exactcdf

exactcdf finds exact p-values

Syntax

Description

Function for finding the exact cdf of each element in the vector x with respect to the empirical distribution, represented by the vector empdist, i.e. the generic element i of the output vector p is the result of:

\[ \frac{ \displaystyle \sum_{j=1}^K I_{empdist(j) \leq x_i}}{K} \] where $I$ is the indicator function and $K$ is the length of vector $empdist$

example

p =exactcdf(x) exactcdf with just one input argument.

example

p =exactcdf(x, empdist) exactcdf with two input arguments.

Examples

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  • exactcdf with just one input argument.
  • k=1000;
    x=randn(k,1);
    p=exactcdf(x);

  • exactcdf with two input arguments.
  • k=10;
    x=randn(k,1);
    K=100000;
    empdist=randn(K,1);
    % Compute empirical cdf for each element of vector x.
    p=exactcdf(x,empdist);
    % Compute theoretical cdf based on normcdf
    pTheo=normcdf(x);
    % Compare empirical cdf with theoretical cdf 
    plot(p,pTheo,'o')
    xlabel('Empirical cdf')
    ylabel('Theoretical cdf')
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

    Related Examples

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  • Using exactcdf for calculating exact p-values.
  • k=10;
    x=randn(k,1);
    K=100000;
    empdist=randn(K,1);
    % Compute empirical cdf for each of element of vector x.
    p=exactcdf(x,empdist);
    % Compute exact p-values for an unilateral right-tailed test
    pval_rt=1-p;
    % Compute exact p-values for an unilateral left-tailed test
    pval_lt=p;

    Input Arguments

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    x — empirical replicates of a test. Vector.

    Vetor of length k containing the empirical realization of a generic test

    Data Types: double

    Optional Arguments

    empdist — empirical distribution of the same test. Vector.

    Vector of length $K$ generally with $K \geq k$ containig the empirical simulated distribution of the test. If this optional argument is not supplied the empirical distribution is taken from input vector x.

    Example: randn(K,1)

    Data Types: double

    Output Arguments

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    p —empirical cdf. Vector

    Vector with the same length of input vector x containing the empirical cdf of each element of input vector x. More precisely: $p(i)$ is computed as \[ \frac{ \displaystyle \sum_{j=1}^K I_{empdist(j) \leq x_i}}{K} \]

    References

    Athey, S., Eckles, D., & Imbens, G. W. (2018). Exact p-values for network interference, "Journal of the American Statistical Association", Vol. 113, pp. 230-240.

    See Also

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