Two Lempel – Ziv goodness-of-fit tests for nonequiprobable random binary sequences

Let the hypothesis $H_p$ mean that elements of the sequence $X_1,\ldots,X_n$ are independent and identically distributed: $\mathbf{P}\{X_i=1\}=p$, $\mathbf{P}\{X_i=0\}=1-p$, where $p\in(0,1)$. Earlier two goodness-of-fit tests for the hypothesis $H_{0.5}$ were proposed based on the possibility of exact computation of Lempel – Ziv statistics distributions. In this paper these tests are generalized for any $p\in(0,1)$. For each test a sequence of length $n=mrT$ is divided into blocks of length $T$, for these blocks Lempel – Ziv statistics $W_1(T),\ldots, W_{mr}(T)$ are computed. The first test for $r=2$ is based on the statistic $\tilde W(2mT)=(W_1+\ldots+W_m)-(W_{m+1}+\ldots+W_{2m})$, its distribution is symmetric about zero. The statistic of the second test is $\tilde \chi^2(mrT)=\max_{1\le k\le m} \chi_{(k)}^2(T)$, where $\chi_{(1)}^2(T),\ldots,\chi_{(m)}^2(T)$ are values of chi-square statistics computed for $(W_{1,1}(T),\ldots, W_{1,r}(T)),\ldots,(W_{m,1}(T), W_{m,2}(T), \ldots, W_{m,r}(T))$ correspondingly. For statistics of both tests limit distributions are found, for the statistic of the first test the rate of convergence to the limit normal distribution is given.