one_combination {Umoments}

Generate symbolic expression for expectation

Description

Generate a string with symbolic expression for expectation of powers and products of non-central (raw) sample moments of an arbitrary order.

Usage

one_combination(powvect, smpsize = "n")

Arguments

powvect

vector of non-negative integers representing exponents j_1, \dots, j_m of non-central moments in expectation (see "Details"). The position (index) of an element of this vector indicates a corresponding moment, e.g. for E(\overline{X}^5 \overline{X^4}), powvect = c(5, 0, 0, 1). Thus the vector will have m elements if m'th is the highest moment.

smpsize

symbol to be used for sample size. Defaults to "n".

Details

For a zero-mean random variable X and a sample X_1, \dots, X_n, find E(\bar{X}^{j_1} \overline{X^2}^{j_2} \overline{X^3}^{j_3} \cdots \overline{X^m}^{j_m}), where overline{X^k} = 1/n \sum_{i = 1}^n X_i^{k} is a k'th non-central sample moment. The expression is given in terms of sample size and true moments \mu_k of X. These expectations can subsequently be used for generating unbiased central moment estimators of an arbitrary order, Edgeworth expansions, and possibly solving other higher-order problems.

Value

A string representing a symbolic expression for further processing using computer algebra (e.g. with Sage or SymPy), for calculating numeric values, or to be rendered with Latex.

Examples

one_combination(c(5, 0, 2, 1))