxj">Wellner, J.A.</a> (1986). Empirical Processes with Applications to Statistics. New York: Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-86725-X.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li> Media related to Empirical distribution functions at Wikimedia Commons</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">A modern introduction to probability and statistics: Understanding why and how. Michel Dekking. London: Springer. 2005. p. 219. ISBN 978-1-85233-896-1. OCLC 262680588.{{cite book}}: CS1 maint: others (link) <a href="978-1-85233-896-1" target="_blank">978-1-85233-896-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 265. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. Springer, p. 36, Definition 2.4. ISBN 978-1-4471-3675-0. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Madsen, H.O., Krenk, S., Lind, S.C. (2006) Methods of Structural Safety. Dover Publications. p. 148-149. ISBN 0486445976 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 265. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 266. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 265. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 266. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 268. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 268. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. p. 268. ISBN 0-521-78450-6. <a href="0-521-78450-6" target="_blank">0-521-78450-6</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"What's new in Matplotlib 3.8.0 (Sept 13, 2023) — Matplotlib 3.8.3 documentation". <a href="https://matplotlib.org/stable/users/prev_whats_new/whats_new_3.8.0.html#axes-ecdf" target="_blank">https://matplotlib.org/stable/users/prev_whats_new/whats_new_3.8.0.html#axes-ecdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Empirical distribution function open-in-new

Empirical distribution function