  In geostatistics, block likelihood offers a balance between statistical
accuracy and computational efficiency when estimating covariance functions.
This balance is reached by dividing the sample into blocks and computing a
weighted sum of (sub) log-likelihoods corresponding to pairs of blocks.
Practitioners often choose block sizes ranging from hundreds to a few thousand
observations, inherently involving matrix-based implementations. An
alternative, residing at the opposite end of this methodological spectrum,
treats each observation as a block, resulting in the matrix-free pairwise
likelihood method. We propose an additional alternative within this broad
methodological landscape, systematically constructing blocks of size two and
merging pairs of blocks through conditioning. Importantly, our method
strategically avoids large-sized blocks, facilitating explicit calculations
that ultimately do not rely on matrix computations. Studies with both simulated
and real data validate the effectiveness of our approach, on one hand
demonstrating its superiority over pairwise likelihood, and on the other,
challenging the intuitive notion that employing matrix-based versions
universally lead to better statistical performance.
