### Revision(s):

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Revision #3 to TR16-015 | 24th March 2016 15:54
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#### The uniform distribution is complete with respect to testing identity to a fixed distribution

**Abstract:**
Inspired by Diakonikolas and Kane (2016), we reduce the class of problems consisting of testing whether an unknown distribution over $[n]$ equals a fixed distribution to this very problem when the fixed distribution is uniform over $[n]$. Our reduction preserves the parameters of the problem, which are $n$ and the proximity parameter $\e>0$, up to a constant factor.

While this reduction yields no new bounds on the sample complexity of either problems, it provides a simple way of obtaining testers for equality to arbitrary fixed distributions from testers for the uniform distribution.

**Changes to previous version:**
This version contains an appendix detailing a comment that appears right after Cor 13.

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Revision #2 to TR16-015 | 10th February 2016 11:22
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#### The uniform distribution is complete with respect to testing identity to a fixed distribution

**Abstract:**
Inspired by Diakonikolas and Kane (2016), we reduce the class of problems consisting of testing whether an unknown distribution over $[n]$ equals a fixed distribution to this very problem when the fixed distribution is uniform over $[n]$. Our reduction preserves the parameters of the problem, which are $n$ and the proximity parameter $\e>0$, up to a constant factor.

While this reduction yields no new bounds on the sample complexity of either problems, it provides a simple way of obtaining testers for equality to arbitrary fixed distributions from testers for the uniform distribution.

**Changes to previous version:**
Haste has caused an error in my claims regarding the open problems posed originally in Section 4. These are only partially addressed by the current results.

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Revision #1 to TR16-015 | 8th February 2016 16:28
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#### The uniform distribution is complete with respect to testing identity to a fixed distribution

**Abstract:**
Inspired by Diakonikolas and Kane (2016), we reduce the class of problems consisting of testing whether an unknown distribution over $[n]$ equals a fixed distribution to this very problem when the fixed distribution is uniform over $[n]$. Our reduction preserves the parameters of the problem, which are $n$ and the proximity parameter $\e>0$, up to a constant factor.

While this reduction yields no new bounds on the sample complexity of either problems, it provides a simple way of obtaining testers for equality to arbitrary fixed distributions from testers for the uniform distribution.

**Changes to previous version:**
The open problems stated in prior version are easy to resolve, to a large extent, using the results of Valiant and Valiant (2011). Details are provided in the revised Sec 4. In addition, some typos were fixed and some clarification were made.

### Paper:

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TR16-015 | 4th February 2016 22:26
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#### The uniform distribution is complete with respect to testing identity to a fixed distribution

**Abstract:**
Inspired by Diakonikolas and Kane (2016), we reduce the class of problems consisting of testing whether an unknown distribution over $[n]$ equals a fixed distribution to this very problem when the fixed distribution is uniform over $[n]$. Our reduction preserves the parameters of the problem, which are $n$ and the proximity parameter $\e>0$, up to a constant factor.

While this reduction yields no new bounds on the sample complexity of either problems, it provides a simple way of obtaining testers for equality to arbitrary fixed distributions from testers for the uniform distribution.