Given dimension \(p\), returns a dataframe containing the position of all derivatives of estimator of moment generating function \(\hat{M}_X(t)\), upto third/fourth order.
mt3_rev_pos(j1, j2, j3, p)mt3_pos(p)
mt4_pos(p)
mt3_rev_pos returns the position of this particular derivative
in the chain of all derivatives, up to third order.
mt3_pos an array contaning all position with respect
to index of \(j_1, j_2, j_3\).
mt4_pos an array contaning all position with respect to
the index of \(j_1, j_2, j_3, j_4\).
Index of the first variables
Index of the first variables, should be at least j1
Index of the first variables, should be at least j2
Dimension
The estimator of multivariate moment generating function is \(\hat{M}_X(t) = \dfrac{1}{n} \sum_{i = 1}^n \exp(t'X_i)\) The chain containing all derivatives up to the third order is $$ Z = \bigg(\hat{M}, \hat{M}^{001}, \dots \hat{M}^{00p}, \hat{M}^{011}, \hat{M}^{012}, \dots \hat{M}^{0pp}, \hat{M}^{111}, \hat{M}^{112}, \dots \hat{M}^{ppp}\bigg)' $$ and $$ \hat{M} = \hat{M}^{000}(t)= \hat{M}_X(t) $$ $$ \hat{M}^{j_1j_2j_3}(t) = \dfrac{\partial^k}{\partial t_{j_1} t_{j_2} t_{j_3}} \hat{M}(t) $$ where \(k\) is the number of \(j_1, j_2, j_3\) different from 0. Similar notation is applied when fourth derivatives is used.
mt3_rev_pos(1, 2, 2, p = 3)
p <- 3
mt3_pos(p)
mt4_pos(p)
Run the code above in your browser using DataLab