[Developers] statistical optimality

[J P Morgan]

Suggested additions to schema in accordance with my posting of 15th April. JP


optimality_criteria_values = element optimality_criteria_values {
    element phi_0 {
        attribute value { xsd:decimal | "not_applicable" } 
    }+
    &
    element phi_1 {
        attribute value { xsd:decimal | "not_applicable" } 
    }+
    &
    element phi_2 {
        attribute value { xsd:decimal | "not_applicable" } 
    }+
    &
    element E_values { 
        cumulative_sums+ 
    }+
    & 
    element no_distinct_canonical_variances {
        attribute value { xsd:positiveInteger } 
    }+
    &   
    element max_min_ratio_canonical_variances {
        attribute value { xsd:decimal | "not_applicable" } 
    }+
    &
    element no_distinct_pairwise_variances {
        attribute value { xsd:positiveInteger | "not_applicable" } 
    }+
    &
    element maximum_pairwise_variances {
        attribute value { xsd:positiveInteger | "not_applicable" } 
    }+
    &
    element max_min_ratio_pairwise_variances {
        attribute value { xsd:positiveInteger | "not_applicable" } 
    }+
}


cumulative_sums = element cumulative_sums { 
    attribute type { "Indexed" }, 
    d+ 
}


efficiency_factors = element efficiency_factors {
    attribute length {xsd:positiveInteger},
    factor+,
    element summaries_of_efficiency_factors {
        element harmonic_mean {
            attribute name {"A"},
            attribute value { xsd:decimal }
        }
        element geometric_mean {
            attribute name {"D"},
            attribute value { xsd:decimal }
        }
        element minimum {
            attribute name {"E"},
            attribute value { xsd:decimal }
        }
        element min_max_ratio {
            attribute name {"Variance Ratio"},
            attribute value { xsd:decimal }
        }
        element arithmetic_mean {
            attribute name {"M"},
            attribute value { xsd:decimal }
        }
        element sum_of_squares {
            attribute name {"S"},
            attribute value { xsd:decimal }
        }
}
          


>===== Original Message From J P Morgan <J.P.Morgan at qmul.ac.uk> =====
>
>
>We currently have a category named efficiency_factors. I propose that we add
>a category under that heading named "optimality_functions_of_efficiency
>_factors" (or something similar), to include geometric mean, harmonic mean,
>etc as Rosemary skecthed out last Wednesday, but limited solely to functions
>of the efficiency factors.  Each is optimized if maximal.
>
>We also include a category at the same level as efficiency_factors named
>optimality_criteria_values. Here we put functions that are minimized for
>optimality. Included are selected functions of the eigenvalues \mu_i of the
>information matrix and other selected criteria not based on eigenvalues.
> Here is my suggested list:
>
>D = \phi_0
>A = \phi_1
>\phi_2
>E = \phi_\infty = 1/\mu_1 = E1
>E2 = E1+1/\mu_2
>E3 = E2 + 1/\mu_3
> .
> .
> .
>E(v-1) = E(v-2) + 1/\mu_{v-1}
>MV = max pairwise variance
># distinct  \mu_i
>var_ratio = \mu_{v-1}/\mu_1
># distinct variances for pairwise comparisons
>pairwise_var_ratio = MV/min pairwise variance
>
>A,D,E,MV are all there, of course, along with several other ways of thinking
>about dispersion of variances and extremal behavior. The Ej's, which are
>components of a Schur-optimality argument, have also been studied on their
>own, albeit not extensively.
>
>
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