Asymptotic behaviour of the weighted Renyi, Tsallis and Fisher entropies in a Bayesian problem

We consider the Bayesian problem of estimating the success probability in a series of conditionally independent trials with binary outcomes. We study the asymptotic behaviour of the weighted differential entropy for posterior probability density function conditional on $x$ successes after $n$ conditionally independent trials when $n\to\infty$. Suppose that one is interested to know whether the coin is approximately fair with a high precision and for large $N$ is interested in the true frequency. In other words, the statistical decision is particularly sensitive in a small neighbourhood of the particular value $\gamma=1/2$. For this aim the concept of the weighted differential entropy introduced in [1] is used when it is necessary to emphasize the frequency $\gamma$. It was found that the weight in suggested form does not change the asymptotic form of Shannon, Renyi, Tsallis and Fisher entropies, but changes the constants. The leading term in weighted Fisher Information is changed by some constant which depends on the distance between the true frequency and the value we want to emphasize.