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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Show that Y is a pivotal quantity [closed]

Let $X_1, X_2,..., X_n$ denote a random sample from $Unif(0,\theta)$. Find a function of the MLE for $\theta$ that is a pivotal quantity. I have the sampling distribution of $X_n$, $f_{x_n}(x)=n[\...
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Find a pivotal quantity (with hint)

Let $X$ be a scalar random sample of from the following density: $$f(x|\theta) = \frac{2(\theta-x)}{\theta^2} \quad \quad \quad \text{for } 0 \leq x \leq \theta.$$ Find a pivotal quantity. (I have ...
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Simple pivot construction using an equivariant estimator

In the statistics course I am taking, there is the following example of a pivot: Given a symmetric distribution $F_0$, let $\mu$ be the parameter of interest and $\hat{\mu}$ be an equivariant ...
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How do I calculate Confidence Interval for Gamma Distributed Pivotal Quantity?

I'm studying confidence intervals and then I came across the following problem: It's said that a random variable X has Skewed Exponencial Distribution with parameters $\alpha >0$ and $v \in \...
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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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Studentization ad pivotal quantity for the slope in simple linear regression

Take the usual linear model: $$ Y=\alpha+\beta X + e$$ In good hypothesis (IID sample, normal error assumption $e \sim N(0,\sigma_e^2) $ and homoschedasticity) the distribution of the sample slope $\...
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Finding the distribution of $2\theta X_i^2$

I need to find the distribution of $2\theta X_i^2$ in order to show that $\sum_{i=1}^n 2\theta X_i^2$ is a pivot, (and thus $\sum_{i=1}^n 2\theta X_i^2 \sim N(0,1)$). $X_1,X_2,...,X_n$ are i.i.d. with ...
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Using Pivot Variable in Bootstrap Procedure

So this question is based on Introduction to Mathematical Statistics, Hogg&Craig In chapter4.9, it introduces bootstrap procedure and also informs that we can improve a pivot random variable ...
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What is a generalized confidence interval?

Quoting Weerahandi, Generalized Confidence Intervals (1993): Confidence interval (Property 1) --- Consider a particular situation of interval estimation of a parameter $\theta$. If the same ...
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Guessing at pivots

I'm trying to get a better idea of the intuition behind finding pivotal quantities. In the Casella & Berger statistical inference text, we have a $beta(\theta,1)$ pdf, $f_X(x)=\theta x^{\theta-1}$...
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Deriving confidence interval by inverting LRT statistic

Consider a random sample of size $n$ from the distribution with pdf $$f(y;\theta)=\theta y^{\theta-1}, 0<y<1, \theta > 0.$$ I want to find a $1-\alpha$ confidence interval for $\theta$ by ...
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Pivot for a confidence interval

I'm looking at an example in my lecture notes where $X_1, X_2,...,X_n$ are iid $N(\mu,\sigma^2)$, where $\sigma^2$ is known. $\bar{X}$ is an unbiased estimator of $\mu$. The pivot for the confidence ...
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Find pivotal quantity based on sufficient statistics

Let $(X_{1}, \dots X_{n})$ be a random sample of a random variable $X$ with pdf: $f(x|\theta) = \exp{(-(x-\theta))}\mathbb{1}_{{(\theta},{\infty)}}(x), \enspace \theta > 0$. How do I find the ...
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Proof: Pivotal Quantity

Can anyone give me a clue of how to address this theorem? Suppose that $T$ is a real-valued statistic. Suppose that $Q(t,\theta)$ is a monotone function of $t$ for each value of $\theta\in \Theta$. ...
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confidence interval and pivotal functions

I am struggling with this question; Given the density function: $$f(x;θ) = \begin{cases}\dfrac{2(θ−x)}{θ^2} &\text{ if }0< x<θ\\ 0 > &\text{otherwise}\end{cases}$$ the upper ...
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Find confidence interval via pivotal quantity?

Suppose $X_1, X_2, ..., X_n$ is a random sample from a population with pdf $$f(x|\theta) = \dfrac{1}{2\theta}e^{-|x|/\theta},x\in \mathbb{R}$$ The pivotal quantity is $\frac{2}{\theta}\sum_{i=1}^n |...
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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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Distribution of pivotal quantity

I'm attempting to determine whether a pivot can be used to construct a confidence interval for $\theta$ given that observations are iid and from the distribution below. Specifically: $f(x \mid \...
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Poisson confidence interval using the pivotal method

I am trying to build a confidence interval for the Poisson distribution using the pivotal method. I have the theory down but I am struggling to come up with $h(Y, \lambda)$, the probability ...
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Picking noninformative priors using pivotal quantities

In 'Bayesian Data Analysis' (Gelman, Carlin, Stern and Rubin) on page 64 it reads: "If the density of $y$ is such that $p(y-\theta|\theta)$ is a function that is free of $\theta$ and $y$, say $f(u)$ ...
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Pivotal to estimate lambda of a exponential

I am studying interval estimation by the method of pivotal quantities. Let $X_1, X_2, ..., X_n$ be a random sample from a p.d.f $f(x;\lambda)=\lambda e^{-\lambda x}, x>0,\lambda >0$. I have to ...
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Constructing a pivot-based confidence interval

So I've been working on a problem in my probability class on which I have become stuck. It involves X1 X2 ... Xn ~ Poisson(lambda) 1 - We were instructed to show ...
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Does pivoting a discrete CDF provide a pivot?

In Section 9.2.3 of Casella's Statistical Inference, they base their confidence interval construction for a parameter $\theta$ on a real-valued statistic $T$ with cdf $F_T(t| \theta)$. They first ...
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Can pivot be used for testing

A pivotal quantity $Q(X, \theta)$ can be used to construct a confidence interval. I was wondering if it can be used to construct a test statistic and rejection region? In simpler cases involving a ...
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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 ...
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pivotal statistic versus distribution free statistic

I was wondering what relations and differences are between pivotal statistic versus distribution free statistic? From Wikipedia a pivotal quantity or pivot is a function of observations and ...
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Pivotal quantities, test statistics and hypothesis tests

We are learning pivot functions, test statistics, and hypothesis testing at university but it makes no sense. I've tried reading my text book/notes, going through examples, etc., but the concepts seem ...