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**Additional info for Statistical Methods for Biostatistics and Related Fields**

**Example text**

Therefore it has to be proved that every linear combination of the ni + nx (Jˆi − Ji ) is also distributed as a normal probability distribution function. 8) corresponds to a weighted sum of independent normal probability distribution functions. 8) is distributed as a normal probability distribution function (Rao, 1973). 5. To characterize the joint probability distribution function Jˆi , we have to compute the covariance between the divergence. 4 Let π1 , . . , πm be m distinct populations. Let assume that the distribution of the populations has the previous regularity conditions.

The formula for estimating the variance of the ﬁxed eﬀects is similar to the computation of variances in the random-eﬀects model. However, the residuals are computed using the within-group estimator β W G Amemiya (1981). Options for the Fixed-eﬀects Model a) Robust standard errors Arelano and Bond (1987) suggests an estimator of the standard errors for β W G that is robust to heteroskedastic and autocorrelated errors uit : −1 N N Xi Xi Var (β W G ) = i=1 −1 N X i ui ui X i i=1 Xi Xi , i=1 where ⎡ ⎢ ⎢ Xi = ⎢ ⎣ ¯i xi1 − x ¯i xi2 − x ..

1) are independent and their distribution is symmetric about zero, then under H0 , f = f0 . In the absence of strong parametric assumptions, the functions f and f0 are not identiﬁable, see Efron, Storey and Tibshirani (2001). 3 A Fully Nonparametric Approach 49 Gamma distributions for f and f0 respectively. Efron, Tibshirani, Goss and Chu (2000) avoided such parametric assumptions and considered a nonparametric empirical Bayesian approach. , µr , Vr ) denotes the normal density function with mean µr and variance Vr , and the πr ’s are mixing proportions.