Generate a string with symbolic expression for expectation of powers and products of non-central (raw) sample moments of an arbitrary order.
one_combination(powvect, smpsize = "n")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.
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.
symbol to be used for sample size. Defaults to "n".
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.
one_combination(c(5, 0, 2, 1))
Run the code above in your browser using DataLab