Conditional Score Functions: Some Optimality Results

Abstract

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The effect of cluster size imbalance and covariates on the estimation performance of quadratic inference functions

“….Y i i /, r D 1; 2; i D 1; 2; : : : N , inducing the need to estimate˛. From Lindsay [14], Hansen [3], and Small and McLeish [11], Qu et al [2] show that because C N .ˇ/ † N p ! 0, the estimating equations given in Equation (6) are fully efficient when the covariance structure is correctly specified and they are in the same class as GEE, and always optimal in the Löwner ordering among estimating equations within their given class.…”

A Simple Method for the Adjustment of Profile Likelihoods

“…In particular, unbiasedness of the estimating equation essentially guarantees consistency, and the condition on the derivative matrix guarantees asymptotic optimality within a limited class of estimating functions. For further details we refer the reader to Godambe (1960), Godambe and Thompson (1974), Lindsay (1982) and Godambe and Heyde (1987).…”

“…Our goal is to adjust the profile log-likelihood so that the mean of the score function is zero and the variance of the score function equals its negative expected derivative matrix. In the terminology of Godambe (1960) and Lindsay (1982), our goal is to adjust the profile log-likelihood score function so that it is unbiased and information unbiased. The hope is that, by making these adjustments, the asymptotic behaviour of the quantities derived from the likelihood (e.g.…”

A bias‐corrected covariance estimate for improved inference with quadratic inference functions

“…ˇ/ is the empirical covariance matrix for the extended score equations. Minimizing the value of Q N .ˇ/ to estimateˇis asymptotically equivalent to solving the optimal estimating equations [6,7,14,15]. In practice, C N .ˇ/ replaces the optimal † N D EOEC N .ˇ/, as C N .ˇ/ † N p !…”