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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.

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.

For example, a Z-score based on normal data (one where you subtract the population mean and divide by population standard deviation) would be a pivotal quantity.

It is often used in the construction of confidence intervals.

Note: don't use this tag for pivot tables - use the pivot-table tag instead.

Reference: Wikipedia: Pivotal quantity

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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 unobservable parameters whose probability distribution…
Tim
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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 referred to as nuisance parameters). Note that a…
Tim
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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 experiment is repeated a large number of times to obtain…
Stéphane Laurent
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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 like a random guess and I'm clueless about how…
user3483
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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 distribution which does not depend on the parameter. Can…
Joel Sinofsky
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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 that X_bar was sufficient. Sol'n: I did this using factorization, and by finding T(x)…
Gosset
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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 assume that $T$ is a continuous random…
Tim
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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 pivotal quantity and an approximated confidence…
Zwart.Schaap
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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 $f(u)$ where $u = y - \theta$, then $y - \theta$ is a…
Taylor
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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 been given the following hint: Consider the…
kburr
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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, $f_X(x)=\theta x^{\theta-1}$, $0 < x < 1$. The text says that $X^\theta$ is a…
user21359
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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 not find any, can anyone give me a tip? EDIT: I can…
user72621
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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: $f(x \mid \theta) = \frac{2x}{\theta^2}$ with $0 < x < \theta$…
PatternMatching
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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 pivotal since $Z\sim N(0, 1)$, which does not…
TrungDung
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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 construct as asymptotic $1-\alpha$ confidence…
Jen Snow
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