In this paper, we propose a general type of estimator based on the idea of adaptive weighting. We shall call the estimator the adaptive estimator.

All the existing sampling graph techniques can be represented as a Bipartite Incident Graph (BIG) at the estimation stage. A bipartite graph is a graph whose nodes are divided into two disjoint sets such that every edge connects two nodes each from a different set. In a BIG, one set of nodes represents the sampling frame and the other the target statistical population. We use the term `elements of interest' to identify the units in the target population and distinguish them from the `sampling unit' in the frame. A sampling unit and an element of interest are connected if the element is selected after the observation of the sampling unit. This observational procedure is called incident.

An element of interest does not have to be a single unit, but it could be a collection of units or, more generally, any subgraph of interest.
Usually, we are interested in estimating some characteristics of the target population expressed as a function defined on its elements.

Under the BIG representation, the same function can be re-defined on the sampling units by using the connections between the frame and the target population. The idea is to `distribute' each element of interest amongst the edge connected to it by associating a weight to each edge. Consequently, a new measure for the sampling unit is defined to be equal to the sum of the weights of its connected edges. This new measure represents the `part' of the population that is transferred to the unit through its connections.

When dealing with estimation of characteristics of populations for which a direct frame is not available, different HT estimators have been proposed, which use the graph structures without making it explicit. Also, they differ with respect to the definition of the weights.

The adaptive estimator that we propose encompasses the estimators that already exist in the literature and it envisages the construction of many more. It generalizes the fixed weights using the concept of adaptive weighting. Given the initial fixed weights, adaptive weighting is the case if the weights are obtained from adjusting the initial weights using information that is not available before the sample graph is observed. We describe two families of adaptive weights. We call the weights adaptive by prioritization if only certain edges are used for the estimation. Also, when auxiliary quantities are used to redistribute the weights amongst the edges, we define them adaptive by redistribution. This classification is not exhaustive.

Throughout this paper, we investigate the conditions under which the adaptive estimator is unbiased and provide a formula of its variance and an estimator of it. Efficiency of the estimator is discussed with regard to the two proposed families of adaptive weights. In light of this discussion, the relative efficiency of the existing estimators is studied. Lastly, numerical illustration about the characteristics of the proposed estimator provided.