Is there a concentration bound for sample mode, when there exists a unique mode for a density $f$ that doesn't depend on $f$ (Essentially I am looking for a Chernoff type bound for mode)?.
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$\begingroup$ How do you define the sample mode? Is the underlying distribution, from which you sample, discrete? Even if so, what other conditions on that distribution are known? In view of re-scaling, it is easy to see that without additional conditions no concentration bound on the sample mode exists. $\endgroup$– Iosif PinelisCommented Aug 30, 2019 at 13:15
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$\begingroup$ $f$ is a discrete distribution. Could you direct me to some type of mode concentration results? I will try to figure out if they can be modified to my problem $\endgroup$– ChandramouliCommented Aug 31, 2019 at 1:19
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1$\begingroup$ Can you state a precise version of your question? $\endgroup$– Yuval PeresCommented Aug 31, 2019 at 5:26
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$\begingroup$ Let $X_{1}, X_{2}, X_{3}, \cdots $ are i.i.d with probability mass function $f:\{1,2,\cdots, L\} \rightarrow [0,1]$. Suppose that $f$ has a unique mode i.e there exists unique $m$ such that $p_m=\displaystyle\max^{L}_{i=1}p_i$. Let $E^{i}_{n}=\{k: X_k=i, 1\leq k\leq n \}$. Let $M_n$ be sample mode of $n$ samples i.e. $M_{n}:=\displaystyle\arg\max^L_{i=1}|E^{i}_{n}|$ (if there is a tie we take the smaller index). Is there a result that bounds $Pr(|M_n-m|>\epsilon)$ interms of $\epsilon$? $\endgroup$– ChandramouliCommented Sep 3, 2019 at 5:02
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$\begingroup$ I am looking for a bound that depends on $\epsilon, n, L$ and preferably not on $f$ $\endgroup$– ChandramouliCommented Sep 3, 2019 at 5:21
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