All Questions
Tagged with sufficient-statistics distributions
25
questions
2
votes
1
answer
86
views
Sufficient Statistic for a family of distributions consisting of Poisson family and Bernoulli family
Suppose $(X_1, . . . ,X_n)$ is an i.i.d. sample from the distribution $f_{\theta,k}(x)$, where $\theta \in (0, 1)$ and $k = 1, 2$. Assume that $$f_{\theta, k}(x)=\begin{cases} \text{Poisson($\theta)$},...
- 207
1
vote
0
answers
37
views
Sufficient Statistic for a finite family of Normal distributions
Suppose we have a finite family of normal distributions $P=\{N(0, 1), N(0, 2), N(1, 2), N(2, 2)\}$ and we want to find a sufficient statistic for this family. Intuitively it is clear that as the means ...
- 207
5
votes
1
answer
188
views
A lemma concerning the distribution of sufficient statistic from exponential family
I understand Lemma 8 in Chapter 1 from Lehmann's Testing Statistical Hypotheses [or Lemma 2.7.2 in Lehmann and Romano] as follows:
If the pdf of an exponential family is $$p_{\theta}(x)=\exp\bigg\{\...
- 65
4
votes
3
answers
824
views
How to prove any one-to-one function of minimal sufficient statistic is minimal sufficient?
So I want to prove that any one-to-one function of minimal sufficient statistic is also minimal sufficient. Here is my proof:
Let $T$ be a minimal sufficient statistic and $f$ is a one-to-one function ...
- 181
0
votes
1
answer
151
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Understanding the Importance of "Sufficiency" within Statistics
I am trying to better understand what it means to be a "sufficient statistic".
"In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown ...
0
votes
1
answer
76
views
Sufficient Statistic for Absolutely Continuous Distribution [duplicate]
The following is a homework problem. Please tell me if my solution is correct and if not please point out my mistakes.
Let $x_{1}, x_{2},...,x_{M}$ be i.i.d. samples from the absolute continuous ...
- 101
0
votes
0
answers
167
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Sufficient statistic for a given distribution from exponential form
Given a particular form, i can verify whether it is sufficient statistic or not using $\frac{p_\theta(x_1,x_2...x_n)}{p_\theta(T(x_1,x_2...x_n))}$ is independendent of $\theta$ then i can say $T(\bar ...
1
vote
0
answers
2k
views
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:
$$ L = (2\pi\sigma^2)^{-n/2} \cdot ...
0
votes
1
answer
41
views
replication of minimal sufficient statistic
Suppose we have a minimal sufficient statistic for observations $X_1, ...,X_n$ that are i.i.d from distribution $f(X|\theta)$, namely $T(X) = (T_1,...,T_k)$ which is a $k$ dimensional statistics. Now ...
- 135
0
votes
1
answer
503
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Finding the conditional distribution of single sample point given sample mean for $N(\mu, 1)$
Suppose that $X_1, \ldots, X_n$ are iid from $N(\mu, 1)$. Find the conditional distribution of $X_1$ given $\bar{X}_n = \frac{1}{n}\sum^n_{i=1} X_i$.
So I know that $\bar{X}_n$ is a sufficient ...
- 51
1
vote
0
answers
65
views
Minimal sufficient statistics
Suppose we have data $X = X_1,\ldots,X_n$, $Y = Y_1,\ldots,Y_n$ that is i.i.d. generated by a distribution $\mathbb{P}_\theta$. Let $T$ be a test statistic such that that $T(X) = T(Y)$ if and only ...
- 11
1
vote
1
answer
222
views
Gaussian sufficient statistic calculation
Consider the Gaussian model
$$
Y_i = \beta + \epsilon_i,\, i = 1, \cdots, n,\; \mbox{where}\; \epsilon_i
\stackrel{i.i.d.}{\sim} \mathcal{N}(0, \sigma^2),
$$
parametrized by $\beta$, with known $\...
- 3,348
3
votes
1
answer
209
views
Proving completeness of highest-order statistic using Leibnitz' Rule
Suppose that $X_1,...,X_n$ are iid with common pdf given by $$f(x;\theta)=2e^{2x}\theta^{-2}I( x<log(\theta)).$$
I am tasked with finding a complete-sufficient statistic for $\theta$, and I have ...
- 1,276
3
votes
2
answers
128
views
Minimal sufficient statistics of increasing dimensionality (not equal to the number of observations)
Restricting the attention to the case of fixed parameters support, it's my understanding that (minimal) sufficient statistics of fixed dimensionality, i.e. a fixed number of of them, exists in, and ...
- 31
4
votes
1
answer
83
views
When family of DF's $\mathcal{P}$ fail to be dominated by a measure $\mu$
On the topic of minimal sufficient statistics, there is an important theorem which requires the family of probability distributions $\mathcal{P}$ is dominated by some measure $\mu$.
As I understand it,...
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