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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
23 views

FInding a complete and sufficient statistic

I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class: Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
Harry Lofi's user avatar
2 votes
1 answer
122 views

Karlin-Rubin theorem: relationship between test statistic having the MLR property vs being sufficient

Let's suppose we are trying to compare two hypotheses for a single parameter $\theta$. The null hypothesis $H_0$ is that $\theta = \theta_0$, and the alternative is that $\theta ≥ \theta_0$. The ...
Mike Battaglia's user avatar
1 vote
0 answers
27 views

Sufficient statistic for the family of PERT distributions?

A beta distribution is one of the form $$ \text{constant}\times x^{\alpha-1} (1-x)^{\beta-1} \, dx \quad \text{ for } 0<x<1. $$ According to this Wikipedia article, the family of "PERT ...
Michael Hardy's user avatar
0 votes
1 answer
92 views

Unbiased estimator for parameter of random variables following a uniform distribution [duplicate]

Suppose $X_i$ are i.i.d. and have density $f_\theta(x) = \frac{1}{\theta}$ if $x \in (\theta, 2\theta)$ for positive $\theta$. $(\min_iX_i, \max_iX_i)$ is a sufficient statistic for $\theta$? To ...
johnsmith's user avatar
  • 335
2 votes
1 answer
111 views

Completeness of Gamma family

Let $X_1,...,X_n$ has a Gamma$(\alpha,\alpha)$ distribution. Find the minimal sufficient statistics. Is this a complete family? My attempt: I found the Minimal sufficient statistics is $T(x)=(\...
Cyno Benette's user avatar
1 vote
0 answers
50 views

Likelihood ratio as minimal sufficient statistics in infinite parameter space

I just read a question from here (Likelihood ratio minimal sufficient) and have some thoughts. Let me restate the question first: Consider a family of density functions $f(x|\theta)$ where the ...
Cyno Benette's user avatar
1 vote
2 answers
218 views

Showing that $X_{(1)}$ is sufficient for shifted exponential distribution

If the pdf of a random sample is $f(x)=e^{-(x-θ)}$ where $x \geq θ$, Show that $T=X_{(1)}$ is a sufficient statistic for $θ$. Can one show that $T$ is a sufficient statistic for $θ$ in the following ...
Wins94's user avatar
  • 11
2 votes
2 answers
132 views

Sufficient/complete statistic $\leftrightarrow$ injective/surjective map?

I can't understand the paragraph in Completeness (statistics) - Wikipedia: We have an identifiable model space parameterised by $\theta$, and a statistic $T$. Then consider the map $f:p_{\theta }\...
Y.D.X.'s user avatar
  • 60
1 vote
0 answers
28 views

How do I know which statistic is for which parameter when calculating joint sufficient statistics using factorization criteria?

For the normal distribution for example, after factorization we get $\mathcal{L} = (2 \pi \sigma^2)^{-\frac{n}{2}}\exp\left(-\frac{n\mu^2}{2\sigma^2}\right) \exp\left(-\frac{1}{2\sigma^2}\left(\sum_{i=...
gununes132's user avatar
2 votes
1 answer
75 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($\theta)$},...
user671269's user avatar
1 vote
0 answers
32 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 ...
user671269's user avatar
0 votes
1 answer
144 views

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, ...
Oscar24680's user avatar
1 vote
1 answer
54 views

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 ...
Stats_Rock's user avatar
3 votes
0 answers
182 views

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 ...
No-one's user avatar
  • 202
5 votes
1 answer
181 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\{\...
rryan's user avatar
  • 53
1 vote
1 answer
36 views

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?
user388375's user avatar
1 vote
0 answers
37 views

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 ...
Hermi's user avatar
  • 687
2 votes
1 answer
481 views

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 ...
Aleph Alpha's user avatar
10 votes
4 answers
311 views

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 ...
Diorne's user avatar
  • 101
2 votes
0 answers
117 views

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 ...
WeakLearner's user avatar
  • 1,461
2 votes
2 answers
450 views

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 ...
virtuolie's user avatar
  • 528
0 votes
0 answers
31 views

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 ...
WeakLearner's user avatar
  • 1,461
0 votes
0 answers
42 views

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 ...
simran's user avatar
  • 377
1 vote
0 answers
123 views

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 ...
mon's user avatar
  • 1,448
3 votes
1 answer
141 views

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 ...
Elisa's user avatar
  • 310
2 votes
1 answer
87 views

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_\...
WinnieXi's user avatar
1 vote
1 answer
99 views

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 ...
mathmicha's user avatar
1 vote
1 answer
118 views

How to prove that this statistic is not sufficient? [duplicate]

Problem. Given $X_1,X_2,X_3$ a random sample from the Bernoulli distribution with success $\theta$, show that the statistic $T= X_1+2X_2+3X_3$ is not sufficient. My attempt When I try to apply the ...
yahiro's user avatar
  • 97
3 votes
1 answer
377 views

Prove that the sum is sufficient using using the definition of sufficiency

If $X_1,\ldots,X_n$ is an IID random sample, with $X_i\sim\,\text{Ber}(\theta)$, prove that $Y = \sum_i X_i$ is sufficient using the definition of sufficiency (not the factorization criterion). Now ...
laurab's user avatar
  • 145
1 vote
0 answers
96 views

Finding UMVUE of a parameter in form of probability of discrete random variables

We have $X$ and $Y$ as independent discrete random variables both in ${1, 2, ...}$. Their pmf's are: $f(x|\alpha)=P(X=x)=\alpha(1-\alpha)^{x-1}, x=1, 2, ...$ $f(y|\beta)=P(Y=y)=-\frac{1}{\log\beta}\...
AlgoManiac's user avatar
1 vote
2 answers
613 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 ...
Alex He's user avatar
  • 121
1 vote
0 answers
94 views

Are there any (exponential) families without a minimal sufficient statistic?

Bahadur's theorem says that if a minimal sufficient statistic exists, then a complete sufficient statistic is also minimal sufficient. Are there any (homogenous, identifiable) families with a complete ...
Christian Chapman's user avatar
3 votes
1 answer
77 views

How to eliminate constant to derive the decision rule in terms of the sufficient statistic $\bar{X}$ for normal distribution means hypothesis test?

Suppose that we have a random sample, of size $n$, from a population that is normally-distributed. Both the mean, $\mu$, and the standard deviation, $\sigma$, of the population are unknown. We want to ...
user avatar
1 vote
1 answer
94 views

MLE of parameters for a difference of two Exponential IID

Suppose I have $X_1 \sim Exp(\theta_1)$ and $X_2\sim Exp(\theta_2)$. Then it is not difficult to show that $Y = X_1 - X_2$ will have density: $f_Y(y) = \frac{1}{\theta_1 + \theta_2}e^{-y/\theta_1}\...
s l's user avatar
  • 87
7 votes
1 answer
137 views

Why sample size is not a part of sufficient statistic?

Following simple example from Wikipedia's definition of sufficient statistic with Bernoulli distribution with parameter $\theta$, where sufficient statistic is a sum of successes $$T(X_n)=\sum_{i=1}^n ...
mikowai's user avatar
  • 118
0 votes
1 answer
25 views

sufficient, minimal, complete

Are all complete statistics functions of each other? For example if I have T and S complete statistics Can you always write T in terms of S and S in terms of T?
statistic-user's user avatar
1 vote
1 answer
2k views

Sufficient statistics for bernoulli distribution

Let $Y_1, \ldots, Y_n $ be a random sample of size $n$ where each $Y_i \sim \textrm{Bernoulli}(p), $ and let $Y = \sum Y_i $ for $i = 1, \ldots, n.$ The estimator is $W= (Y+1)/(n+2). $ Is the ...
asjndna999's user avatar
1 vote
1 answer
313 views

Is this a sufficient statistic for variance?

I have $X_1,\dots,X_n,X_{n+1}\overset{iid}{\sim}F_X(x)$, where $F_X$ has a finite mean $\mu$ and variance $\sigma^2$. If I calculate $\bar X_n = \dfrac{1}{n}\sum_{i=1}^n$ and $S^2_n = \dfrac{1}{n-1}\...
Dave's user avatar
  • 61k
2 votes
1 answer
150 views

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 $$\...
The Pointer's user avatar
  • 1,782
1 vote
2 answers
50 views

How does the result $\dfrac{1}{n^T} \dfrac{T!}{\prod_{i = 1}^n Y_i!}$ tell us what distribution $T(\mathbf{Y})$ is?

This follows on from my question here. I have the following problem: Let $Y_1, \dots, Y_n$ be a random sample from a Poisson distribution $\text{Pois}(\lambda)$. Recall, the $\text{Pois}(\lambda)$ ...
The Pointer's user avatar
  • 1,782
11 votes
1 answer
633 views

Is there a standard measure of the sufficiency of a statistic?

Given a parametrical model $f_\theta$ and a random sample $X = (X_1, \cdots, X_n)$ from this model, a statistic $T(X)$ is sufficient if the distribution of $X$ given $T(X)$ doesn't depend on $\theta$. ...
Pohoua's user avatar
  • 2,436
0 votes
0 answers
119 views

Minimal sufficient statistic with parameter-dependent support on density function

I'm having trouble finding a minimal sufficient statistic for this particular population. It has the pdf defined as $$f_\theta (x) = 4\theta^4 x^{-5}$$ if $\theta \leq x$, and 0 otherwise. We take $n$ ...
user avatar
0 votes
0 answers
52 views

Rao Cramèr Lower Bound problem

Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at ...
Cooper's user avatar
  • 21
1 vote
0 answers
53 views

Does this distribution belong to the exponential family? [duplicate]

I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal ...
Yeison Augusto Quiceno Duran's user avatar
0 votes
0 answers
22 views

Is the ordering of a sufficient statistic meaningful (soft question, maybe)?

Suppose we have distribution with a vector of parameters, and a random sample of size n from said distribution. Is the writing order of a sufficient statistic meaningful? For a normal distribution, we ...
Winston's user avatar
  • 217
2 votes
0 answers
98 views

Sufficiency for Truncated Geometric

Here is a deviant of a question I feel like I have seen several times on truncated exponentials and similar distributions for finding sufficient statistics: Let $$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
pSrIoGcNeAsLs's user avatar
1 vote
0 answers
127 views

Bayesian definition of sufficient statistics

Some time ago I wrote a question about what I think/thought (up to my understanding) is an ambiguity of the common definition of sufficient statistics : Conditioning in the definition of sufficient ...
Thomas's user avatar
  • 870
4 votes
1 answer
185 views

Rao-Blackwellisation using non-sufficient statistics

The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1") Now, I do understand that the ...
abhishek's user avatar
  • 236
1 vote
2 answers
457 views

Sufficient Statistic for $f(x,\theta)=\dfrac{2}{\theta^{2}} (\theta-x) \cdot 1_{(0,\theta)}(x), \;\forall \theta \in (0,\theta) $

Let $X_{1},\ldots, X_{n}$ be random variables independent and identically distributed; show that the following density function is in the exponential family and find the sufficient statistic for $\...
Darlyn LC's user avatar
2 votes
1 answer
309 views

Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution

Let be $X_{1},\ldots , X_{n}$ random variables independent and identically distributed with density function: $$ f_{\theta}(x)=\dfrac{1}{2}e^{-|x-\mu|}, \quad x,\mu \in \mathbb{R} $$ Find the joint ...
Darlyn LC's user avatar

1
2 3 4 5
10