The use of summary statistics beyond the two-point correlation function to analyze the non-Gaussian clustering on small scales is an active field of research in cosmology. In this paper, we explore a set of new summary statistics -- the $k$-Nearest Neighbor Cumulative Distribution Functions ($k{rm NN}$-${rm CDF}$). This is the empirical cumulative distribution function of distances from a set of volume-filling, Poisson distributed random points to the $k$-nearest data points, and is sensitive to all connected $N$-point correlations in the data. The $k{rm NN}$-${rm CDF}$ can be used to measure counts in cell, void probability distributions and higher $N$-point correlation functions, all using the same formalism exploiting fast searches with spatial tree data structures. We demonstrate how it can be computed efficiently from various data sets - both discrete points, and the generalization for continuous fields. We use data from a large suite of $N$-body simulations to explore the sensitivity of this new statistic to various cosmological parameters, compared to the two-point correlation function, while using the same range of scales. We demonstrate that the use of $k{rm NN}$-${rm CDF}$ improves the constraints on the cosmological parameters by more than a factor of $2$ when applied to the clustering of dark matter in the range of scales between $10h^{-1}{rm Mpc}$ and $40h^{-1}{rm Mpc}$. We also show that relative improvement is even greater when applied on the same scales to the clustering of halos in the simulations at a fixed number density, both in real space, as well as in redshift space. Since the $k{rm NN}$-${rm CDF}$ are sensitive to all higher order connected correlation functions in the data, the gains over traditional two-point analyses are expected to grow as progressively smaller scales are included in the analysis of cosmological data.
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