All Questions
Tagged with sufficient-statistics normal-distribution
28
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Show minimal sufficient statistic is not complete in normal distribution
Let $Z_i$ for $1 \leq i \leq n$ be a sample from the $N(ap, bp(1-p))$ density, where $a \gt 0, b \gt 0$ are known but $p \in (0,1)$ is an unknown parameter.
I have shown that $T = (\sum^n_{i = 1} Z_i, ...
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A lemma concerning the distribution of sufficient statistic from exponential family
I understand Lemma 8 in Chapter 1 from Lehmann's Testing Statistical Hypotheses [or Lemma 2.7.2 in Lehmann and Romano] as follows:
If the pdf of an exponential family is $$p_{\theta}(x)=\exp\bigg\{\...
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1
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Can the dimension of a (potentially) sufficient statistic exceed the dimension of the parameter it estimates?
I understand that if the dimension of a sufficient statistic exceeds that of the parameter it estimates, then that particular sufficient statistic won't be minimal. Now, in the following case, I ...
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How is $\text{Cov}(\bar{Y}, Y_i - \bar{Y}) = \dfrac{1}{n^2} \text{Cov} \left( \sum_{j = 1}^n Y_j, nY_i - \sum_{j = 1}^n Y_j \right)$?
I have this example of sufficiency:
Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. Hence
$$\...
- 2,096
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2
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Mean is not a sufficient statistic for the normal distribution when variance is not known?
According to the PDF here: https://www.math.arizona.edu/~tgk/466/sufficient.pdf, the sum of a sample of data is not a sufficient statistic for the normal distribution when the variance is unknown. ...
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2
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Sufficient statistics and randomized estimator in TPE
I've been reading Lehmann and Casella's Theory of Point Estimation 2nd Edition (TPE). In Chapter 1 Section 6 (pp.32-33), they introduce the idea "randomized estimator". Their explanation is, ...
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For iid $X_1, \dots, X_n \sim N(0,\sigma^2)$, get sufficient statistic $T = \sum_{i=1}^nX_i^2$, how to find unbiased estimator of $\sigma^a$
For $X_1, \dots, X_n \sim N(0,\sigma^2)$, we define a sufficient statistic $T = \sum_{i=1}^nX_i^2$. There is a positive number $a$. My question is how to find unbiased estimator of $\sigma^a$ using ...
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Do sufficient statistics for parameters of interest depend on whether nuisance parameters are known?
The definition of sufficient statistic is as follows:
A statistic $T(X_1,...,X_n)$ is sufficient for parameter $\theta$ if the conditional distribution of $X_1,...,X_n$, given that $T=t$, does not ...
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Sufficient Statistic for Variance (Normal Distribution)
Suppose $X_1, \dots, X_n \sim N(\mu, \sigma^2)$.
If $\mu$ is known and $\sigma^2$ is unknown, prove that $S^2$ is a sufficient statistics for $\sigma^2$.
Likelihood:
$$ L = (2\pi\sigma^2)^{-n/2} \cdot ...
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1
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Sufficient Statistic for Normal Distribution | Mean, Variance & Kurtosis
I have seen multiple times that a normal distribution is fully specified by mean and variance. It is obvious that the third moment is not necessary for a perfect normal distribution as it is 0. I ...
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Sufficient statistics in the uniform distribution case
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \mathbf{y})...
- 2,096
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$Y_1 + \dots + Y_n \sim N(n\mu, n\sigma^2)$ implies that $\bar{Y} \sim N(\mu, \sigma^2/n)$?
I have this example of sufficiency:
Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. Hence
$$\...
- 2,096
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Bayesian Linear Regression and the Exponential Family
In a straight forward linear regression model, assuming a fixed input $\mathbf{x}$, and additive noise with unit variance we can write:
\begin{equation}
p(y\mid \mathbf{x,w})=\frac{1}{\sqrt{2\pi}\...
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2
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Finding the form $g(T(\mathbf{y}), \lambda) \times h(\mathbf{y})$ for sufficiency statistic examples
I'm studying some notes that present examples of sufficiency:
Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y}...
- 2,096
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1
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503
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Finding the conditional distribution of single sample point given sample mean for $N(\mu, 1)$
Suppose that $X_1, \ldots, X_n$ are iid from $N(\mu, 1)$. Find the conditional distribution of $X_1$ given $\bar{X}_n = \frac{1}{n}\sum^n_{i=1} X_i$.
So I know that $\bar{X}_n$ is a sufficient ...
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