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Inequalities compared

We know that the partial means m_rof a sequence of i.i.d. standardized random variables tend to 0 with probability 1. If we want P{m_k≥εfor some k ≥r}≤δ for given positive ε and δ, how large should we take r? Several (strong) inequalities for the distribution of partial sums providing an answer to this question can be found in the literature (Hájek -Rényi Robbins , Khan ). Furthermore there exist wellknown (weak) inequalities (Bienaymé -Chebyshev , Bernstein , Okamoto ) that give us values of rfor which P{m_r≥ε}≤δ. We compare these inequalities and illustrate them with numerical results for a fixed choice ofε and δ. After a general survey and introduction in section 1, the normal and the binomial distribution are considered in more detail in the sections 2 and 3, while in section 4 it is shown that the strong inequality essentially due to Robbins can give an inferior result for particular distributions.     

On Optimal Instrumental Variables Generators, with an Application to Hedge Fund Returns

In this paper, we propose a new benchmarking procedure lying on cumulants for computing the factor loadings in financial models of returns. We apply this technique to the well-known augmented Fama and French (J Fin Econ 43(2):153–193, 1997) model and compare it with another technique of ours based on higher moments. Our new procedure confirms the fact that the alpha is supposed to decrease when we disaggregate HFR indices to the level of individual funds while correcting for specification errors. Our new technique is therefore useful for hedge funds selection or ranking based on the alpha of Jensen corrected for specification errors. This technique will also be useful for calibrating other financial models of returns like the simple market model or the conditional alpha and beta models.