By J. O. Irwin (auth.), Walter Joh. Ziegler (eds.)
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Additional resources for Contribution to Applied Statistics: Dedicated to Professor Arthur Linder
An example of Plan II. The problem is to estimate the total number N of fish that traverse a channel in a season. A shunt provides an alternate path, attracting into the shunt some average fraction P of the fish. It is a fairly simple matter throughout the season to count the fish that traverse the shunt, but not so easy to count the fish that traverse the channel. However, it is possible to count on a few selected days the fish that traverse the channel. Comparison of the counts of fish that traverse shunt and channel provides an estimate of P.
4) That is, X is an unbiased estimator of A, and x is an unbiased estimator of a. (1 1) a VarX = N N-n 2 - - -0'2 - == N2 - - N-1 n n N Varx = N - n N-1 ~ == (~_~) 0'2. n n N 2 ' (5) (6) Variances of Estimators of a Total Population 47 For the reI-variances Cx2 C x2 = = (1 IV1) --;; - C2. (7) All this is well known. The proofs are in any book on sampling. Plan II. P fixed, N may be known or unknown; na random variable. To select the sample, start with sampling unit No. 1. Accept it or reject it, depending on a side-play of random numbers.
Remark 2. We note that for Plan II, E x7 = (liN) 1: a7 = a 2 a2 for any member i of the sample. Hence any in the sample is an unbiased estimator of a 2 a 2 , and a sample of size n = 1 provides an estimate of Var X (noted privately by my friend and colleague the late William N. Hurwitz). x; + Remark 3. The appendix shows for illustration all the possible samples of n = 1 for P = Q = 1/2 that can be drawn from a frame of N = 2 sampling units, along with calculations and comparisons with some of the formulas just learned, and with some that will appear in section 5.
Contribution to Applied Statistics: Dedicated to Professor Arthur Linder by J. O. Irwin (auth.), Walter Joh. Ziegler (eds.)