Consider $K$ independent Laplace variables $X_i$ ($1 \leq i \leq K$) with mean 0 and scale $\lambda$. Let $X′$ be the variable taking the value of the Laplace variable whose absolute value is the minimum among all $X_i$'s. Due to the randomness of $X_i$'s, $X′$ may not always equal a fixed $X_i$. I would like to know what the CDF of $X′$ is. Does it also follow Laplace distribution? How to prove or disprove that? Many thanks!
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Consider a set of random variables $Y_i\stackrel{\text{iid}}{\sim}\text{Exp}(\lambda)$ (this is the scale parameterization not the rate parameterization, so it has scale parameter $\lambda$). Let $B_i\stackrel{\text{iid}}{\sim}\text{Bernoulli}(\frac12)$, where the $Y$'s and $B$'s are all mutually independent.
Let $Z_i=2B_i-1$, so the $Z$'s are random $+1$'s and $-1$'s ("random signs").
Let $X_i=Z_iY_i$. Clearly $X_i\stackrel{\text{iid}}{\sim}\text{Laplace}(0,\lambda)$.
Let $Y_{(1)}$ be the first order statistic of $Y_1,Y_2,...,Y_n$.
It's a simple matter to show that $Y_{(1)}\sim\text{Exp}(\lambda/n)$.
$P(Y_{(1)}>y)=P(Y_{1}>y,Y_{2}>y,...,Y_{n}>y)$
$\qquad=\exp(-y/\lambda)\cdot \exp(-y/\lambda)\cdot....\cdot\exp(-y/\lambda)$
$\qquad=\exp(-ny/\lambda)$
Hence $Y_{(1)}\sim\text{Exp}(\lambda/n)$.
Let $W_1=X_k$ be the particular Laplace observation corresponding to $Y_{(1)}$; that is $Y_{(1)}=Y_k$.
Then $W_1=Z_k\cdot Y_{(1)}$. Since the $Z$'s are independent of the $Y$'s, this attaches a random sign to $Y_{(1)}$, making $W_1\sim\text{Laplace}(0,\lambda/n)$.
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$\begingroup$ I guess we did not explain the problem clearly. X' can be negative. Its value is the value (NOT absolute value) of the Laplace variable $X_i$ whose absolute value is minimum. We tried the Monte Carlo method, which shows that $X'$ is very likely to follow the Laplace distribution. $\endgroup$– NeedHelpCommented Jul 7, 2015 at 3:30
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$\begingroup$ Yes, your question could be clearer. No, your Monte Carlo method doesn't indicate how likely it is to follow the Laplace distribution (since it could be similar to but subtly different from it) -- only that it's not inconsistent with it being Laplace. What's the context in which this arises? $\endgroup$– Glen_bCommented Jul 7, 2015 at 3:48
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$\begingroup$ Cool. Thanks, Glen! It is a question arisen from a research problem. We tried to prove it by establishing the symmetry of $X'$ from the $X_i$'s symmetry of $X_i$ and making use of the minimum property of the exponential distribution, but were not 100% sure. $\endgroup$– NeedHelpCommented Jul 7, 2015 at 4:22
self-studytag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. $\endgroup$