Qn robust estimator of scale (first quartile of interpoint distances $|x_i-x_j|$)




y =Qn(X, dim) Qn with all default otpions.


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  • Qn with all default otpions.
  • X = [1 2 4 4  7;
    3 4 6 6  8;
    5 6 8 8  10;
    5 7 10 12 1500];
    y1 =
        2.2752    2.2752    2.2752    4.5505    3.4128
    y2 =

    Input Arguments

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    X — Input array. Vector | matrix | 3D array.

    Input array, specified as a vector, matrix, or 3D array.

    For vectors, Qn(X) is the scale estimator of the elements in X. For matrices, Qn(X) is a row vector containing the scale estimator value of each column. For 3D arrays, Qn(X) is the robust scale estimator of the elements along the first non-singleton dimension of X.

    Data Types: ingle | double | int8 | int16 | int32 | int64 |uint8 | uint16 | uint32 | uint64

    dim — Dimension to operate along. Positive integer scalar.

    Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.

    Data Types: ingle | double | int8 | int16 | int32 | int64 |uint8 | uint16 | uint32 | uint64

    Output Arguments

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    y —robust estimator of scale. Scalar | Vector or 3D array

    Qn(X,dim) takes the robust estimator of scale along the dimension dim of X.

    More About

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    Additional Details

    $Q_n$ is the first quartile of the distances { $|x_i-x_j|$; $i <j$} Note that $Q_n$ does not need any location estimate. More in detail, let $d_{(1)} \leq d_{(2)} \leq ... \leq d_{(m)}$ the ordered values of the $m$ differences $|x_i-x_j|$ with $i>j$ and $m = {n \choose 2}$. $Q_n=d_{(k)}$ where $k= {[n/2]+1 \choose 2}$. Since $k$ is approximately $m/4$, $Q_n$ is approximately the first quartile of the ordered distances $d_{(1)} \leq d_{(2)} \leq ... \leq d_{(m)}$. $Q_n$ is multiplyed by $c$ and $c_n$.

    $c$ is the so called asymptotic consistency factor and is equal to 2.2219 while $c_n$ is a finite sample correction factor to make the estimator unbiased.


    Rousseeuw P.J. and Croux C., (1993), Alternatives to the median absolute deviation, "Journal of American Statistical Association", Vol. 88, pp. 1273-1283

    Croux C. and Rousseeuw P.J.(1992) Time-efficient algorithms for two highly robust estimators of scale, in "Computational Statistics", Vol. 1, eds. Y . Dodge and J. Whittaker, Heidelberg: Physika-Verlag, 41 1-428.

    See Also

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