All Questions
Tagged with estimators inference
51
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What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves? [duplicate]
!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
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Cramer-Rao lower bound for the variance of unbiased estimators of $\theta = \frac{\mu}{\sigma}$
Let $X_1, \cdots, X_n$ be a sample from the $N(\mu, \sigma^2)$ density, where $\mu, \sigma^2$ are unknown.
I want to find a lower bound $L_n$ which is valid for all sample-sizes $n$ for the variance ...
- 63
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1
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Fisher Information for $\bar{X}^2 - \frac{\sigma^2}{n}$ with $X_1, \dots, X_n$ normally distributed
I need to find the Fisher Information for $T = \bar{X}^2 - \frac{\sigma^2}{n}$ with $X_1, \dots, X_n$ normally distributed sample with unknow mean $\mu$ and know variance $\sigma^2$. For this I'm ...
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I am looking for a method to estimate a threshold function for binary outcome data
A literature search yielded no obvious answers, so I wonder here if there any feasible methods to estimate the following.
Suppose I have data $Y_i, \vec X_i$ indexed by $i = 1, \cdots, N$.
Note that $...
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Extending Minimal sufficient statistics to arbitrary dimension
I am wondering if the following reasoning is correct regarding minimal sufficiency and dimension. Given $X_1,\dots,X_n$ i.i.d. $N(\mu,1)$, we know that the sample mean $S = \bar{X}$ is a minimal ...
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How can we compare biases of two estimators with no parametric form?
I was reading in my textbook that the bias of a statistical estimator $\hat{\theta}_n$ can be quantified as $B(\hat{\theta}_n,\theta)=E[\hat{\theta}_n-\theta]$. This expectation seems to be w.r.t. to ...
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How do we know the true value of a parameter, in order to check estimator properties?
For example, we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate. However, if we already know the true value of the ...
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If the bias of an estimator is expressed as a difference, what do you call the ratio of the estimator and true value?
If $Bias(\hat{\beta}) = (\beta - \mathbb{E}[\hat{\beta}])$, is there a term to describe the quantity $\frac{\mathbb{E}[\hat{\beta}]}{\beta}$?
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When is it better to have an unbiased estimator instead of one that has a smaller risk?
I just learned that for $X_1, \ldots X_n \sim N(\mu, \sigma^2)$ i.i.d, the sample variance $\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2$ is unbiased, and it is in fact UMVUE.
However, it is not ...
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350
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What is Bayes estimator of $\theta$ when loss function is $L(\theta,a)=I(|\theta -a|>\delta)$?
Suppose $X$ given $\theta$ has pdf $f(x\mid \theta)=e^{-(x-\theta)}I(x>\theta)$ and there is a standard Cauchy prior on $\theta$. As part of an exercise, I am trying to find a Bayes estimator of $\...
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286
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Fisher matrix for a discrete distribution
Let $\mathbf{X} = \{X_1, \ldots, X_n\}$ be a sample of i.i.d. variables following a discrete distribution with parameters $\mathbf{p}^T = (p_1, p_2, p_3)$. How can I find the Fisher information matrix ...
- 405
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119
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Statistical Inference: Definition of contrast function
Reading a paper recently regarding results on parameter estimation and I came across the terminology "contrast function" which was a function constructed out of a sample.
If I compare it to ...
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Estimating $\theta$ based on censored data when $X_i\sim \text{Uniform}(0,\theta)$ with $\theta\ge 1$
Suppose $(X_i)_{1\le i\le n}$ are i.i.d $\text{Uniform}(0,\theta)$ random variables where $\theta \ge 1$. We observe $Y_i=\min(X_i,1)$ instead of $X_i$. I wish to estimate $\theta$ based on the data $(...
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64
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Instrumental variables - OLS - estimation
I have a question regarding the OLS estimation, in the case of an estimation with instrumental variables:
We assume the linear model $𝒚= 𝑿\beta+𝒖$ with $Z$ = instrumental variables.
Multiplying the ...
- 95
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141
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Should a Test Statistic Consist of a Consistent Estimator for the Parameter of Interest?
Suppose that we want to test the following hypothesis:
$$H_0: \theta \in \Theta_0\quad vs \quad H_1: \theta\in \Theta_0^c.$$
Suppose that our test statistic is $T_n$.
Then, should the test statistic $...
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