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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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What is a good journal for submitting my article on a conjecture in theoretical statistics, re: ancillary complement for correlation?

I'm working on a draft of a statistics article, and I'd like to plan for the journal where I'll ultimately submit. My problem is, the article topic is somewhat abstract—it's a conjecture in ...
2 votes
2 answers
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Pivotal quantity and confidence interval [closed]

Let $X$ be a random variable with p.d.f.: $$f(X|\theta) = \frac{e^{x-\theta}}{(1+e^{x-\theta})^2}$$ where $-\infty<x<\infty$ and $-\infty<\theta<\infty$ Use the pivotal method to verify ...
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1 vote
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Interpretation of distribution that appears when calculating CI for population mean

Let $X \sim \mathcal{N}(\mu, \sigma)$ be the model for a normally distributed population, described by the probability density function $f_{X}(x; \mu, \sigma)$. We can denote $\mathbf{X} = (X_1, X_2, \...
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Is it useful to find a $\chi^2$ pivotal quantity?

My statistics class makes a big deal out of the following fact: for any random sample $X_1, \ldots, X_n$ with continuous, invertible cdf $F(x;\theta)$, $$-2\sum_i\ln F(X_i;\theta) \sim \chi^2(2n).$$ ...
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3 votes
2 answers
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Question on solution of Casella and Berger Exercise 9.10: Showing that $Q(t,\theta)$ is a pivot

My question concerns Exercise 9.10 of Statistical Inference by Casella and Berger: On page 428 the authors say In general, suppose the pdf of a statistic $T$, $f(t|\theta)$, can be expressed in the ...
Leonidas's user avatar
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2 votes
0 answers
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Pivotal question about bootstrap confidence intervals

I'm reading through Efron's work on bootstrapping and I have a few questions. There are a few assumptions that are eluded to but not really explicitly stated. (1) Do we have to have a statistic that ...
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1 vote
2 answers
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"Centered" linear regression in point-slope form: pivot distribution and notation

I'm a "pure math" probabilist who's been roped into teaching an undergraduate statistics course, despite little experience with statistics per se, and I'm trying to stay one chapter ahead of ...
Nate Eldredge's user avatar
1 vote
0 answers
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Pivotal quantity of exponential model [closed]

I'm attempting to solve this problem but it's been driving me crazy. The goal it's to find a pivotal quantity of the model below and a symmetric confidence interval for $\theta$ with an $(1-\alpha)$ ...
Mariano Peñas's user avatar
1 vote
0 answers
33 views

Using pivotal quantities for transformed parameters

Setup Suppose we have some iid data $X_1, \ldots, X_n$ that arises from a distribution with parameters $\theta$ and $\rho$. $\theta$ is the parameter of interest, and $\rho$ is a nuisance parameter. ...
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2 votes
1 answer
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Why is this a generalized pivotal quantity?

I am reading "Generalized Confidence Intervals" by Weerahandi, and I'm trying to get my head around the definition of a generalized pivotal quantity. I understand what a (regular) pivotal ...
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Can the distribution of a pivot depend on known parameters?

I have a question about the distribution of a pivot. Different sources give slightly different definitions of a pivot. Some of them define it as a random variable $Q(\mathbf{X}, \theta)$ whose ...
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In general, do we have any strategy to find a pivotal statistic?

I have solved a number of exercises where I am asked to prove that a particular quantity is pivotal. The most popular example is the $Z$-score. If $Y\sim N(\mu, \sigma^2)$, then $Z=(Y-\mu)/\sigma$ is ...
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Pivotal quantity inference statistics of Exponential distribution?

Bus waiting times are distributed like this (they are independent) I know the average time is 8 minutes. I need to find the pivotal quantity of Theta parameter and after it of P. (P is the ...
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Did Fisher consider a joint fiducial distribution for the Gaussian model?

Consider the Gaussian model $y_i \sim_{\text{iid}} \mathcal{N}(\mu,\sigma^2)$, $i = 1, \ldots, n$, with unknown mean $\mu$ and unknown standard deviation $\sigma$. The random variable $t = \tfrac{\...
Stéphane Laurent's user avatar
2 votes
1 answer
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Confidence Interval Pareto Dist

Let $X_1,...,X_n$ be iid random variables from Pareto distribution with the following distribution $\theta a^{\theta} x^{-(\theta+1)}$, $x>a, \theta >1, a>0$ I have to find a $100(1-a)\%$ CI ...
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Multiplying by a constant (T random variable)

Suppose that $T$ has a distribution $t(n-1)$. If we were to multiply $T$ by $\frac{1}{\sqrt{n}}$, what would be the distribution of $\frac{T}{\sqrt{n}}$?
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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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1 answer
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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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1 answer
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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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0 votes
1 answer
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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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How do I calculate Confidence Interval for Gamma Distributed Pivotal Quantity? [closed]

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 ...
WorldGov's user avatar
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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 ...
Quality's user avatar
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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 $\...
omega's user avatar
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1 vote
1 answer
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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 ...
Silvia Rossi's user avatar
0 votes
1 answer
1k views

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 ...
HyeonPhil Youn's user avatar
7 votes
2 answers
898 views

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 ...
Stéphane Laurent's user avatar
4 votes
2 answers
93 views

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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3 votes
1 answer
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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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1 answer
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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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5 votes
1 answer
3k views

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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2 votes
1 answer
1k views

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 $\theta\in \Theta$. ...
Héctor Garrido's user avatar
3 votes
1 answer
772 views

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 ...
Lauren's user avatar
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1 vote
1 answer
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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 |...
Gejun's user avatar
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2 answers
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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 ...
user avatar
4 votes
1 answer
3k views

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 \...
PatternMatching's user avatar
6 votes
1 answer
4k views

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 ...
Joel Sinofsky's user avatar
5 votes
1 answer
273 views

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 $f(u)$ ...
Taylor's user avatar
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1 vote
2 answers
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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 ...
clarkson's user avatar
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5 votes
1 answer
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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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1 answer
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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 ...
Tim's user avatar
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2 votes
1 answer
352 views

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 ...
Tim's user avatar
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8 votes
1 answer
2k views

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 ...
Tim's user avatar
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8 votes
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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 ...
Tim's user avatar
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6 votes
1 answer
9k views

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 ...
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