The seven columns produced by this function summarize the results where the signs of
stochastic dominance order values (+1 or -1) are weighted by `wt=c(1.2,1.1, 1.05, 1)`

to
compute an overall result for all orders of stochastic dominance by a weighted sum for
the crietria Cr1 and Cr2. The weighting is obviously not needed for the third criterion Cr3.

1 2 |

`mtx` |
The data matrix in the first column is paired with all others. |

`dig` |
Number of digits for reporting (default |

`verbo` |
Make |

`rnam` |
Make |

`wt` |
Allows user to choose a vector of four alternative weights for SD1 to SD4. |

`sumwt` |
Sum of weights can be changed here =4(default). |

The reason for slightly declining weights on the signs from
SD1 to SD4 is simply that the local mean comparisons
implicit in SD1 are known to be
more reliable than local variance implicit in SD2, local skewness implicit in
SD3 and local kurtosis implicit in SD4. The source of slightly declining sampling
unreliability of higher moments is the
higher power of the deviations from the mean needed in their computations.
The summary results for all
three criteria are reported in one matrix called `outVote`

:

typ=1 reports ('Y', 'X', 'Cause', 'SD1apd', 'SD2apd', 'SD3apd', 'SD4apd') nameing variables identifying 'cause' and measures of stochastic dominance using absolute values of kernel regression gradients (or amorphous partial derivatives, apd-s) being minimized by the kernel regression algorithm while comparing the kernel regresson of X on Y with that of Y on X.

typ=2 reports ('Y', 'X', 'Cause', 'SD1res', 'SD2res', 'SD3res', 'SD4res') and measures of stochastic dominance using absolute values of kernel regression residuals comparing regresson of X on Y with that of Y on X.

typ=3 reports ('Y', 'X', 'Cause', 'r*x|y', 'r*y|x', 'r', 'p-val') containing generalized correlation coefficients r*, 'r' refers to. Pearson correlation coefficient p-val is the p-value for testing the significance of 'r'

Prints three matrices detailing results for Cr1, Cr2 and Cr3. It also returns a grand summary matrix called ‘outVote’ which summarizes all three criteria. In general, a positive sign for weighted sum reported in the column ‘sum’ means that the first variable listed as the input to this function is the ‘kernel cause.’ For example, crime ‘kernel causes’ police officer deployment (not vice versa) is indicated by the positive sign of ‘sum’ (=3.175) reported for that example included in this package.

The output matrix last column for ‘mtcars’ example has the sum of the scores by the three criteria combined. If ‘sum’ is positive, then variable X (mpg) is more likely to have been engineerd to kernel cause the response variable Y, rather than vice versa.

The European Crime data has all three criteria correctly suggesting that high crime rate kernel causes the deployment of a large number of police officers.

Prof. H. D. Vinod, Economics Dept., Fordham University, NY.

'Generalized Correlation and Kernel Causality with Applications in Development Economics' in Communications in Statistics -Simulation and Computation, 2015, http://dx.doi.org/10.1080/03610918.2015.1122048

See Also `somePairs`

1 2 3 4 5 6 7 8 9 10 11 | ```
## Not run:
some0Pairs(mtcars) # first variable is mpg and effect on mpg is of interest
## End(Not run)
## Not run:
data(EuroCrime)
attach(EuroCrime)
some0Pairs(cbind(crim,off))
## End(Not run)
``` |

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