# one_combination: Generate symbolic expression for expectation In innager/Umoments: Unbiased Central Moment Estimates

## Description

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

## Usage

 1 one_combination(powvect, smpsize = "n") 

## Arguments

 powvect vector of non-negative integers representing exponents j[1], ..., 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], ..., X[n], find E(X-bar^{j_1} {X^2}-bar^{j[2]} \cdots {X^m}-bar^{j[m]}), where {X^k}-bar = 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 μ[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

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

innager/Umoments documentation built on Aug. 11, 2018, 10:12 p.m.