By De Cicco V., Fusco N., Verde A.
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Extra resources for A chain rule formula in BV and application to lower semicontinuity
M n(n − 1) i< j ˆ are exact formulae, It is important to observe that M and M unlike the approximations M =N N − n1 N −1n N (Y i − R X i ) 2 1 ˆ = N N ( N − n) M n(n − 1) (Y i − Rˆ X i ) 2 i∈s where R = Y/X, Rˆ = y/x and 1 1 y= Yi, x = Xi n i∈s n i∈s due to COCHRA N (1977). For the approximations n is required to be large and N much larger than n. cls 20 dk2429˙ch02 January 27, 2005 11:19 Chaudhuri and Stenger hard to calculate even if X i is known for every i = 1, . . , N . ˆ ˆ it is enough to know only X i for i ∈ s, but to use M To use M one must know X i for i ∈ / s as well.
A resulting q is called a controlled design and a corresponding scheme of selection is called a controlled sampling scheme. Quite a sizeable literature has grown around this problem of finding appropriate controlled designs. The methods of implementing such a scheme utilize theories of incomplete block designs and predominantly involve ingeneous devices of reducing the size of support of possible samples demanding trials and errors. But RA O and NIGA M (1990) have recently presented a simple solution by posing it as a linear programming problem and applying the well-known simplex algorithm to demonstrate their ability to work out suitable controlled schemes.
4 Therefore, with p SRSWR of size n, = Y (1 − Y ) ≤ V p ( N y) = N 2 ≤ σ yy n N2 4n . From Ep y = Y we derive that the random interval N y±3 N2 3 1 = N y± √ 4n 2 n covers the unknown N Y with a probability of at least 8/9. It may be noted that Y is regarded as fixed (nonstochastic) and s is a random variable with a probability distribution p(s) that the investigator adopts at pleasure. It is through p alone that for a fixed Y the interval t ± 3σ p (t) is a random interval. In practice an upper bound of σ p (t) may be available, as in the above example, or σ p (t) is estimated from survey data d plus auxiliary information by, for example, σˆ p (t) inducing necessary changes in the above confidence statements.