A goodness-of-fit Lempel-Ziv test for equiprobable binary sequences

Abstract

Let according to hypothesis H 0 the elements of random sequence X 1 ,...,X n be independent and have equiprobable distribution on the set {0,1}. We propose a goodness-of-fit test for the hypothesis H 0 based on the Lempel-Ziv statistics. A sequence of length n = 2mT is divided into 2m blocks of (equal) length T, for these blocks we compute values W 1 (T),...,W 2m (T) of Lempel-Ziv statistics and if the hypothesis H 0 is true, these values are independent and identically distributed. The test is based on the statistic ˜W(2mT) = (W 1 + W 2 + ... + W m ) − (W m+1 + W m+2 + ... + W 2m ). For this test we find limit distribution of statistic ˜W(2mT) and also obtain an estimate of the rate of convergence to the limit normal distribution. To compute the distributions of statistics W k (T) we apply formulae proposed in a previous paper by V.G. Mikhailov (these formulae may be found in [4])