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Results of Daskalakis, Kamath, and Wright {{cite|DaskalakisKW-18}} show that the ''worst-case'' sample complexity remains $\Theta(\sqrt{n}/\varepsilon^2)$.  Moreover, due to the quadratic dependence between Hellinger and total variation distances, both instance-specific bounds mentioned above apply, yet with possibly a quadratic gap between upper and lower bounds in terms of $\varepsilon$: leading to bounds on the instance-specific sample complexity $\Psi_{\rm H}$ of Hellinger identity testing of
 
Results of Daskalakis, Kamath, and Wright {{cite|DaskalakisKW-18}} show that the ''worst-case'' sample complexity remains $\Theta(\sqrt{n}/\varepsilon^2)$.  Moreover, due to the quadratic dependence between Hellinger and total variation distances, both instance-specific bounds mentioned above apply, yet with possibly a quadratic gap between upper and lower bounds in terms of $\varepsilon$: leading to bounds on the instance-specific sample complexity $\Psi_{\rm H}$ of Hellinger identity testing of
$$\Psi_{\rm TV}(q,\varepsilon) \leq \Psi_{\rm H}(q,\varepsilon) \leq \Psi_{\rm TV}(q,\varepsilon^2).$$
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$$\Psi_{\rm TV}(q,\varepsilon) \leq \Psi_{\rm H}(q,\varepsilon) \leq \Psi_{\rm TV}(q,\varepsilon^2)$$.
  
 
What is the right dependence on $\varepsilon$ of $\Psi_{\rm H}$?
 
What is the right dependence on $\varepsilon$ of $\Psi_{\rm H}$?
  
 
''Note that in both instance-specific bounds obtained for $\Psi_{\rm TV}$, there exist (simple) examples of $q$ where $\varepsilon$ ends up ''in the exponent,'' so this quadratic gap is not innocuous even for constant $\varepsilon$.''
 
''Note that in both instance-specific bounds obtained for $\Psi_{\rm TV}$, there exist (simple) examples of $q$ where $\varepsilon$ ends up ''in the exponent,'' so this quadratic gap is not innocuous even for constant $\varepsilon$.''

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