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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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Asymptotic normality and multiplication by limit $1$ sequences

Suppose $X_n \sim \text{AN} (a,q^2/v_n^2)$. This means $v_n(X_n-a)\rightarrow_D N(0,q^2)$, where $\rightarrow_D $ stands for convergence in distribution, $a,q \in \mathbb{R}$ and $v_n$ is a real ...
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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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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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Why is simple random sampling not sufficient for exponential distribution [on hold]

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 [on hold]

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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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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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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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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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
23 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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If $\sigma$ is a scale parameter, i.e. $f_\sigma(x)=\frac{1}{\sigma}f_1(\frac{x}{\sigma})$ then $\bar X/\sigma$ is a pivotal quantity

This was taken from Casella-Berger. In page 427, it is asserted that, if $X_1\dots,X_n$ is a random sample of a population with density $f_\sigma(x)=\frac{1}{\sigma}f_1(\frac{x}{\sigma})$, then the ...
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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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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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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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28 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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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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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
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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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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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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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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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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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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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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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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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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{...
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1answer
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Likelihood ratio test for both sided alternative in $U(0,\theta)$ distribution

Suppose $X_1,X_2,...,X_n$ is a random sample from $U(0,\theta)$. We need to construct a likelihood ratio size $\alpha$ test to test $H_0:\theta=\theta_0$ against $H_{1}: \theta \neq \theta_{0}$ My ...
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1answer
47 views

Moment Generating Function of beta ( Hard )

Given $X$ is a random variable ~ $Beta ( a , b)$ distribution and $X$ belongs in (0,1) Does the (MGF ) $E[e^{tx}]$ exist for every value of $a , b$ ? (Mgf must not be equal to infinity in order ...
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1answer
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Pivotal Quantity for the location parameter of a two parameter exponential distribution

Let $X$ be a random variable with probability density function $f(x,\theta, \beta)=\beta e^{-\beta(x-\theta)} \mathbb{1}_{(\theta,\infty)}$ with $\beta>0, \theta \in \mathbb{R}$ (a two parameter ...
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Hypothesis Test for Simple Linear Regression coefficients

Given $\hat{Y}=0.3 + 1.7X$, would I be right if the hypothesis test shows that the intercept 0.3 is not significantly different from zero and the slope 1.7 was not significantly different from one. ...
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compute quadratic risk bayesian statistics

I don't understand how can we move from the line 1 to line 2 and 3. The mean $\bar X_n$ following $\theta$ distribution is for me just $\bar X_n$ but it seems to be another thing here
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posterior risk bayesian statistics

This is the expression of the posterior risk but I don't understand the step of the line 2. How can we developped the first line in this way ? $T^*$ is defined as $E[\theta | X]$
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How d I compute $E(T|\bar{X})=2\bar{X}$?

Let $X_1,X_2,...,X_n$ be iid observations from a normal distribution with mean $\mu$ and variance $\sigma^2$, $\sigma^2>0$ is known and $\mu$ is an unknown real number. Let $g(\mu)=2\mu$ be the ...
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Estimating parameter of binomial distribution

We have some solution containing a compound A. We find that when we mix $1$ $\mathrm{mm}^3$ of our solution A with some amount of some compound B, we get a reaction 200 out of 185 times when we check ...