Questions tagged [sufficient-statistics]
A sufficient statistic is a lower dimensional function of the data which contains all relevant information about a certain parameter in itself.
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Sufficient statistics and parametric bootstrapping [closed]
Does the resampling step of the bootstrap method require to have the sample entirely, or a sufficient statistic suffices?
In general, the bootstrapping can be nonparametric or parametric. In the ...
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Is $\log(X_1+X_2)$ a sufficient statistic for $\beta$?
I have trouble finding the following sufficient statistics.
How do you do this?
$$X\sim \Gamma(\alpha, \beta)$$
$$f(x;\alpha, \beta)=\frac{e^{-x/\beta}x^{\alpha-1}}{\Gamma(\alpha)\beta^\alpha}$$
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Sufficient statistic for function of parameter
Suppose we are estimating $\tau(\theta)=\theta e^{-\theta}$ from $X_1,...,X_n \sim \mathrm{G}(\theta,r)$ (G is the gamma distribution) then it is easily shown that $T=\sum_{i=1}^n \ln(X_i)$ is ...
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Data normalization and sufficient statistic
I was taught that when we feed our data to machine learning algorithm (e.g. SVM), we should first normalize our data.
Suppose I have a set of data $X = \{x_1,x_2,...,x_n\}$, I knew two-way of ...
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Understanding Sufficient Statistics
As I began my study of sufficient statistics I stumbled upon a definition that puzzled me. The conditional probability distribution of the sample values given an estimator $\hat{\Theta}=\hat{\theta} $ ...
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3 parameter Exponential Family and sufficient statistics
This is a homework problem. I've derived the following distribution from an earlier part in the problem $$ f_{X_1,X_2}(x_1,x_2) = \dfrac{\Gamma(x_1+x_2+r)\alpha_1^{x_1}\alpha_2^{x_2}\theta^r}{\Gamma(r)...
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Finding a sufficient statistic
Consider an i.i.d. sample $(X_{1},\ldots, X_{n})$ where the $X_{i}$ have density
$f(x) = k \cdot \exp(−(x − θ)^4)$ with $x$ and $\theta$ real, obtain the sufficient statistic and its dimension. What ...
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Complete sufficient statistic
I've recently started studying statistical inference. I've been working through various problems and this one has me completely stumped.
Let $X_1,\dots,X_n$ be a random sample from a discrete ...
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Kolmogorov's paper defining Bayesian sufficiency
I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper
Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
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Does Fisher's factorization theorem provide the pdf of the sufficient statistic?
From Wikipedia
Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is $ƒ_θ(x)$, then $T$ ...
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Likelihood estimation using iid normal samples
Given an i.i.d. sample $X = (x_{1}, \dots, x_{n}) \sim N(\mu, 1)$. I have been asked to show that the likelihood of $\mu$ based on the whole sample is proportional to the likelihood based on $\bar{x}$ ...
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Are these statistics sufficient?
Question (Casella and Berger 6.5):
Let $X_1 \ldots X_n$ be independent random variables with pdfs:
$f(x_i|\theta)= \begin{cases} \frac{1}{2i\theta}, & -i(\theta - 1)<x_i<i(\theta+1) \\ 0,...
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Sufficient statistics for layman
Can someone please explain sufficient statistics in very basic terms? I come from an engineering background, and I have gone through a lot of stuff but failed to find an intuitive explanation.
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Showing that the order statistic $X_{(n)}$ is sufficient
I have some trouble showing sufficiency for largest order statistic ${x}_{n}$.
This is from Casella's text, problem 1.6.3.
Let ${p}_{\theta}$ be a density function.
${p}_{\theta}(x)=c({\theta})f(x)$ ...
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Sufficient Statistic
I have a question:
Does a sufficient statistic have to be one to one? For example, can $T(x) = x^2$ or $T(x) = |x|$ be sufficient statistics? I know that one to one functions of sufficient ...
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