The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Question on sufficient complete statistics proof and estimators of zero

I am trying to prove theorem 7.3.23 in Casella and Burger. Theorem: Let T be a complete sufficient statistic for a parameter $\theta$, and let $\phi(T)$ be any estimator based only on T. Then $\phi(...
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15 views

evaluation of type1 and 2 error for hypothesis test of variance (two methods with different results)

$\newcommand{\tend}[1]{\oalign{\mbox{\boldmath$#1$}\crcr\hidewidth$\scriptscriptstyle\sim$\hidewidth}}$ Each of $n=10$ persons used the same instrument to measure the same object; the true value ...
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16 views

How to compare two tests according to the power of the test?

enter image description here Can the rejection region calculated from the problem? I'm a little bit confused by all this staff.
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24 views

t distribution : formula for the degrees of freedom

I understood why we are using a t distribution in this case , because the sample isn't big enough to approximate the true standard deviation of the population by the sample's . But what I can't find ...
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20 views

Casella and Berger Likelihood Ratio Tests statistic vs Wasserman LRT

It seems like there is a discrepancy between these two authors on what a LRT is. Casella and Berger state on pg. 375. That the LRT statistic is: $\lambda(x)=\frac{L(\hat{\theta}_0|x)}{L(\hat{\theta}|...
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31 views

Help with the Maximum Likelihood Estimator?

I'm really struggling to understand this and am trying to learn it for my upcoming exam. The question I'm trying to do is Write down the likelihood function and then find the Maximum likelihood ...
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29 views

Where can using two different test statistics in a hypothesis test lead you?

Do two test statistics at the same $\alpha$ value give the same TYPE 1 error rate and same decision? I think it is clear that they do give the same TYPE 1 error rate by definition, but do they always ...
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1answer
31 views

Expectation of the conditional density

What is the difference between E[$X_1$|$X_n$ = $x_n$] and E[$X_1$|$X_n$]? I have found the first one, by integrating x*$f_{X_{(1)}|X_{(n)} = x_{(n)}}$ (x). If anyone has pointers for finding E[$X_1$|$...
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14 views

Help with Hidden Markov model and SMC methods

So its quite a long background i don't really know where to start but here goes. The background is as follows: Background Observation model As the target is moving, it measures the signal (RSSI) ...
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1answer
22 views

Testing the Uniformly Most Powerful Test against the alternative

Hi I am working on the following problem A single observation $X$ is made from one of three densities listed below with parameter space $\Theta=\{0,1,2\}$. \begin{align*} x=0\hspace{0.4cm}x=1\hspace{....
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20 views

Paired T-Tests vs Independent

The effectiveness of a training course is examined, and performance of each individual in a group is taken both before and after, and the differences are used in a paired T test. Would it be possible ...
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36 views

Finding Uniformly Most Powerful(UMP) tests of size $\alpha$

Hi I am working on the following problem: Let $X_1,X_2,\ldots,X_n$ be a random sample from a distribution with PDF given by $$f(x\mid\theta)=\frac{c}{\theta^c}x^{c-1}e^{-(\frac{x}{\theta})^c}\,\,\,(\...
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81 views

Finding the joint distribution and covariance matrix of a function.

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \...
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11 views

Expectation Maximization (EM) for 3-dimensional parameter $(\alpha,\mu_2,{\sigma}^2)$.

Let $x_i$ where $i=1,...,100$ are iid observations from a mix of two normal distributions with means $\mu_1=0$ and $\mu_2$ and the same variance ${\sigma}^2$. If $\alpha$ is the proportion of the ...
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18 views

Can the original function be derived from its $k^{th}$ order Taylor polynomial?

Coming from a statistics background, I'll provide an example related to fitting a model to an analysis dataset. Let's suppose I suspect the relationship between the mean value of the outcome variable (...
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22 views

How to find sampling distribution S.D in this case

The distribution of the weights of 1000 students is normal with a mean of 55kg and a variance of 25. 100 random samples of size 16 are taken from this population. Determine the mean and standard ...
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1answer
40 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatθ = X _{(1)} + X_ {(n)}$ for a U(0, θ) distribution comes from? Where: $X _{(1)}$ = min$_i (X_i)$ $X_ {(n)}$ = max$_i (X_i)$ I know it is not the MLE, ...
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34 views

Divergence of Chi-squared statistic

I want to write "proof" that a ${\chi}^2$ statistic becomes larger and larger as the sample size increases. I have come up with the following: For ${\chi}^2=\sum_{j=1}^{n} \frac{(O_j - E_j)^2}{E_j}\...
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27 views

Finding MLE and UMVUE for $\theta$ for the following distribution??

I was working on following problem: Let $X_1,X_2,...,X_n$ be a random sample from a distribution with PDF given by $$f(x|\theta)=\theta^{-c}cx^{c-1}e^{-(\frac{x}{\theta})^c}$$ a) Find the MLE for $\...
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1answer
14 views

Finding minimal sufficient for the following distribution???

Hi I was working on finding the minimal sufficient for the following distribution $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}}I(0\le x\le 1),\,\,\,\,\,\,\theta>0$$ By factorization theorem ...
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25 views

Expectation in Bayes rule

Let $L(\theta,\delta)=(\theta-\delta )^2e^{\frac{(\theta-100)^2}{900}}$ with $X\sim N(\theta,100)$ and $\theta\sim N(100,225)$ find the bayes rule. I already founded that posterior is $f(\theta|x)\...
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32 views

expected value of the average cubed

I can not resolve an issue of the book Mathematical Statistics of Shao, is as follows: If $E|X_{1}|^3$ is finite, get $E(\bar{X}^3)$ and $Cov(\bar{X},S^2)$ If $E|X_{1}|^4$ is finite, get $Var(S^{2})$...
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1answer
25 views

Linear Regression with Paired Data

For a sample of paired data (x,y), t tests are performed for the slopes of the population regression lines of y on x and of x on y. The null hypothesis in both tests is $H0:β=0$. Is it possible for ...
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1answer
44 views

Maximum Likelihood of single observation

I'm stumped on this problem... I keep getting an undefined answer of having to solve -20 = 0. The likelihood function I get is $e^{-20\alpha}$. So I have $y_i=$ $ \begin{cases} 1& w/...
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hypothesis testing on exponential family

In class, the professor did present the next problem: Suppose you have a random sample $x_1,...,x_n$, but is unknown if the original distribution of the sample is gamma or exponential. You also have, ...
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32 views

Clarifying the assumptions about a paired t-test

I've wrote my question in red ink (see links). There are two questions that I have. Primarily I want to know why they concluded that "there is some evidence that there is some difference in mean ...
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1answer
27 views

Hypothesis test to justify a claim

So I have a question regarding hypothesis tests where i have to justify a claim with statistical evidence. It is as follows: The average number of accidents in previous years in a city has been 15 ...
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2answers
47 views

Determine who is the best seller

The numbers below show the number of lollipops Betty and Sharon each month for a total of 12 months or a year. Using the data and plot below, can you determine who is the bestseller? Would it be ...
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1answer
28 views

How to estimate the max of a population using the normal distribution equation on a small sample

I recently watched a documentary on Mathematics. In the show they managed to estimate the weight of the largest fish that the fisherman was likely to of ever caught in his career just by analysing one ...
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24 views

How to calculate survey bias due to preference for first answer?

I was recently given the results to a survey in which participants chose answers to questions they would be likely to randomly answer, and in which the survey population is known to have a preference ...
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Convergence of Sample Moments and Weak Law of Large Numbers

Given $\{X_i\}_{i=1}^n$, sequence of indepedent random variables such that $E[X_i]=\mu$, prove: $$a_r=\frac{\sum X_i^r}{n}\xrightarrow{p}E[X^r]$$ and prove: $$m_r=\frac{\sum (X_i-X_m)^r}{n}\...
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1answer
38 views

How to formulate and test this statistical hypothesis

first of all I'd like to clarify that my biggest problem with this topic is probably my inability to formulate it correctly, maybe the answer is found trivially on the internet but I'm not able to ...
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16 views

Hypothesis testing - comparing data with a function

I am a beginner in statistics. My problem is as follows: I have a set of $N$ observations $\{(x_i,y_i,e_i)\}$, where $e_i$ is the error in the $i^\mathrm{th}$ observation. The individual observations ...
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2answers
45 views

An Estimator Based on Exponential RVs

Let $X_1$, $X_2$, $\cdots$, $X_n$ be $n$ random variables independently sampled from the exponential distribution $\text{exp}(1)$. Suppose $k \leq n$, and $X_{(k)}$ is the $k$-th order statistic, i.e.,...
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1answer
37 views

Interpter P-value. Is the following statement true or false, and where is the mistake?

I have the following question, Statement: A given exercise has the p-value of 0.08 and my alpha is 5%. The exercise was using a linear regression model to predict some future value ...
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3answers
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Can the correlation of a random variable $X$ and $g(X)$ be $0$?

Question: Can the correlation of a random variable $X$ and $g(X)$ be $0$? My attempt: I don't believe it can because they are dependent by definition therefore $Cov(X,g(X)) \ne 0$ which means the ...
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66 views

Deriving Joint probability density functions

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 ...
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23 views

$U$ as a random variable? And what's with this integration becoming $1$?

I'm confused about two things. For the set up, I am told that Likelihood function is $L(\theta)=\Pi f(y_i;\theta)$ for a distribution with pdf $f(y;\theta)$. Log likelihood function is $l(\...
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13 views

Horseshoe estimator posterior

Suppose given the Horseshoe estimator: $Y|\beta,\sigma^2 \sim N(X\beta,\sigma^2 I)$ $\beta|\sigma^2,\tau_{1}^2,...,\tau_{p}^2 \sim N(0,\sigma^2 D)$ $\tau_{j} \sim C+(0,1)$ $\sigma^2 \sim \pi (\...
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19 views

how to find AIC values for both models using R software?

I'm studying survival analysis. I estimated both Cox regression model and Buckley&James regression model. In order to determine which model is better for my dataset, I used Akaike Information ...
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1answer
33 views

Using head-to-head results and Bayes' Theorem to modify predictions of sport/game contests that are initially derived from Elo-type ratings

I am working on an extension of the Glicko2 rating system to use in predicting the outcome of sport/game contests that uses the actual head-to-head results of previous meetings of competitors to ...
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1answer
29 views

Determining the degree of freedom for a $\chi$-squared test

I have read that the degree of freedom is calculated by subtracting $1$ from the number of states a random variable can be in. I am performing a goodness of fit test on a $64\times 32$ matrix where ...
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Help: SPSS and Data Interpretation of Voters. Republican vs. Democrats (1993 election)(Almost finished)

Hello everyone, I am Julieta this time I get stuck in the following exercise. It is a statistical analysis of pools, the statement is quite long I will try to keep it short and put some links. Note: ...
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10 views

Comparing expected counts to observed counts

The question is as follows: A restaurant offers 7 different dishes and predicts the dishes will be ordered in the following proportions: 1 (25%), 2 (20%), 3 (10%), 4 (15%), 5 (5%), 6 (6%), 7 (15%). ...
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1answer
15 views

$MLE$ of a multinomial. [CORRECTED]

It might be a very silly question, but I just can't figure it out. Let $X_1,...,X_n$ be random variables with pmfs: $$f(k,p)= \begin{cases} p_1, \hspace{0.5cm} \text{if } k=a\\ p_2, \hspace{0.5cm} \...
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23 views

Existence of asymptotic variance for an estimator when it doesn't converge to normal distribution.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,\ldots)$, where: $U_i=U_i(X_1,\ldots,X_i)$, if for a sequence of constants $\{k_n\}$: $$...
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14 views

How to find n=# of samples

My question is what is the equation to find n if you know standard deviation, and the CI of 95%. Like what is the equation for n?
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18 views

Sample Estimate of the Mean of a Population Lying in [a,b]

Suppose I have a population of $N\ge 1$ real numbers, all known to lie in the real interval $[a,b]$, where $a,b\in\mathbb{R}$, $a\le b$, and $a$ and $b$ are known values. I know nothing else about the ...
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24 views

Two sample testing, when all the parameters are unknown

Suppose $(X_1,...,X_p)$ is a sample of independent $\mathcal{N}(μ_1, σ_{1}^2)$ random variables, and $(Y_1, . . . , Y_q)$ is a sample of independent $\mathcal{N}(μ_2, σ_2^2)$ random variables. If all ...
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12 views

Existence of asymptotic variance for an estimator when it doesn't converge to normal.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,...)$, where: $U_i=U_i(X_1,...,X_i)$, if for a sequence of constants $\{k_n\}$: $$k_n(U_n-\...