# Questions tagged [absolute-value]

Question concerns the consequences of using the absolute value function in part of the definition of one or more statistics.

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### Expected absolute deviation greater than standard Laplace

Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that \begin{equation*} E[|x-a|] \geq |a|+...
21 views

### Probability that a random variable has a greater absolute value than the sample mean of iid random variable

I have encountered a problem which involves finding the probability that an observation of a random variable has a greater absolute value than the sample mean of an independent set of observations of ...
149 views

### Representation of the expectation of absolute value of the difference $Y-X$

Given the representations of the mean values $$E[Y]=\int(1-F(x))\,dx$$ and $$E[X]=\int(1-G(x))\,dx$$ where $F$ and $G$ are the distributions of $Y$ and $X$ respectively, can I use them to find the ...
117 views

### Expectation of the absolute value of the product of correlated jointly gaussians?

I am reading the Performer paper https://arxiv.org/abs/2009.14794. To understand their ReLU kernel used to approximate softmax attention, I need to evaluate $\mathbb{E}[ReLU(x^T w) \cdot ReLU(y^T w)]$ ...
68 views

### Convergence in probability to a constant and absolute value (?)

I am a bit loss with the convergence in probability and the absolute value. Let $X_n$ be a random variable defined in $\mathbb{R}$ with $\lim_{n \rightarrow \infty} E[X_n] = a$ and $V[X_n] = O(n^{-1})$...
23 views

### Sampling distribution of the slope estimator of a stationary non-linear AR(1)

Consider a stationary non-linear AR(1) model $$x_t = |\theta x_{t-1}|+w_t,$$ where $w_t$ is i.i.d. standard normal. Given a sample $\{x_0, \cdots, x_n\}$, I want to find an estimate of $\theta$ and ...
118 views

### Solve an inequality finding the upper bound

Suppose that there exists a constant $C$ such that the following relation holds for all $G$: \begin{equation*} \vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert \end{equation*} Suppose that ...
54 views

### Test which time series is closer to zero

I have two time series, A and B, which represent reactions of subject A and B to a certain medication measured at different point in times. I want to formally test that the reaction of patient A is ...
216 views