Geostatistical Tools for the Determination of Fundamental Sampling Variances and Minimum Sample Masses

Influence Diagnostics for Elliptical Multivariate Linear Regression Models

“…As considered in the Appendix, to derive the asymptotic covariance matrix of the maximum likelihood estimators derived in the previous section, we have to compute the matrices V n and E[A n (q)] which involves first and second derivatives of h i [Z i ; q] with respect to q, i = 1, …, n. Then, defining we compute, after some algebraic manipulation, where and Furthermore, defining it can be shown that (9) Thus, it follows that the maximum likelihood estimator q = (â,b)A is asymptotically normally distributed with asymptotic covariance matrix n 21 W n (q), where (10) with and where and w i as in eqn. (6), i = 1, …, n. To obtain a consistent estimator of the matrix W n (q) it suffices to replace q by its consistent estimator q .…”

“…For example, Riu and Rius 1 claim that the distribution of the F statistic in their expression (9) follows an F distribution since it is the ratio of two chi-square distributions. A description of several methods is reported in Riu and Rius 1 who advocated the use of bivariate least squares considering joint confidence intervals for the intercept and slope parameters.…”

Detection of analytical bias

“…As considered in the Appendix, to derive the asymptotic covariance matrix of the maximum likelihood estimators derived in the previous section, we have to compute the matrices V n and E[A n (q)] which involves first and second derivatives of h i [Z i ; q] with respect to q, i = 1, …, n. Then, defining we compute, after some algebraic manipulation, where and Furthermore, defining it can be shown that (9) Thus, it follows that the maximum likelihood estimator q = ( â, b)A is asymptotically normally distributed with asymptotic covariance matrix n 21 W n (q), where (10) with and where and w i as in eqn. ( 6), i = 1, …, n. To obtain a consistent estimator of the matrix W n (q) it suffices to replace q by its consistent estimator q.…”

“…These functions were obtained using Pierre Gy's sampling equations, 8 calibrated by studies performed after François-Bongarçon's suggestions. 9 The estimates are as follows: The scatterplot of the points, as well as the standard deviations and regression lines are displayed in Fig. 7.…”

“…However the estimation of the covariance matrix is proposed in an ad hoc manner, and is based on results borrowed from ordinary least squares (OLS) without formal justification. For example, Riu and Rius 1 claim that the distribution of the F statistic in their expression (9) follows an F distribution since it is the ratio of two chi-square distributions. However, the weights w i are also random since they depend on the estimator b of the slope parameter b.…”