# asycovTkTl: Asymptotic Covariance between T_k and T_l Values In nnspat: Nearest Neighbor Methods for Spatial Patterns

## Description

This function computes the asymptotic covariance between T_k and T_l values which is used in the computation of the asymptotic variance of Cuzick and Edwards T_{comb} test, which is a linear combination of some T_k tests. The limit is as n_1 goes to infinity.

The argument, n_1, is the number of cases (denoted as `n1` as an argument). The number of cases are denoted as n_1 and number of controls as n_0 in this function to match the case-control class labeling, which is just the reverse of the labeling in \insertCitecuzick:1990;textualnnspat.

The logical argument `nonzero.mat` (default=`TRUE`) is for using the A matrix if `FALSE` or just the matrix of nonzero locations in the A matrix (if `TRUE`) in the computations.

See page 80 of (\insertCitecuzick:1990;textualnnspat) for more details.

## Usage

 `1` ```asycovTkTl(dat, n1, k, l, nonzero.mat = TRUE, ...) ```

## Arguments

 `dat` The data set in one or higher dimensions, each row corresponds to a data point. `n1` Number of cases `k, l` Integers specifying the number of NNs (of subjects i and m in a_{ij}(k) a_{mj}(l)). `nonzero.mat` A logical argument (default is `TRUE`) to determine whether the A matrix or the matrix of nonzero locations of the A matrix will be used in the computation of N_s and N_t. If `TRUE` the nonzero location matrix is used, otherwise the A matrix itself is used. `...` are for further arguments, such as `method` and `p`, passed to the `dist` function.

## Value

Returns the asymptotic covariance between T_k and T_l values.

Elvan Ceyhan

## References

\insertAllCited

`covTkTl`, `covTcomb`, and `Ntkl`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```n<-20 #or try sample(1:20,1) Y<-matrix(runif(3*n),ncol=3) cls<-sample(0:1,n,replace = TRUE) #or try cls<-rep(0:1,c(10,10)) n1<-sum(cls==1) k<-1 #try also 2,3 or sample(1:5,1) l<-1 #try also 2,3 or sample(1:5,1) c(k,l) asycovTkTl(Y,n1,k,l) asycovTkTl(Y,n1,k,l,nonzero.mat = FALSE) asycovTkTl(Y,n1,k,l,method="max") ```