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Example 9.6: Hawkins-Bradu-Kass Data

The first 14 observations of this data set (refer to Hawkins, Bradu, and Kass 1984) are leverage points; however, only observations 12, 13, and 14 have large h_ii and only observations 12 and 14 have large MD_i values.

   title "Hawkins, Bradu, Kass (1984) Data";
      aa = { 1  10.1  19.6  28.3   9.7,
             2   9.5  20.5  28.9  10.1,
             3  10.7  20.2  31.0  10.3,
             4   9.9  21.5  31.7   9.5,
             5  10.3  21.1  31.1  10.0,
             6  10.8  20.4  29.2  10.0,
             7  10.5  20.9  29.1  10.8,
             8   9.9  19.6  28.8  10.3,
             9   9.7  20.7  31.0   9.6,
            10   9.3  19.7  30.3   9.9,
            11  11.0  24.0  35.0  -0.2,
            12  12.0  23.0  37.0  -0.4,
            13  12.0  26.0  34.0   0.7,
            14  11.0  34.0  34.0   0.1,
            15   3.4   2.9   2.1  -0.4,
            16   3.1   2.2   0.3   0.6,
            17   0.0   1.6   0.2  -0.2,
            18   2.3   1.6   2.0   0.0,
            19   0.8   2.9   1.6   0.1,
            20   3.1   3.4   2.2   0.4,
            21   2.6   2.2   1.9   0.9,
            22   0.4   3.2   1.9   0.3,
            23   2.0   2.3   0.8  -0.8,
            24   1.3   2.3   0.5   0.7,
            25   1.0   0.0   0.4  -0.3,
            26   0.9   3.3   2.5  -0.8,
            27   3.3   2.5   2.9  -0.7,
            28   1.8   0.8   2.0   0.3,
            29   1.2   0.9   0.8   0.3,
            30   1.2   0.7   3.4  -0.3,
            31   3.1   1.4   1.0   0.0,
            32   0.5   2.4   0.3  -0.4,
            33   1.5   3.1   1.5  -0.6,
            34   0.4   0.0   0.7  -0.7,
            35   3.1   2.4   3.0   0.3,
            36   1.1   2.2   2.7  -1.0,
            37   0.1   3.0   2.6  -0.6,
            38   1.5   1.2   0.2   0.9,
            39   2.1   0.0   1.2  -0.7,
            40   0.5   2.0   1.2  -0.5,
            41   3.4   1.6   2.9  -0.1,
            42   0.3   1.0   2.7  -0.7,
            43   0.1   3.3   0.9   0.6,
            44   1.8   0.5   3.2  -0.7,
            45   1.9   0.1   0.6  -0.5,
            46   1.8   0.5   3.0  -0.4,
            47   3.0   0.1   0.8  -0.9,
            48   3.1   1.6   3.0   0.1,
            49   3.1   2.5   1.9   0.9,
            50   2.1   2.8   2.9  -0.4,
            51   2.3   1.5   0.4   0.7,
            52   3.3   0.6   1.2  -0.5,
            53   0.3   0.4   3.3   0.7,
            54   1.1   3.0   0.3   0.7,
            55   0.5   2.4   0.9   0.0,

            56   1.8   3.2   0.9   0.1,
            57   1.8   0.7   0.7   0.7,
            58   2.4   3.4   1.5  -0.1,
            59   1.6   2.1   3.0  -0.3,
            60   0.3   1.5   3.3  -0.9,
            61   0.4   3.4   3.0  -0.3,
            62   0.9   0.1   0.3   0.6,
            63   1.1   2.7   0.2  -0.3,
            64   2.8   3.0   2.9  -0.5,
            65   2.0   0.7   2.7   0.6,
            66   0.2   1.8   0.8  -0.9,
            67   1.6   2.0   1.2  -0.7,
            68   0.1   0.0   1.1   0.6,
            69   2.0   0.6   0.3   0.2,
            70   1.0   2.2   2.9   0.7,
            71   2.2   2.5   2.3   0.2,
            72   0.6   2.0   1.5  -0.2,
            73   0.3   1.7   2.2   0.4,
            74   0.0   2.2   1.6  -0.9,
            75   0.3   0.4   2.6   0.2 };

       a = aa[,2:4]; b = aa[,5];

The data are listed also in Rousseeuw and Leroy (1987, p. 94).

The complete enumeration must inspect 1,215,450 subsets.

The following output reports the robust parameter estimates for MVE.

Output 9.6.4 shows the classical Mahalanobis and robust distances obtained by complete enumeration. The first 14 observations are recognized as outliers (leverage points).

The following two graphs show

• the plot of standardized LMS residuals vs. robust distances RD_i
• the plot of standardized LS residuals vs. Mahalanobis distances MD_i

The output follows.

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