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Questions tagged [statistical-inference]

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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What does this kind of mu mean in statistics

I know how to work with it but it have absolutely no clue wat is means. I mean the mu with the ~ on top
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Is the following always true: $\mbox{Var}[\mbox{Range}(X_1,\cdots,X_n)] = O(n^{-B})$ with $0\leq B \leq 2$?

Here $X_1,\cdots,X_n$ are i.i.d. The two extremes $B=0$ and $B=2$, and the standard case $B = 1$ are illustrated in the picture below. For the reference, see here.
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Proving independence between Beta estimated and Delta in OLS

I know that in ordinary least squares $b$(beta estimated) and $\delta^2$(variance estimated) are independent, but how do I prove that?
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Show Consistency for every component

For $j=1,...,k$ let $t_{n,j}:\Omega_n \rightarrow \mathbb{R}$ be an estimator for $h_j(\theta) \in \mathbb{R}$. Show that $t_n(X)=(t_{n,1}(X),...,t_{n,k}(X))$ is a consistent estimator of $h(\theta)=(...
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1answer
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How to argue why one dice is more rigged than the other?

Let $\omega$ be a finite set and $P : \Omega \rightarrow \mathbb{R}$ be a probability measure. You are given a set of three dices $\{A, B, C\}$. The following table describes the outcome of six ...
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42 views

Asymptotic Normality lemma (Serfling - 1980)

I'd like some assistant on the proof of the following Lemma: If $X_n$ is $AN(\mu,\sigma_n^2)$, then also $X_n$ is $AN(\overline\mu,\overline\sigma_n^2)$ if and only if $\frac{\overline\sigma_n}{\...
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Why wont the mean height of 70 fall under 99.7 percent when using formula: mean+/-3(SD)

Sample problem: In general, the mean height of women is 65″ with a standard deviation of 3.5″. What is the probability of finding a random sample of 50 women with a mean height of 70″, assuming the ...
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How easy is it to create false evidence for a biased coin?

I have a biased coin which comes up heads with probability $p$. I know the value of $p$, but I want to falsely claim that the coin has a different probability of heads, $q$, where $q > p$. To ...
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link function interpretation for models

In the categorial regression using the logit link function, the estimated coefficients are used to calculate the odds ratios or the ratio between two odds, that is, the model is interpreted through ...
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1answer
51 views

Finding the UMVUE of $\frac{1}{\lambda}$

I have been given the pdf: $$f_X (x; \lambda) = \left(\frac{\lambda}{\pi}\right)^{\frac{1}{2}} x^{-\frac{3}{2}} e^{-\frac{\lambda}{x}} $$ with support $x>0$ and $\lambda>0$. I am asked to ...
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Confusion with Gauss Markov

Consider the the linear regression model Yi = β xi + ei , where the numbers x1, . . . , xn are known, the independent random variables e1, . . . , en have the N(0, σ 2 ) distribution, and the ...
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Why is the KL divergence the number of bits required to represent the error of an estimator?

I am familiar with several interpretations of the KL divergence, last week I heard of a new one, mentioned in a lecture on probabilistic graphical models. It was stated kind of offhandedly, so I hope ...
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Conditional Interpretations of Linear Regression

We estimate a linear regressor in the 1 dimensional with x and y random variables with zero mean: y/x = $\alpha$ x We can rewrite this using the variance of the variables as: y/x = $\rho \frac{\...
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81 views

Can I do statistical analysis over a MILP problem?

I'm trying to solve a delivery problem which involves transportation of goods from a set of sources to a set of destinations ...
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12 views

Why is simple random sampling not sufficient for exponential distribution [closed]

For exponential distribution, why is a simple random sampling not the best design to yield a precise estimate for the population paremeters such as mean and variance? Suppose there was an auxiliary ...
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Question about the equality of variances of two populations [closed]

When finding the confidence interval for the mean difference between two groups of sample, under which assumption about the equality of their population variances should we take if no information ...
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24 views

Rao Blackwell and sufficient statistics

Suppose that X1, . . . , Xn are independent identically distributed random variables with a B(m, θ) distribution where m is a known positive integer and θ is unknown. I have shown that θ* = X1/m is ...
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Does $X_1,…,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$?

Does $X_1,...,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$? If so does the above imply that a standard normal divided by the ...
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19 views

Goodness of fit test

I have the following exercise that shows $n=6$ numbers: $$ 1.40, 1.55, 1.35, 1.50, 1.29, 1.64 $$ Is data normally distributed at the 5% significance level? Surely $\overline{x} = 1.455$, $s=0....
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28 views

If $X_i \sim U(\theta-\frac{1}{2};\theta+\frac{1}{2})$, show that $[X_{(1)},X_{(n)}]$ is a confidence interval

Let $X_1,...X_n$ random sample from $f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$. a) Show that $[X_{(1)},X_{(n)}]$ is a confidence interval for $\theta$. b) Compute the ...
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$U$~$N(3,16)$ $V$ ~$\chi_{9}^{2}$ U and V are independent random variables. Find $P(U-3<4.33\sqrt{V})$

$U$~$N(3,16)$ $V$ ~$\chi_{9}^{2}$ U and V are independent random variables. Find $P(U-3<4.33\sqrt{V})$ (The notes I'm working through don't seem to approach this rigorously...) The answer is $P(...
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Equality regarding the square of the sample mean

Given that $X_1,...,X_n$ is an i.i.d sample and its sample mean is $\overline X_n$, I have to prove the following equation: \begin{equation*} \frac{n-1}{n} \sum_{i=1}^n(\overline{X}_{n-1,i}^2 ...
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Generating random samples from a posterior distribution

Let $$p(D \mid \mu,\sigma^2) \sim \mathcal{N}(\mu,\sigma^2)$$ where $D=(x_1\ldots x_n)$ is my data. I imposed a normal prior on the mean as $$\pi(\mu) \sim \mathcal{N}(\mu_0,\sigma_0^2)$$ Using Bayes, ...
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Find $k \in R$ such that $P\left(\max\left\{\frac{{S_x}^2}{{S_y}^2}, \frac{{S_y}^2}{{S_x}^2}\right\} > k\right)= 0.05$

Let $\overline{X}$ and $\overline{Y}$ sample means and ${S_x}^2, {S_y}^2$ unbiased estimators for the variance of 2 independent random samples of size 7 with normal distribution with mean unknown and ...
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1answer
45 views

Finding a confidence interval for shifted exponential distribution

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\sigma ,\theta)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}, x\gt \theta$$ where $\sigma \gt 0 $ and $\theta \in R$ . a) if ...
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econometrics: weighting frequency over time

I want to measure a model like: Score = arunning_sum_of_all_exercises + brunning_sum_of_identical_exercise + c*exercise_density_value_in_period_x --other variables -- Lets say I want to test ...
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1answer
24 views

Finding shortest Confidence Interval for an Exponential Distribution

Let $X$ such that $f_{X}(x\mid\theta) = \theta e^{-\theta x} I_{(0, \infty)}(x)$, where $\theta > 0$. If $[X, 2X]$ is a confidence interval for $\frac{1}{\theta}$: a)Find the confidence ...
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Which of $s^2$ and $S^2$ is a better estimator of $\sigma^2$ in the sense of the mean squared error?

Let's recall that: $$s^2=\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar X)^2\quad\&\quad S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar X)^2$$ We actually know that $S^2$ is an unbiased estimator of the ...
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Party Invitations Problem [closed]

There is a party on some Sunday next year, it is open to all. a)Everyday at some point, a person comes and drops a slip with contact details in a dropbox. b)A person on a given day could comeback ...
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31 views

Require guidance to proceed further in learning statistics

In my undergraduate course I learnt introductory level statistics and I really enjoyed it. With that background I decided to follow predictive analysis further and decided to take up this book . But I ...
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28 views

Why isn't the chi-squared test appropriate in this case?

I was trying to solve the following exercise: Two researchers studied the relationship between infant mortality and environmental conditions in Dauphin County, Pennsylvania. As a part of the study, ...
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29 views

Finding aconfidence interval for $\theta$ of the uniform distribution on $(0, \frac{1}{\theta})$

Suppose $X_1,\ldots, X_n$ are i.i.d. random variables $Uniform (0, 1/ \theta)$. Find a 95% confidence interval for $\theta$. What I tried: $f_{X} (x) = \frac{1}{1/\theta}=\theta, F_{X} (x) = \frac{...
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36 views

Show that $\min\{X_{1},X_{2},\ldots,X_{n}\}$ is sufficient for $\mu$ when $\sigma$ is fixed

Let $X_{1},X_{2},\ldots,X_{n}$ be a sample from a population with density $p(x,\theta)$ given by \begin{align*} p(x,\theta) = \frac{1}{\sigma}\exp\left\{-\left(\frac{x-\mu}{\sigma}\right)\right\} \...
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1answer
33 views

Find distribution of test statistic under $H_0$

I have a shifted double exponential distribution with density $$f(x;\theta)=\frac{1}{2}e^{-|x-\theta|}$$ Now I have a test statistic given by $$T(x_1,...,x_n;\theta_0) = \sum_{i=0}^n |x_i - \theta_0| -...
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1answer
39 views

Finding confidence interval for $\frac{kx^{k-1}}{\theta^k}$

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\theta)=\frac{kx^{k-1}}{\theta^k}, x\in (0,\theta)$$ where $\theta \gt 0 $ and $k$ is a positive integer. Find a $100(1-\alpha)% $% ...
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61 views

Coverage probability for Uniform$(0, \theta)$

Let $X_1 \dots X_n$ denote a random sample from a uniform $(0, \theta$) distribution. PROBLEM: Compute the coverage probability for the CI: $$\left(\frac{X_{(n)}}{0.95}, \frac{X_{(n)}}{0.25}\right)...
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Poisson Generalised Likelihood Ratio Test

I am attempting Q19H from this document: https://www.maths.cam.ac.uk/sites/www.maths.cam.ac.uk/files/pre2014/undergrad/pastpapers/2014/ib/PaperIB_1.pdf I am alright with the first part, however, it ...
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1answer
10 views

Distribution of expectation operator when computing mgf of X bar

I'm trying to work through the proof for the moment generating function of $\overline{X}$. The proof below looks fairly straightforward but I'm having trouble understanding getting from the 2nd to ...
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29 views

Confidence interval from p value

The question is: given that $H_0: \mu=34, H_a:\mu<34$ gives p-value $p$, find the largest confidence level, $c$, that does not include $34$. The answer does this:$1-2p=c$ But I do t understand ...
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Do we have to assume normality of the data, even when we conduct z-test or t-test with large samples?

I read this lecture note and found that it assumes normality of the data when we conduct z-test or t-test. I can accept that when we have small samples we have to assume normality of the data, ...
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Compute the profile log likelihood for $\alpha$ giving your answer in terms of $\hat{\alpha}_{\beta}$.

We have $X_{1}, X_{2},...,X_{n}$ IID random variables from a Poisson distribution with mean $\mu_{i}=\exp{(\alpha + \beta z_{i})}$. i) For fixed $\beta$, find $\hat{\alpha}_{\beta}$, the maximum ...
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22 views

Some questions on the nature of statistical/probabilistic deduction

I have been doing some reading and thinking on the nature of statistical inference and the way formal statistics models an event seems a bit strange to me. Here is how I like to think about it: In ...
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40 views

The continuous probability dilemma

If X is a random variable with a mean of 85.43 and a standard deviation of 17.23, what is the mean of the sample means calculated from samples of size 12 drawn from this population? Give your answer ...
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53 views

What does E_n mean

I came across the following symbol in this paper: I am a little bit confused by the symbol . Does it simply mean population average?
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16 views

How to infer the underlying distribution of a statistic (Bayesian inference?)

I have a list of approximately 30,000 venues in a major US city. These venues hold all kinds of events, sports, conferences, concerts etc. I want to know the distribution of the 'capacity' of these ...
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Covariance and trace ( Mallow's statistics )

From Elements of Statistical Learning Exercise 6.10 ( cross-validation and estimator for the in-sample prediction error ) The difference between in-sample prediction error and the average sum of ...
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B-Splines and sum of uniform variables

Exercise 5.2 in Elements of Statistical Learning Goal is to show that an order $M$ B-Spline basis function is the density function of a convolution of $M$ uniform random variables. Although I feel ...
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11 views

Conflicting unilateral and bilateral tests. How to proceed?

Suppose I have $H_0: \beta = 0$ vs $H_1: \beta \neq 0$. My data tells me that I don't reject the null, with a p-value of approximately 0.06. Then out of curiosity, since by the scientific theory one ...
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40 views

Question about a new type of confidence interval

I came up with the following result, tested on many data sets, but I do not have a formal proof yet: Theorem: The width $L$ of any confidence interval is asymptotically equal (as $n$ tends to ...
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2answers
44 views

How to find MLE of this piecewise pdf?

Suppose $X_1,\ldots, X_n$ are i.i.d. random variables having pdf $$ f_{\theta}(x)=\left\{\begin{array}{ll}{\theta,} & {0 \leqslant x \leqslant 1} \\ {1-\theta,} & {1<x \leqslant 2}\end{...