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 ...
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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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