I encountered the following problem in my research:
Suppose there are $N$ random variables that are independent and identically distributed (IID). The probability density function (PDF) of these random variables $f(x)$ is a unimodal function symmetrical about $0$ (i.e., $f(x)$ is non-decreasing within $(-∞,0)$, and for any $x$, $f(x) = f(-x)$ holds. for example, the distribution can be uniform distribution, normal distribution, Cauchy distribution with mean $0$, etc.).
For a given real number $x_0$, Sort these random variables as $X_1, X_2, …, X_N$ such that $$|X_1-x_0|\leq |X_2-x_0| \leq … \leq |X_N-x_0|$$
For example, if $N = 3$, the $N$ random variables are randomly chosen as $-0.5, 1.5, 5$, and $x_0 = 1$, then $X_1 = 1.5, X_2 = -0.5, X_3 = 5$.
Let $Y_i = |\frac{X_1+X_2+…+X_i}{i}-x_0|^r (i=1,…,N, r = 1\ or\ 2)$, then for any $x_0$ and $f(x)$, does the inequality
$$EY_1 \leq EY_2 \leq… \leq EY_N$$
always hold? Where $E$ denotes the expected value.
The inequality above is tested via the Monte Carlo method for cases where the distributions are uniform distribution, normal distribution, and Cauchy distribution. Details can be seen in https://math.stackexchange.com/questions/4039555/an-inequality-of-expected-value-of-random-variables since I cannot post figures here...
Moreover, is it possible to derive the PDF of $Y_i$?
Answers or ideas for either $r=1$ or $r=2$ would be so grateful!
