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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Bayesian Estimation of CDF

i'm getting pretty confused by the following problem, hope anyone can clarify my mind: Using a bayesian approach obtain a posteriori and interval estimations for $\mathbf{F}_{X}(x)$ using a Uniform(0,...
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Best way to play 20 questions

Background You and I are going to play a game. To start off with I play a measurable function $f_1$ and you respond with a real number $y_1$ (possibly infinite). We repeat this some number $N$ of ...
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How to find the reliability of a sample mean?

I am trying to solve the following question (based on Student's t-test) but I can't even understand what the problem is actually asking. Find out the reliability of the sample mean of the following ...
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Randomised Response Technique

Context : To get answers to sensitive questions, we sometimes use a method called the randomized response technique. Suppose, for instance, that we want to determine what percentage of the students at ...
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Finding UMVUE estimator [closed]

Suppose that we have a random variable whose distribution is exponential with parameter $\theta$. We know that the sum of the sample observations is a sufficient statistic for $\theta$. Then, I have ...
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Help with visualising the confidence interval.

I’m trying to understand the confidence interval by reference to the diagram below: • I want to guess a parameter that we know to be 0.88. The vertical line represents the parameter’s value. • I ...
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Gaussian Processes - What to condition on?

I am quite confused about the conditional dependencies of Gaussian Processes and wondered if someone could explain them to me. My understanding thus far is as follows: A GP is a distribution over ...
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Bayesian inference and bayesian updating

If $H$ and $E$ are events in a probability space with probability measure $P$, then we have $$P(H | E) = \frac{P(E | H) P(H)}{P(E)}$$ Let's think of $H$ as a hypothesis and $E$ as evidence. I have ...
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How to test nonlinear hypothesis of the form mx+b?

Suppose I have a multiple linear regression of the form: $$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} +\beta_3 x_{i3}+\epsilon$$ Then suppose I am interested in testing this set of hypotheses: $$...
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Derivation for conditional power formula

I am studying conditional power, and was told that this is a general formula for conditional power when you have a continuous outcome: $$CP = P\bigg(Z > \frac{c_2\sqrt{I_2}-z_1\sqrt{I_1}-(I_2-I_1)\...
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How can you derive score test under nonlinear null hypothesis with nuisance parameter? [duplicate]

Suppose I have a multiple linear regression of the form: $$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} +\beta_3 x_{i3}+\epsilon$$ Then suppose I am interested in testing this set of hypotheses: $$...
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How to derive test statistic under composite null hypothesis?

Suppose I have a multiple linear regression of the form: $$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} +\epsilon$$ Then suppose I am interested in testing this set of hypotheses: $$H_0: \beta_1 = ...
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How to prove $\sqrt n \sigma^{-1}(\overline{X}_n -c) \Rightarrow N$ and $\sqrt n (f(\overline{X}_n) - f(c))/\sigma |f'(c)| \Rightarrow N$

How to prove that $\sqrt n \sigma^{-1}(\overline{X}_n -c) \Rightarrow N$, where $\overline{X}_n = n^{-1}\sum_{k=1}^n X_k$. If$f(x)$ has a nonzero derivative at $c$, then $\sqrt n (f(\overline{X}_n) - ...
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The Conditional MLE of $\theta=\frac{\mu}{\mu+\tau}$, where $N_1$ ~ Poi($\mu v$) and $N_2$ ~ Poi($\tau v$) (Estimating Equations)

Problem Setting: Knowing that $N_1\sim\operatorname{Poisson}(\mu v)$ and $N_2\sim \operatorname{Poisson}(\tau v)$, where $\mu$, $\tau$, and $v$ > 0 are all unknown. We set $\theta = \frac{\mu}{\mu +...
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Finding an Unbiased estimator of $λ^2$ for a Poisson distribution [closed]

I found this question difficult to attempt: Let $x_1$ , $x_2$ be a random sample from a Poisson distribution with parameter λ. Find an unbiased estimator of $λ^2$ . $x_1$ and $x_2$ are independent....
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1answer
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independence of sets of a sigma algebra

I have been trying to solve this question all day, i am not good at probability theory so i am not able to solve this. Let (Ω, A, P) be a probability space and A, B, C ∈ A and A, B, C are independent ...
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Based on a sample of size $n$, find a sufficient and complete statistics for $\theta$ and the MLE for $\theta$

Let $X_1, ...., X_n$ be iid random variables from the following distribution $x$ $-1$ $0$ $1$ $P(X=x)$ $\theta / 3$ $(1- \theta)$ $2\theta/3$ for $0\leq \theta \leq 1$. Based on a sample of size $n$...
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How to interpret minimal sufficient statistics?

I know the definition of minimal sufficient statistics is: A sufficient statistic T(X) is called a minimal sufficient statistic if, for any other sufficient statistic T'(X), T(X) is a function of T'(X)...
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Expected Value of MSR in General Linear Model

The setting of my problem is as follows: Consider the general linear model $Y = X\boldsymbol{\beta} + \epsilon$, where $\beta = \begin{bmatrix} \beta_0 \\ \boldsymbol{\beta_1} \end{bmatrix}$ and $X = \...
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Let $A$ be a $n\times m$ random matrix with entries from finite field $\mathbb{F}_q$. What is the probability that rank of $A$ is full.

Let $A$ be a $n\times m$ random matrix with entries from finite field $\mathbb{F}_q$. What is the probability that rank of $A$ is full. I know that complex-valued random matrix say $\mathbf{A} \in \...
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Rao blackwellised form of unbiased estimator of $a^3$

Let n iid random variables $X_1,X_2,..,X_n$ follow normal distribution $N(a,b^2)$. Find the rao blackwellised form of unbiased estimator of $a^3$ I am having problem in calculating conditional ...
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Undergraduate Statistics books suggestions [closed]

What are some topic wise good books to become a competent statistics graduate? The topics I've for my under graduation are 1)Descriptive probability 2)Probability distributions 3)Statistical methods 4)...
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Geometric distribution from exponential estimation

Problem: $X \sim \exp(\lambda)$ Now $X$ is descretized to $Y$ using floor function $Y_1, Y_2, \dots,Y_n$ is available from $Y$ distribution I need to find method of moment estimator for $\lambda$ ...
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Estimation in uniform distribution

Is there a case when uniform distribution posses unbiased and umvue estimate of $\theta$? Suppose $X_1, X_2, \dots, X_{45}$ ~ uniform on interval $[\theta-1/2\ , \theta+1/2]$ My views: I know $\max(...
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books similar to '100 page machine learning book' but on probability and statistics

I would like to find a technical book that goes through the theory behind probability and statistics, although does not read like a textbook. Similar to the writing style of Burkov in '100 page ...
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Having a problem getting the Intuition behind t-tests and z-tests

Say we have a population of $n$ rodents and that the mean of the population is $\mu$ with standard deviation $\sigma$, the standard error of means in this population would be $\frac{\sigma}{\sqrt n}$ ....
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Sufficiency and completeness in two distribution

I have two variables $X_1$ and $X_2$ which follows exponential and double exponential with rate $\theta$ $X_1\sim\frac1\theta \exp((-x/\theta)),\quad X>0$ $X_2\sim\frac2\theta \exp(-2x/\theta)),\...
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Covariance, matrix inverse, and second order derivatives of log-likelihood

On page 35 of the book Analysis of survival data by D.R. Cox and D. Oakes, I see the following idea: the observed information matrix is defined as the matrix of minus the second derivatives of $l,$ ...
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For sampling to be independent(2 random samples), do the two populations have to be independent as well?

I was trying to think of a scenario where we draw two random samples from two different populations. Can it be a case that the samples are independent but the populations are not? EDIT: By independent ...
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Confidence Interval for difference in means. Am I doing something wrong?

I have to do the following question but I've tried it multiple times but my confidence interval doesn't match with the answer key. Am I doing something wrong? An economist believes that a typical ...
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The estimator for the risk function

The Question Let $\{\mathbb{P}_\theta\}_{\theta\in\Theta},\Theta \subseteq \mathbb{R}$ be an identifiable parametric family of distributions with common support, where card $(\Theta) \ge 2.$ Consider ...
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Resource recommendations for beta distributions, rule of succession and Wilson score interval

As a high school assignment, I'm writing a 4000 word essay on the topic of a blog post by John D. Cook, which poses the question: I was buying a used book through Amazon this evening. Three resellers ...
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Show that the statistic $T(x_1,…,x_n)=\sum_{i=1}^nx_i^2$ is complete.

Let $X_1,...,X_n$ be iid random sample form $N(\theta,c\theta)$, where $c$ is a known constant. Show that the statistic $T(x_1,...,x_n)=\sum_{i=1}^nx_i^2$ is complete. In other words I have to show ...
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Question about error analysis for a normally distributed data

Let us suppose we have one constant variable $b \pm \delta b = 20 \pm 1$ and one function that depends on $x$, such as, $a(x) \pm \delta a$ The problem is I want the difference between $a(x)$ and $b$ ...
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Show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete.

I just proof that $T=(X_{(1)},X_{(n)})$ is not complete but now I want to show a more general case. To be more specific I want to show that if a function of a sufficient statistic is ancillary, then ...
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Expected squared deviation of the James Stein estimator

Let $z\sim\mathcal{N}_N(\mu,I)$, the multivariate $N$-dimensional normal with mean $\mu=(\mu_1,\dots,\mu_N)^T$ and the identity matrix $I$ as the covariance matrix. In the textbook that I'm reading, ...
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Two statistics called equivalenet

Here is some step that I did. However, the hint from my professor is that drop the condition $\mathcal{P}-a.s$ and just try to establish the implication as(ie, "everywhere"). You can do this ...
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How does a difference under the integral sign turns into a definite integral of just a dummy variable?

I'm trying to follow trhough the proof of a statistics theorem due to Pratt (Full proof here, from Casella-Berger's Statistical Inference, 2nd Ed., p. 447), but I'm stuck with this passage: $$\int_{\...
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Finding the form of the uniformly most powerful test

i am trying to find the form of the uniformly most powerful test for a variable with a certain pdf. the PDF is $f(x;\theta) = \theta \cdot (1 - x)^{\theta - 1}$ and $0 < x < 1$ where $\theta >...
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Show that $T(X)=(R,V)=\left( X_{(n)}-X_{(1)},\frac{X_{(n)}+X_{(1)}}{2} \right)$ is a minimal sufficient statistic for $\theta$.

Let $X_{1}, X_{2}, ..., X_{n}$ be a random sample from $\text{Uniform}(\theta,\theta+1)$ population with $-\infty<\theta<\theta+1< \infty$ show that $T(X)=(X_{(1)},X_{(n)})$ is a minimal ...
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MLE of $\frac{1}{\theta}$ Uniform distribution.

Let $X_1,X_2...X_n$ be a random sample from $U(\theta-5,\theta+5)$ where $\theta \in(0,\infty) $ is unknown. Let $T=\max(X_1,X_2...,X_n)$ and $U=\min(X_1,X_2...,X_n)$. Then which of the following ...
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finding UMVUE for $\theta_x$/$\theta_y$

Let Xi ~ Exp($\theta_x$), Yj ~ Exp($\theta_y$), i = 1; ... ; n1, j = 1;...;n2. Find UMVUE of $\theta_x$/$\theta_y$. Since $\bar{X}$ and $\bar{Y}$ are compelete sufficient statistic, by using Lehmann-...
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Bayesian statistical inference problem continuous case

This question is about updating distribution of unknown random variable with observed data x. I took a photo of what I have done and my reasoning. I am newbie to probability and statistics and I do ...
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Mysterious claim in Casella's book

I'm studying by Statistical Inference by Casella and Berger and on page 225, I didn't understand why he expected the $F$ distribution behaved in this way (see the highlighted part in the end of the ...
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1answer
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Hypothesis Testing concerning difference in means. Misprinting Error?

I was solving this question in my book and I realized the answer at the back and my answer doesn't match. I wanted to verify whether my process is correct or not? The question is as follows: Samples ...
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Uncertainty quantification Frequentist vs Bayesian

Is it actually possible to quantify the uncertainty in a frequentist setting? (e.g. using Maximum Likelihood Estimator). Say that we have a dataset $\mathcal{D} = \{(x_i,y_i)\}_{i=1}^n$ and assume ...
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Why these variances are calculated differently?

I'm studying by this book Data Analysis and Graphics Using R: An Example-Based Approach and on chapter 5, page 150 they wrote: There are two types of predictions: prediction of points on the line, ...
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Missing all superscripts in bayesian inference ELBO

I'm a student studying bayesian inference. When I read this paper(https://arxiv.org/pdf/1312.6114.pdf), I had one question in eq2. $$log_{p_\theta}(x^{(i)}) = D_{KL}(q_\phi(z|x^{(i)})||p_{\theta}(z|x^{...
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Null Hypothesis: Is it being rejected?

I was doing this particular question that is as following: A drugs manufacturer claims that the amount of paracetamol in tablets is 60 mg. A sample of 10 tablets is taken and the amount of paracetamol ...
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Uniformly Most Powerfl test: Student t-test

Let $X_1, \ldots, X_n \sim \mathcal{N}(\mu, \sigma^2)$ be i.i.d. random variables, where the variance $\sigma^2$ is unknown. We're interested in the following test problem: $$H: \mu = \mu_0 \quad \...

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