The estimation of \(\hat{j}_0\) is achieved via a moderate deviation-based approach. The probability that an estimator, computed from a pilot sample size \(\nu\), exceeds a value z, the deviation above z is said to be a moderate deviation if its associated probability is polynomially small as a function of \(\nu\), and to be a large deviation if the probability is exponentially small in \(\nu\). The values of \(z=z_\nu\) that are associated with moderate deviations are \(z_\nu\equiv\bigl(C\,\nu^{-1}\,\log\nu\bigr)^{1/2}\), where \(C>\frac{1}{4}\). The null hypothesis that \(p_k=\frac{1}{2}\) for \(\nu\) consecutive values of k, versus the alternative hypothesis that \(p_k>\frac{1}{2}\) for at least one of the values of k, is rejected when \(\hat{p}_j^\pm-\frac{1}{2}>z_\nu\). The probabilities \(\hat{p}_j^+\) and \(\hat{p}_j^-\) are estimates of \(p_j\) computed from the \(\nu\) data pairs \(I_\ell\) for which \(\ell\) lies immediately to the right of j, or immediately to the left of j, respectively.
The iterative algorithm consists of an ordered sequence of "test stages" \(s_1, s_2,\ldots\) In stage \(s_k\) an integer \(J_{s_k}\) is estimated, which is a potential lower bound to \(j_0\) (when \(k\) is odd), or a potential upper bound to \(j_0\) (when \(k\) is even).
compute.stream(Idata, const=0.251, v, r=1.2)A named list containing:
Is the estimated index for which the Idata degenerate into noise
\(k=j0_est-1\)
The reason why the computation has ended - convergence or break condition
Is the sequence of estimated \(j_0\) in each iteration run, also showing the convergence behaviour
Is the preselected value of the parameter \(\nu\)
Input data is a vector of 0s and 1s (see prepare.idata)
Denotes the constant C of the moderate deviation bound, needs to be larger than 0.25 (default is 0.251)
Denotes the pilot sample size \(\nu\) related to the degree of randomness in the assignments. In each step the noise is estimated from the Idata as probability of 1 within the interval of size \(\nu\), moving from \(J_{s_{k-1}} -r \nu\) if \(k\) is odd or \(J_{s_{k-1}} +r \nu\) if \(k\) is even, until convergence or break (see r)
Denotes a technical constant determining the starting point from which the probability for \(I=1\) is estimated in a window of size v (see v, default is 1.2)
Eva Budinska <budinska@iba.muni.cz>, Michael G. Schimek <michael.schimek@medunigraz.at>
prepare.idata
set.seed(465)
myhead <- rbinom(20, 1, 0.8)
mytail <- rbinom(20, 1, 0.5)
mydata <- c(myhead, mytail)
compute.stream(mydata, v=10)
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