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Abstract.--An illustration of the Rao-Hartley-Cochran (RHC) sampling strategy for a simulated forest inventory is presented. The objective of this illustration is to promote the consideration of the RHC strategy when a probability proportional to size with replacement strategy is being considered in a forest inventory. The illustration is based on a sampling frame of partitioned selection areas of the forest floor, and comparing the
RI-IC strategy to a Hansen-Hunvitz probability proportional to size strategy.
INTRODUCTION
This note presents an illustration of the Rao-Hartley-Cochran sampling strategy in the context of forest sampling when the selection units are partitioned areas of a forest floor. In forest surveys, the goal i s to obtain reliable estimates of characteristics of a forest based on a sampling strategy. Naturally, one would like to use a sampling strategy that results in unbiased estimates and 'small' standard errors of those estimates. The Rao-Hartley-Cochran strategy can provide smaller standard error estimates as compared to a with replacement, probability proportional to size strategy.
Sampling Frame
There are various approaches to sampling a forest based on particular sampling frames (de Vries 1989), which depend on the objectives of the survey. The sampling frame used in this illustration is based on Roesch, Green and Scott
(1993). This sampling frame is constructed by partitioning the forest floor into mutually exclusive segments based on the (overlapping) trees' K-circles (a circle centered at the tree with a fixed radius). The resulting segments are then selected, using some randomization mechanism, with probabilities proportional to the size of the segments. The trees whose selection areas contain the randomly selected
. segment are included in the sample.
Following the notation in Roesch, Green and Scott (1993), let ((1,
GI),
(2, y,),
( N ) denote the finite population of trees with their unit labels and

associated values of a characteristic of interest. Let f Al , A2,
. . ., AN } be the set of
(A , j
=
1, 2,
. . ., M } , determined by the overlapping areas, Ai, or Aj some nonoverlapping selection areas, forms the sampling frame.
=
Ai for
A REVIEW OF THE SAMPLING STRATEGIES
A brief review of the Hansen-Hunvitz and the Rao-Hartley-Cochran strategies using the above sampling frame is presented.
Hansen-Hurwitz Estimation
Roesch, Green and Scott (1993) constructed a sampling strategy that incorporated Hansen-Hurwitz (1 943) {HHJ estimation. The probability that selected segment j adds tree i to the sample is p6
=
(Aj / Ai)ZG, where 2,
=
1, segment j is part of the tree's K-circle
0, otherwise
, and the value associated with segment j is y, =
N
-,
C pij yi. An estimator of the i=l parametric total Y
=
C y ,
=
C y, is j-1 j=I where pj
=
AjIAT and AT is the area of the ground in the forest to be surveyed.
The variance of the estimator in Eq. 1 is with variance estimator

Rao-Hartley-Cochran Estimator
Estimators based on without replacement sampling are generally more efficient
(i.e., the estimators have smaller variance) than estimators based on with replacement sampling. The well-known Rao-Hartley-Cochran sampling strategy
(Rao, Hartley and Cochran 1962) {RHC} applied to the segmentation sampling frame is as follows. Partition the universe 24 of M segments into a random grouping {Gl, G2, ..., G,, ..., Gm}, where 1 5 M,, the groups are selected with equal probabilities, and M
=
77). 2 g= 1
(SRSWOR) can be used to assign segments to groups by selecting a SRSWOR of
Ml segments from M (for group GI), then selecting a SRSWOR of M2 segments fiom U-GI, then selecting a SRSWOR of M3 segments from 24- (GI U G2),
..., and, finally, assigning the remaining segments to Gm. Then, independently from each group, select one segment with probability proportional to the normed size measures Aj within each group. Specifically, those probabilities are p, = pj/Pg, g
=
1, 2,
. . . , M, , for segment j in group Gg, where pj =
A j I A ~ P,
=
C
Pj
&Gg
Let y, and A, be those m values of yj and Aj, respectively, selected fiom group g. The unbiased RHC estimator of the parametric total Y is
Chauduri and Mitra (1 992) indicated that the variance derived by Rao, Hartley and
Cochran (1962) usually performs as well as an alternative variance estimator proposed by Ohlsson (1989), so the RHC variance estimator was used in the simulation illustration. Rao, Hartley and Cochran (1962) gave several group size
A r\ conditions for comparing the variance of Y , to the variance of Y,. In the illustration (see next section), the variance minimizing condition MI
=
M2
=
. . . =
M,
=
Mlm applies, so the form with variance estimator

was used. Note that there has been some discussion about how well the efficiency of the RHC strategy holds as compared to some with replacement sampling strategies (Deshpande 1 984, Singh and Kishmore 1975). The simulation illustration provides an additional view on this issue.
AN ILLUSTRATIVE COMPARISON OF THE STRATEGIES
A small simulation study was designed to investigate some of the empirical properties of the RHC and the HH strategies for the above segmentation sampling frame. A dispersion pattern of N
=
100 "trees" was obtained by generating the coordinates (x, y) using independent random values from Poisson(60)
+
Uniform(0, 1) (figure 1). The volume (m3) of each tree was generated using independent normal(8, 2) random values based on the timber volumes in an example of de Vries (1989, p. 77). Note that the use of the Poisson and normal distributions give 'well-behaved' characteristics to the data. The segmentation sampling frame was laid out as described above, the M
=
156 areas A, computed, and the pairs entered into a data list. Here, AT is the sum of the Aj .
Using the Mlm criterion with M
=
156, the following set of nontrivial pairs((m,
M,)} was available for RHC sampling ((2, 78), (4, 39), (6, 26), (12, 13), (26, 6),
(39, 4), (78, 2)). The RHC and HH strategies were simulated 1000 times for each sample size m =
6, 26 or 78 using XLISP-STAT (Tierney 1991). The estimators in Eq.'s 1, 2, 3 , and 4 were used in computations. Boxplots of the estimates of
\ 1 total tree volume, Y,, or Y,, and estimates of their standard errors, or [ v(Y,)
'
1.
, are used to summarize the simulation results (figures 2 and 3).
Overall, both strategies resulted in the median of the estimates close to the parametric total
T =
793 m3, and the RHC estimates were slightly less variable
\ than the HH estimates (figure 2). Because both Y,, and Y , are unbiased estimators with respect to their sampling designs, these results were anticipated.
As the sample size increased, both sets of estimates became less variable.
Variability of the estimates was similar for n2 =
6, but the RHC strategy produced estimates with proportionately less variability as compared to the HH strategy as the sample size increased.
The estimates of the standard errors became smaller as sample size increased, and the estimates from the RHC strategy became proportionately smaller than the estimates from the HH strategy as the sample size increased. The variability in the

Figure 1.-Geometric representation of the dispersion of "trees." Each tree was generated by two independent random values obtained from Poisson(60)
+
Uniform@, 1).
Circles illustrate the K-circles of the trees.
RHC standard error estimates decreased proportionately relative to the HH standard error estimates as sample size increased. Thus the standard error of the
RHC strategy (Eq. 2) tends to give smaller standard error estimates relative to the
HH estimates (Eq. 4), although the practical difference appears minimal for the this illustration.

+
- RHC in = 6
HH RHC m = 26
HH RHC m = 78
HH
C3
+
- m
1
.
8 3
- tree volume
Figure 2.-Boxplots of the Rao-Hartley-Cochran (RHC) and Hansen-
Huwitz (HA) estimates of total tree volume. The total tree volume is 793.6 cubic meters, indicated by the horizontal line. Sample size is m.
RHC RHC HH RHC HH
Figure 3.-Boxplots of the Rao-Hartley-Cochran (RHC) and Hansen-
Hurwitz (HH) standard error estimates of estimated total tree volume.
Sample size is m.

REFERENCES
Chauduri, A. and Mitra, J. 1992. A note on two variance estimators for
Rao-Hartley-Cochran estimator. Communications in Statistics - Theory and
Methods, 21: 3535-3543. de Vries, P. G. 1989. Sampling theory for forest inventory. New York, NY:
Springer-Verlag . 3 99 p.
Deshpande, M. N. 1984. A note on Rao, Hartley and Cochranfs method. sankhya,
36: 114-116.
Hansen, M.H. and Hunvitz, W.N. 1943. On the theory of sampling from finite populations. Annals of Mathematical Statistics, 20: 426-43 2.
Ohlsson, E. 1989. Variance estimation in the Rao-Hartley-Cochran procedure. sankhya, B, 5 1 : 348-361.
Rao, J.N.K., Hartley, H. O., and Cochran, W. G. 1962. On a simple procedure of unequal probability sampling without replacement. Journal of the Royal
Statistical Society B, 24: 482-49 1.
Roesch, F. A., Jr., Green, E. J., and Scott, C. T. 1993. An alternative view of forest sampling. Survey Methodology, 19: 199-204.
Singh, R. and Kishmore, L. 1975. On Rao, Hartley and Cochranfs method. sankhya, C, 37:88-94.
Tierney, L. 1990. LISP-STAT: An object-oriented environment for statistical computing and dynamic graphics. New York, NY: John Wiley. 397 p.
BIOGRAPHICAL SKETCH
Jeffrey S. Pontius is an Assistant Professor in the Department of Statistics and the Center for Applied Statistics, Kansas State University. His research interests are sampling and stochastic processes in ecology.