  We study the multivariate integration problem for periodic functions from the
weighted Korobov space in the randomized setting. We introduce a new randomized
rank-1 lattice rule with a randomly chosen number of points, which avoids the
need for component-by-component construction in the search for good generating
vectors while still achieving nearly the optimal rate of the randomized error.
Our idea is to exploit the fact that at least half of the possible generating
vectors yield nearly the optimal rate of the worst-case error in the
deterministic setting. By randomly choosing generating vectors $r$ times and
comparing their corresponding worst-case errors, one can find one generating
vector with a desired worst-case error bound with a very high probability, and
the (small) failure probability can be controlled by increasing $r$
logarithmically as a function of the number of points. Numerical experiments
are conducted to support our theoretical findings.
