LP.moment: Finds LP moments of a random variable or comoments of two random variables
Description
Evaluates $m$ LP moments of a random variable.
Estimates LP-comoment matrix of order $m \times m$ between $X$
and $Y$ , i.e., covariance between the LP
transformations of $X$ and $Y$; where the random variables could be discrete or continuous.
Usage
LP.moment(x, m)
LP.comoment(x, y,zero.order = TRUE, m)
Arguments
x
The observations on the variable $X$.
y
The observations on the variable $Y$.
zero.order
Logical argument set to TRUE if zero-order LP comoments are required.
m
The number of LP moments to be found using LP.moment;
or
The order of LP - comoment matrix.
Value
A vector of LP moments.
A matrix of LP co-moments between X and Y.
Details
LP moments of a general random variable (discrete or continuous) is defined as
$$\mbox{LP}[j;\, X] = \mbox{LP}[j, 0; X, X] = \mbox{E}[X \; T_{j}(X; X)].$$
LP comoments are the cross-covariance between higher-order orthonormal LP
score functions $T_j(X; X)$ and $T_k(Y ; Y )$
$$\mbox{LP}[j, k; X, Y ] = \mbox{E}[T_j(X; X)\,T_k(Y ; Y )].$$
Zero-order LP-comoments are defined as
$$\mbox{LP}[j, 0; X, Y] = \mbox{E}[T_j(X; X)\,Y],$$and
$$\mbox{LP}[0, k; X, Y] = \mbox{E}[X\, T_k(Y; Y)].$$
References
Mukhopadhyay S. and Parzen E. (2014). LP approach to
statistical modeling.arXiv:1405.2601.
Parzen E. and Mukhopadhyay S. (2013a). LP Mixed Data
Science:Outline of Theory. arXiv:1311.0562.