Necessary sample sizes for categorial data

This paper deals with the problem how to determine the necessary sample size for the estimation of the parameter π=(π1,...,πk) (πj ≥ 0, Σjπj=1) based on the vector f=(f1,...,fk) of relative frequencies with sample size n. The vector n-f has a multinomial distribution. For a given precision c, 0≤c≤1, and a given confidence number β, 0≤β≤1, there exists a smallest positive integer N⁰=N⁰(β, c, k) with P{|fj−πj|≤c; j=1, ...,k}≥β for all sample sizes n≥N⁰ and for all π. As results are given in this paper exact upper bounds for N⁰ and an improved asymptotical upper bound for N⁰ which is derived from the asymptotical multinormal approximation for the distribution of f.