Compute U-statistics HOIF-type from Order 2 to 6 in pure R code. — calculate_u_statistics_pure_r

This function serves as a core computational component in higher-order influence function (HOIF) estimators in pure R code.

calculate_u_statistics_pure_r_six(Vector_1, Vector_2, A1, A2, A3, A4, A5)

Arguments

Vector_1: Numeric column vector of length $n$.
Vector_2: Numeric column vector of length $n$.
A1, A2, A3, A4, A5: Numeric $n \times n$ kernel matrices of the same dimension.

Value

A named list containing numeric U-statistic estimates:

U2: Second-order U-statistic
U3: Third-order U-statistic
U4: Fourth-order U-statistic
U5: Fifth-order U-statistic
U6: Sixth-order U-statistic

Details

Internally, the function constructs kernel matrices for orders 2 through 6 using recursive matrix operations and removes diagonal contributions to ensure degenerate U-statistics.

All diagonal elements of intermediate kernel matrices are removed to avoid self-interactions. Matrix multiplications are performed via `eigenMapMatMult()` and element-wise products via `hadamard()`. The exact formula of the output is: $$ \mathbb{U}_{n,m} = \frac{1}{\binom{n}{m} m!} \sum_{i_1 \ne \cdots \ne i_m} Vector_1[i_1] \cdot A1[i_1,i_2] \cdot A1[i_2,i_3] \cdots A1[i_{m-1},i_{m}] \cdot Vector_2[i_{m}] $$