UniformFhatY

UniformFhatY is a nonparametric estimator for the distribution function F_Y of a real-valued random variable Y. It is a smoothed alternative to the empirical distribution function (EDF), designed to produce a monotone, bounded estimate of F_Y by applying a uniform kernel smoothing on the data. The construction uses a sample Y_1, ..., Y_n and a bandwidth parameter h > 0 to control smoothing.

One practical form of UniformFhatY uses the cumulative uniform kernel. For a given y, the estimator is defined as

F_hat_Y(y) = (1/n) sum_{i=1}^n max(0, min(1, (y - Y_i)/h)).

Intuitively, each observation contributes a linearly increasing amount to F_hat_Y over an interval of width h, with full contribution once y exceeds Y_i. The resulting F_hat_Y is nondecreasing in y and lies between 0 and 1, reducing the jaggedness of the raw EDF while retaining the basic cumulative structure.

As with other kernel-based methods, UniformFhatY depends on the bandwidth h. Smaller h decreases bias and more closely approximates F_Y, while larger h reduces variance but increases smoothing. Under standard regularity conditions, and with h shrinking appropriately as n grows, UniformFhatY converges to F_Y. The estimator is particularly useful for constructing confidence bands for the distribution function and for downstream nonparametric inference that requires a smooth, monotone CDF estimate.

Relation to related methods: it is a smoothed variant of the empirical distribution function achieved via a uniform kernel, sharing goals with kernel CDF estimators while emphasizing the cumulative distribution form. See also empirical distribution function, kernel density estimation, and probability integral transform.