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G. Casella and Berger (1990)). The confidence set W(X_,Y_) with the confidence coefficient 1 — 7 then becomes W(X,Y) = {R : -2lnL(R\X,Y) < x$(l)} , where x ^ l ) , 0 < 7 < 1, is the upper 1007-percent point of the x 2 distribution with 1 degree of freedom. This approach has been implemented by Madansky (1965) and Easterling (1972). The advantage of the asymptotic algorithms of interval estimation is that they can be carried out for practically any distribution. However, a shortcoming is that these techniques very often run into serious difficulties and provide crude unreliable results when R is close to zero or one and the sample sizes are relatively small.

In this situation, the samples X_ and Y_ may be of unequal sizes and R = 30 The Theory and Some Useful Approaches Readers interested to learn more about empirical Bayes models and methods are referred to Carlin and Louis (2000) or the paper by Casella (1985). 1 Interval Estimation The Theory In many applications just knowing a point estimator is not sufficient. For illustration, consider a medical application where X and Y represent responses by an old treatment and a new treatment A and B, respectively; the aim is to decide whether one should abandon the old treatment in favor of the new one.

Thus, constructing an estimator a\ for a2R, one can assert that P{—z1/2 < (R — R)/(TR < z1/2) « 1 - 7 for arbitrary small 7 > 0 provided n\ and n2 are fairly large. Here, za is the (1 — a) percentile, or the upper a-cut-off point of the standard normal distribution, namely, za is the solution of the equation r = a. 62) Confidence intervals for estimators of R based on normal approximations have been studied by Church and Harris (1970), Nandy and Aich (1994a) and Gupta et al. (1999) among others.

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A hierarchical Bayesian approach to modeling embryo implantation following in vitro fertilization (2 by Dukk V.

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