Answer the following questions a$,$ b$,$ c$,$ d with true or false

We consider the sample $x_1,$dots,$x_n$ as a realization of $x_1,$dots,$x_n$ which are i$.$i$.$d$.$ with distribution $P$ on $x.$

We are interested in the distribution of

hat$()=t(x_1,$dots,$x_n)$

where $t$:$x^{n}R$ is a sample function.

Determine for each of the following statements whether it is true or false.

a$.$ The sampling distribution of hat$()$ is given by the image measure $t(P^n)$

b$.$ When performing non$-$parametric bootstrap, the sampling distribution of hat$()$ is approximated by the empirical

measure

$lo{n}_n=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{}_{x_i}$

c$.$ When performing non$-$parametric bootstrap, the number $B$ of bootstrap samples must not be greater than

the number $n$ of observations in the dataset.

d$.$ When performing non$-$parametric bootstrap, new datasets $x_{b,1}^{**},$dots,$x_{b,n}^{***}$ are drawn which are merely a

permutation of $x_1,$dots,$x_n.$