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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Proving Incompleteness of joint sufficient statistic
Let $X_1, ..., X_n$ be a sample from the continuous density $C~exp(-(x-\theta)^4)$ (for $ -\infty < x < \infty$) with $\theta$ as unknown parameter. Show that the minimal sufficient statistic is ...
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Minimal sufficient statistic: a measurability issue in a well-known theorem
Given a statistical model $\{\mathbb{P}_\theta\,|\,\theta\in\Theta\}$ on $(\Omega,\mathscr{F})$, and given a real-valued random variable $X$, we say a real-valued random variable $T=T(X)$ is a ...
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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 $$p_{\theta}(x)=\exp\bigg\{\...
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Prove covariance between sufficient statistic and logarithm of base measure in exponential family is equal to zero
Exponential family form is
$$f_X(x) = h(x)\exp(\eta(\theta)\cdot T(x) - A(\theta))$$
I know
$$\operatorname{Cov}(T(x), \log(h(x)) = 0.$$
But how can I prove it?
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Show that $T=\sum_{i=1}^n X_i$ is a sufficient statistic for $p$ [duplicate]
I try to use the definition of sufficient statistic to prove that
Suppose that $X_1,\dots, X_n$ is an iid random sample from $X\sim \mathrm{Bernoulli}(p)$. Show that $T=\sum_{i=1}^n X_i$ is a ...
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Verifying the statistics are complete and sufficient for two parameter Pareto distribution
Let$(X_1,...,X_{n})$ be a random sample from the Pareto distribution
with pdf density $\theta a^{\theta} x^{-(\theta+1)}I_{(a,\infty)}(x),$ where $\theta>0$ and $a>0$
$\textbf{(i)}$ Show that ...
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Why is median not a sufficient statistic? [duplicate]
Suppose a random sample of $n$ variables from $N(\mu,1)$, $n$ odd. The sample median is $M=X_{(n+1)/2}$, the order statistic of the middle of the distribution.
How to prove that sample median is not a ...
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Solving the Neyman-Scott problem via Conditional MLE
In section 2.4 of the book Essential Statistical Inference by Boos and Stefanski, the authors discuss the idea conditional likelihoods and illustrate their usefulness by describing how they can be ...
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Why is the weak likelihood principle not a theorem?
The weak likelihood principle (WLP) has been summarized as: If a sufficient statistic computed on two different samples has the same value on each sample, then the two samples contain the same ...
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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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Is $ T = X_1 +5 X_2 $ sufficient estimator of $p$? [duplicate]
If $ X_1 $ and $ X_2$ are $\textrm{Ber}(p)$ random variables, examine the sufficiency of $ T_1 = X_1 + 5 X_2 $ for $ p .$
I have no idea on how to proceed, I tried to use the conditional ...
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Concrete example of what Sufficient Statistics is [closed]
Having read articles to try to understand Sufficient Statistics.
Sufficient statistics for layman
A sufficient statistic summarizes all the information contained in a sample so that you would make ...
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The equivalence between two sufficient statistics for the same parameter $\theta$
Exercise. Let $X_1,\cdots,X_{n}$ be i.i.d.r.v.'s from $N(\theta,1),$ where $\theta$ is unknown.Show the statistic $T(\mathbf{X})=\sum_{i=1}^{n}X_{i}/n=\bar{X} $ is sufficient for $\theta$.
The answer ...
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Sufficiency and completeness of truncated distribution
[From Theory of Point Estimation (Lehmann and Casella, 1999, Exercise 6.37)]
Let $P=\{P_\theta:\theta \in \Theta\}$ be a family of probability
distributions and assume that $P_\theta$ has pdf $p_\...
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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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