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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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### 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 ...
419 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 ...
274 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 ...
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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, $f_X(x)=\theta x^{\theta-1}$...
368 views

### 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 ...
879 views

### 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 ...
768 views

### Proof: Pivotal Quantity

Can anyone give me a clue of how to address this theorem? Suppose that $T$ es a real-valued statistic. Suppose that $Q(t,\theta)$ is a monotone function of $t$ for each value of $\theta\in \Theta$. ...
280 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 ...
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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 ...
153 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 $f(u)$ ...
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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 ...
741 views

### 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 ...