91Ó°ÊÓ

\({X_1},...,{X_n}\) a random sample from a normal distribution with unknown mean \(\theta \) and variance \({\sigma ^2}\)

The asymptotic distribution of\(g\left( {{{\bar X}_n}} \right)\)

Where,

\(g\left( x \right) = {x^3}\)

The distribution\({\bar X_n}\)is normally distributed with mean\(\theta \)and variance\(\frac{{{\sigma ^2}}}{n}\)

According to the delta method,

The asymptotic distribution of\(g\left( {{{\bar X}_n}} \right)\)should be the normal distribution with the mean,

\(g\left( \theta \right) = {\theta ^3}\)

The variance is,

\(\begin{array}{c}Variance = \left( {\frac{{{\sigma ^2}}}{n}} \right){\left[ {g'\left( \theta \right)} \right]^2}\\ = \frac{{9{\theta ^4}{\sigma ^2}}}{n}\end{array}\)

Therefore, the asymptotic distribution of\({\bar X_n}^3\)is follows a normal distribution with mean\({\theta ^3}\)and variance\(\frac{{9{\theta ^4}{\sigma ^2}}}{n}\)