Stat.UStatistic.Order­M

For a tuple length and a number of available sample positions, the collection of ordered injective tuples consists of all ordered selections of the specified length whose entries are distinct positions among the first specified number of observations.

For a tuple length and a number of available sample positions, the ordered injective-tuple count is the real-valued number of ordered selections of that length with distinct entries among the first specified number of observations.

For a domain size and a codomain size, the equivalence between injective maps and embeddings identifies every injective map from an $m$-element index set to an $n$-element index set with the corresponding embedding, and conversely.

Given an independent and identically distributed sample on a measurable sample space, with a specified sample-space measure and observation-space measure, an order-$m$ real-valued kernel, and a sample size, the fixed-order U-statistic maps each sample outcome to the average of the kernel over all ordered $m$-tuples of distinct observations among its first $n$ observations.

Given a measurable observation space and an order-$m$ real-valued kernel on it and a measure on that space, the population mean of the kernel is its integral under the $m$-fold product of that measure.

Given an order $m$, a distinguished coordinate, a value in the observation space, and values for every remaining coordinate, the completed coordinate tuple assigns the given value to the distinguished coordinate and the supplied remaining values to all other coordinates.

Given an order-$m$ real-valued kernel, a distinguished coordinate, and a measure on the observation space, the coordinate-specific first Hoeffding projection maps a proposed value at that coordinate to the kernel integrated over the product measure for all other coordinates, minus the kernel's population mean.

Given a positive order $m$, an order-$m$ real-valued kernel, and a measure on the observation space, the first Hoeffding projection is the coordinate-specific first projection obtained by treating the first coordinate as distinguished.

Given a positive order $m$, an order-$m$ real-valued kernel, and a measure on the observation space, the higher-order residual kernel maps each $m$-tuple to the original kernel value minus its population mean and minus the sum of all coordinate-specific first Hoeffding projections at that tuple.

Given an observation space and a real-valued binary kernel, the corresponding two-coordinate kernel maps an ordered pair to the binary kernel evaluated at its first and second entries.

For a finite coordinate family and a sample size, the collection of injective sample assignments consists of all assignments sending distinct coordinates to distinct positions among the first $n$ observations.

Given an independent and identically distributed sample on a measurable sample space, with a specified sample-space measure and observation-space measure, a finite coordinate family, a real-valued kernel indexed by that family, and a sample size, the normalized finite-kernel statistic maps each sample outcome to the average kernel value over every injective assignment of the coordinate family to the first $n$ sample positions. This is an average when the family has at most $n$ members; otherwise there are no such assignments, the normalizing count is zero, and the value is zero by the inverse-of-zero convention.

Given an observation space, an order, and one real-valued function of an observation for each coordinate, the ordered-product kernel maps an ordered $r$-tuple to the product of its coordinate-specific function values.

Given an independent and identically distributed sample on a measurable sample space, with a specified sample-space measure and observation-space measure, an order, one real-valued function of an observation for each coordinate, and a sample size, the normalized ordered-product statistic maps each sample outcome to the average, over all injective ordered $r$-tuples from its first $n$ observations, of the product of the corresponding coordinate-specific function values. For $r > n$ there are no such tuples and the value is zero by convention.

For a measurable observation space, an order $m$, a real-valued kernel of that order, and an observation distribution, the first-order influence function maps each observation to the sum of the kernel's first Hoeffding projections over all $m$ coordinate positions.

For an i.i.d. sample on a measurable sample space, with observations in a measurable observation space and their probability distribution, a positive integer order, a real-valued kernel of that order, and a sample size, the higher-order Hájek remainder statistic is the order-$m$ U-statistic of the kernel remaining after all of its first-order Hoeffding projections have been removed.

For an i.i.d. sample on a measurable sample space, with observations in a measurable observation space and their probability distribution, a positive integer order, and a real-valued kernel of that order, the order-$m$ degeneracy-negligibility condition asserts that the kernel's higher-order Hájek remainder, multiplied by $\sqrt n$, converges in probability to zero as the sample size tends to infinity.

For a partial matching, its matching size is the number of selected left coordinates, equivalently the number of paired coordinates.

For two nonnegative coordinate-set sizes, the empty partial matching selects no coordinate on either side and therefore contains no pairs.

For a partial matching between coordinate sets of sizes $r$ and $s$, the merged coordinate set contains every left coordinate together with precisely those right coordinates that are not selected by the matching.

For a partial matching and a left coordinate, the left-coordinate injection assigns that coordinate its own position in the merged coordinate set.

For a partial matching and a right coordinate, the right-coordinate injection assigns the coordinate to its matched left-coordinate position when it is matched, and otherwise to its own separate position in the merged coordinate set.

For two nonnegative coordinate-set sizes, the partial-matching representation equivalence bijects partial matchings with a selected subset of each coordinate set and a bijection between the two selected subsets.

For two finite sets, the collection of bijections between them is itself a finite collection.

For a finite set, the collection of all its finite subsets is itself a finite collection.

For two nonnegative coordinate-set sizes and a prescribed number of pairs, the fixed-size partial-matching family is the finite set of all partial matchings having exactly that prescribed number of paired coordinates.

For a measurable observation space, a measure on that space, a nonnegative integer kernel order $m$, and a real-valued kernel of $m$ observations, the order-$m$ kernel second moment is $\int g(z)^2\,dP^m(z)$, where $P^m$ is the product measure of $m$ independent draws from the given measure.

For two nonnegative orders, a family of real-valued left factors indexed by the first order, a family of real-valued right factors indexed by the second order, and a partial matching between their indices, the merged product kernel assigns to each collection of observations indexed by the merged coordinates the product of all left and right factors, evaluating matched factors at their common observation and unmatched factors at separate observations.

For two nonnegative orders, a measure on a measurable observation space, families of real-valued left and right factors, and a partial matching between their indices, the merged product moment is the integral of the associated merged product kernel under independent draws from that measure, one draw for each merged coordinate.