# Questions tagged [pivot]

In statistics a pivot, or pivotal quantity is a function of unknown parameters and data whose distribution doesn't depend on the values of the unknown parameters - used to construct confidence intervals.

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### Pivotal Quantities for confidence intervals - Why does it work?

I try to get an intuition on, why pivotal quantities are used to construct confidence intervals. First, I show how I understand the algorithm: For example let $x_1,...,x_n \in \mathbb{R}$ be ...
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### How can I construct an asymptotic confidence interval using a specified pivotal quantity and the score test?

Let there be a random sample $X_1,...,X_n \sim Poison(\theta)$, where $\theta>0$ is unknown. Show that $P(\mathbf{X},\theta)=\frac{\bar{X}-\theta}{\sqrt{\bar{X}/n}}$ is asymptotically pivotal, then ...
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### How can I compare the lengths between these two confidence intervals? [closed]

Let $X\sim Beta(\theta,1)$. I was asked to find two confidence intervals and compare their lengths. The first confidence interval is for $Y=-(logX)^{-1}$ over the set $[y/2,y]$. The second ...
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### How can I use this pivotal quantity to find the shortest length confidence interval for $\theta$?

Let $X_1,\cdots,X_n \sim f(x|\theta)=\frac{\theta}{x^2}, x> \theta$ be a random sample where $\theta>0$ is unknown. I want to use $\frac{\theta}{X_{(1)}}$ as a pivotal quantity. How can I use ...
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### (Definition) Why is Z not a pivotal quantity while estimating the mean of a normal distribution when variance is unknown?

The definition of a pivotal quantity as given in Wikipedia is: In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's ...
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### How to get this confidence interval from a pivotal quantity

Suppose we have $n$ iid samples from $Exp(1,\eta)$ This distribution is $e^{-x+\eta}$ for $x \ge \eta$ I want to understand why the following is a correct symmetric $100 \gamma$ confidence ...
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### Deriving an expression for a confidence interval for σ^2 using the asymptotic distribution of √n(σ̂^2−σ^2)

We have We have $X1,…,Xn i.i.d N(μ,σ^2)$where $μ$ is known and $σ^2$ isn't known. $σ̂^2=(\frac{1}{n})∑(X_i−μ)^2$. First of all what I did, I derived an equitailed 95% confidence interval for $σ^2$....
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### Find the confidence interval, uniform distribution

Let $X_1,..,X_n$ a random sample of $X$~$U[-\theta,\theta]$, $\theta>0$. Find the confidence interval for $\theta$. I'm trying to find a pivotal quantity with the maximum and minimum, but I can ...
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### Why is a pivot quantity not necessarily a statistic?

From Wikipedia In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on the unknown parameters 1 (also ...