91Ó°ÊÓ

Comment. For a random variable X, the skewness, denoted by

\({\gamma _1}\)

is defined as

\({\gamma _1} = E\left( {\frac{{{{(X - \mu )}^3}}}{{{\sigma ^3}}}} \right)\)

where\(E(X) = \mu ,Var(X) = {\sigma ^2}\$ ,and\$ E\left( {{X^3}} \right) < \infty .\)

Assume that\({\bar X^*}\)is the random variable taken from the distribution\({F_n}\). Then, it is true that

\(E\left( {{{\bar X}^*}} \right) = \bar X = \mu \)

and the variance is given by

\(Var(X) = \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} \)

Next, the expected value of\({\left( {{X^*} - \mu } \right)^3} = {\left( {{X^*} - \bar X} \right)^3}\) is

\(E\left( {{{\left( {{X^*} - \mu } \right)}^3}} \right) = \frac{1}{n}{\left( {{X_i} - \bar X} \right)^3}.\)

Finally, the sample skewness\({M_3}\)and the skewness of the sample distribution\({F_n}\)are identical because

\(\begin{aligned}{l}{\gamma _1} &= E\left( {\frac{{{{(X - \mu )}^3}}}{{{\sigma ^3}}}} \right)\\ &= \frac{{\frac{1}{n}{{\left( {{X_i} - \bar X} \right)}^3}}}{{{{\left( {\frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} } \right)}^{3/2}}}} = {M_3}\end{aligned}\)

This is true because the third moment is finite.

To manually compute the skewness, you may use the given data and formula

\({M_3} = \frac{{\frac{1}{n}{{\left( {{X_i} - \bar X} \right)}^3}}}{{{{\left( {\frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} } \right)}^{3/2}}}}.\)

As software is used to simulate the bootstrap estimate of the bias of the sample skewness, you may also use the code. To simulate the \(\nu = 1000\) bootstrap samples, they should first compute the sample skewness of the initial sample. Denote \({M_3}\) the initial data skewness and \(M_3^{*(i)},i = 1,2,...,\nu \) the sample skewness of the bootstrap samples. The estimated bootstrap bias is then the mean of value\(M_3^{*(i)} - {M_3}\)

The estimate of the standard error is the sample standard deviation of \(M_3^{*(i)},i = 1,2,...,\nu .\)

The result obtained is the following. The sample skewness of the initial data is\({M_3} = 1.218\) , the bootstrap estimate of the bias is -0.282, and the estimate of the sample standard deviation is 0.547. Note that the values should change every time you run the software below. The given code is from RStudio.