Projection-Based Approach. The projection-based method for approximating the likelihood ratio described by Li [44] depends solely on the first two moments of the data. Thus, this approach does not require any additional knowledge about the moment structure of the feature vector than would be required to construct the estimating functions. In past work, McLeish and Small(1992)[55] worked directly with the likelihood ratio and projected it onto i=1 the subspace of L2 spanned by ∏n Xi, where the Xi, i = 1, . . . , n represent indepen- dent observations. In contrast, Li considers projecting the log likelihood ratio onto a subspace linear in the observations X1, . . . , Xn. Unlike in the work of McLeish and Small, Li’s approach applies even when the Xi, i = 1, . . . , n are dependent observa- tions; thus, it can be used for longitudinal studies. Unfortunately, the log likelihood ratio is not amenable to projection when only the first two data moments are known. Thus, rather than projecting the log likelihood ratio directly, Li obtains a linear ap- proximation of the log likelihood ratio via a Xxxxxx series expansion before conducting the projection. Briefly, assume that Y = (Y1, . . . , Yb)T is a b × 1 vector of possibly dependent observations with distribution pω. Further, suppose only the mean and covariance matrix of Y are known. Specifically, let µω be the mean vector of Y and Vω be the variance-covariance matrix of Y under ω. Now, for simplicity of notation, let a = pω(Y) and b = pν(Y) and note that 2log [a ] = log [a/b ] b b/a b a = log [a ] − log [ b ] b a = log [b + a − b ] − log [ a + b − a ] b a = log [1 + a − b ] − log [1 + b − a ] . Then, a Xxxxxx series expansion yields the following linear approximation of the log likelihood ratio 2log [a ] ≈ a − b + a − b + (b − a)3(b + a) 2a2b2 b b a ≈ a − b + a − b. Thus, by substituting back in a = pω(Y) and b = pν(Y), the following approximation p (Y) of log pω (Y) ν can be obtained using Li’s approach: pν(Y) 2pν(Y) 2pω(Y) log [pω(Y)] ≈ pω(Y) − pν(Y) + pω(Y) − pν(Y). 2pν (Y) pν (Y) Next, the approximation of the log likelihood ratio is projected onto a suitable Xxxxxxx subspace. Let R1 = pω (Y)−pν (Y) = 1 { p ω (Y) − 1}. Consider the Xxxxxxx space L2 and the closed subspace of L2 defined as Lν = span {Y1 − µ1ν, . . . , Yb − µbν} with inner product ⟨g1, g2⟩ν = Eν (g1g2). Since R1 is a member of L2 it can be projected onto Lν. Also, all elements of Lν are linear in Y and, thus, take the form aT (Y − µν ) for some vector a. Denote the projecti...
