What test is appropriate for comparing the difference in accuracy between recognition tools? I have developed a tool that recognizes a set of six classes, and then tested and evaluated its ability to recognize these classes by using the F-score (aka F-measure). I then tested two other tools that recognize the same set of six classes and evaluated them on the same test set on which my tool is evaluated. The following table shows the f-measures calculated for the tools in the six classes (not the actual values):
        | My tool | Tool A  | Tool B
------------------------------------
class 1 |  0.431  |  0.297  |  0.327
class 2 |  0.388  |  0.348  |  0.334
class 3 |  0.979  |  0.826  |  0.790
class 4 |  0.290  |  0.389  |  0.238
class 5 |  0.990  |  0.730  |  0.642
class 6 |  0.886  |  0.516  |  0.566

Since the tools are all tested on same test set, I cannot use Kruskal-Wallis, or Mann–Whitney U test, as the data fails the independence assumption (as suggested by the accepted answer of THIS question). What test should I use to check if my tool is significantly different/better (pair-wise) than the other two tools?
EDIT: I need the test to be performed (pair-wise) based on accuracy measures, like the f-measure, not the raw result data used to calculate them.
EDIT 2: The problems I am trying to solve are binary classification problems, and I have six of such problems each of which is handled and tested separately. Each class type has its own and separate test dataset, which is used to test how accurate the three tools are in recognizing the corresponding class-type.
 A: You can probably produce a list of pairs with (estimated, original classes) for each tool instead of just F-measure. In that case I would suggest to use McNemar significance test to check improvements. Good paper which describe how to perform this test is
Salzberg, On Comparing Classifiers: Pitfalls to Avoid and a Recommended Approach, Data mining and knowledge discovery 1,(1997)
A: 1) In your approach, you want to perform significant test on some quality metric (in this case F-measure) for each of the classes of the problem. The much more common approach is to perform the statistical significance test on sub-samples of the data. In this case you would peroform a k-fold, and measure a quality metric for each test fold, for each of the three tools. 
One problem is that you want the quality metric to be F-measure and it is not obvious how to extend the F-measure idea for a multi-class problem (but I am sure someone already solved this problem). Once you solve how to use a single quality measure for each sub-sample and each tool, the canonical thing to do is to perform a Friedman test (your data is paired), and if the test result in a p-value low enough (traditionally 0.05 or below) then perform a Wilcoxon signed-rank pairwise comparison  followed by a p-value adjustment (for the multiple comparisons). In this case you want to know whether your tool is better than the other two, so there are only 2 pairwise comparisons, so I would use the Bonferroni correction (instead of using 5% as the p-value threshold, use 2.5%). 
2) But from a statistical significance point of view, there is nothing wrong with performing the test on the data as you presented. Each toll generates 6 sets of measures and you can perform the procedure above on these sets of measures: Friedman + Wilcoxon signed rank + Bonferroni adjustent. I do not see the problem you stated that the 6 sets of measures for each toll are not independent. At a first approximation the 6 values for each toll are independent.  In terms of your table, the data in each column are independent (the lines are not - they are paired and that is why you should use Friedman and Wilcoxon signed-rank).
