Edge Estimation in the Population of a Binary Tree Using Node-Sampling

Abstract

Suppose a finite population of several vertices, each connected to single or multiple edges. This constitutes a structure of graphical population of vertices and edges. As a special case, the graphical population like a binary tree having only two child vertices associated to parent vertex is taken into consideration. The entire binary tree is divided into two sub-graphs such as a group of left-nodes and a group of right-nodes. This paper takes into account a mixture of graph structured and population sampling theory together and presents a methodology for mean-edge-length estimation of left sub-graph using right edge sub-graph as an auxiliary source of information. A node-sampling procedure is developed for this purpose and a class of estimators is proposed containing several good estimators. Mathematical conditions for minimum bias and optimum mean squared error of the class are derived and theoretical results are numerically supported with a test of 99% confidence intervals. It is shown that suggested class has a sub-class of optimum estimators, and sample-based estimates are closer to the true value of the population parameter.

Keywords:

• Bias
• Binary tree
• Confidence interval
• Estimator
• Graph
• Left edge group (LEG)
• Mean squared error (MSE)
• Optimum choice
• Right edge group (REG)
• Simple random sampling without replacement (SRSWOR)
• Vertices (nodes).

Mathematics Subject Classification:

• 62D05.

Acknowledgment

The authors are thankful to the referee for suggestions.