Simulation studies to examine bias and precision of some estimators that use auxiliary information in design-based sampling in forest inventory

New Zealand Journal of Forestry Science

Table 2 Bias (%) in estimates of the population mean from simulations of inventory in five example populations

Example	Methods with complete enumeration of N sampling units as first phase					Methods with first-phase sample size = f
Example	Ratio of means	Mean of ratios	Model-assisted	PPS sampling	QPPS sampling	Simple random sampling	Ratio of means	Mean of ratios	Model-assisted	QPPS sampling
Second-phase sample size = 100
1	0.00	0.01	0.00	0.01	0.01	0.00	0.00	0.01	0.00	− 0.61
2	− 0.01	− 0.01	− 0.02	0.02	0.03	− 0.01	− 0.02	−0.02	− 0.02	0.13
3	− 0.01	− 0.01	− 0.01	0.00	0.01	− 0.01	− 0.01	− 0.01	− 0.01	− 0.47
4	− 0.02	− 0.02	− 0.02	− 0.03	− 0.01	− 0.02	− 0.02	− 0.02	− 0.02	− 0.72
5	0.01	0.00	0.01	0.00	− 0.01	0.02	0.01	0.00	0.01	− 0.65
Second-phase sample size = 40
1	0.02	0.03	0.02	0.02	0.04	0.02	0.02	0.03	0.02	− 0.55
2	0.00	0.00	0.01	0.04	0.01	− 0.02	0.00	0.00	0.01	0.03
3	0.00	0.00	0.00	0.03	0.00	0.01	0.00	0.00	0.00	− 0.47
4	0.02	0.02	0.02	− 0.02	− 0.01	0.03	0.01	0.01	0.01	− 0.70
5	0.04	0.04	0.05	0.01	0.03	0.05	0.04	0.03	0.04	− 0.57
Second-phase sample size = 10
1	− 0.02	− 0.01	− 0.02	0.06	− 0.07	0.01	0.00	0.01	0.00	− 0.76
2	− 0.03	− 0.04	− 0.03	− 0.01	− 0.01	− 0.08	− 0.02	− 0.03	− 0.02	− 0.03
3	0.01	0.01	− 0.02	0.00	0.00	− 0.03	− 0.02	− 0.02	− 0.04	− 0.55
4	0.03	0.03	0.02	− 0.02	− 0.03	0.02	0.04	0.04	0.03	− 0.84
5	0.02	0.02	0.01	− 0.02	0.00	0.01	0.02	0.01	0.00	− 0.75

Bias was determined as the difference between the mean of the simulation estimates of the population mean and the population true mean, expressed as a proportion of the true mean. Results are shown for each of the methods applied in this work both when a complete enumeration of the auxiliary variable was done in the first-phase for the entire population or when the first-phase sample was smaller, of size f (<N). Each value shown in the table is the mean of 45,000 simulations of each estimator, those being made up of 5000 simulations for each of nine levels of target-auxiliary variable correlation. Results are given also when three different second-phase sample sizes were used. The examples are numbered as 1 – Basal area of Japanese rainforest, 2 – Stocking density of Japanese rainforest, 3 – Tree stem diameters, P. radiata, South Australia, 4 – Tree stem diameters, rainforest planting, New South Wales, 5 – Stand sawlog volume, Victorian eucalypt forest. Values shown as zero were actually <0.005%