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 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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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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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 ...
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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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"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 ...
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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 $(1-\alpha)$ ...
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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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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{\...
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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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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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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 ...
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