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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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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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single variable is significant but overall test is not

I do a multiple regression with 3 independent variables $X_1$, $X_2$ and $X_3$. The correlation between $Y$ and $X_1$, $Y$ and $X_2$, and $Y$ and $X_3$, are each large and statistically significant. ...
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Divergence based robust inference

The term 'divergence' means a function $D$ which takes two probability distributions $g,f$ as input and puts out a non-negative real number $D(g,f)$. I have learnt that the inference based on ...
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The distribution of the ith order statistic for discrete random variables

Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$. We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim Ber(...
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(Conditional) uniform asymptotic inference

Let $(\Omega,\mathcal{F})$ be measurable space and $\mathcal{P}$ be a family of probability measures on $(\Omega,\mathcal{F})$. A "typical" statistical problem is to show that $$\tag{1}\label{1} \...
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Likelihood function & MLE without known values of observed data

Question: Let $X_1,\dots,X_n$ be iid exponential rate $\lambda$. Suppose we don't know the observed values of our experiments, but we know that $k$ values were $\le M$ and the remaining $n-k$ were $&...
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Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also $\operatorname{...
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Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
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Are there generalised rules for generating heuristics from data?

Heuristics seems to be more of an art than a science, like a gut-feel supported by data; I might be wrong. Are there algorithms for mathematically generating heuristics from data, like pruning a ...
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Estimate median of Cauchy distribution

Motivated by this question, assume we have independent samples $(X_i)_{i=1}^{\infty}$ from a Cauchy distribution with unknown median $a\in\mathbb{R}$ and scale parameter $b$. What is the best way to ...
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UMVUE and Cramér-Rao lower bound

I'm trying to solve the following problem: Let $X_1 , \dots , X_n \sim$ Bernoulli$(\theta)$, iid. Find $T_n$, the UMVUE of $\theta(1-\theta)$, and show that Var($T_n$) does not attain the Cramér-...
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What does a confidence interval tell you about the relationship between two things?

So I have found a 95% confidence interval for the odds ratio of developing cancer with vitamin A intake. My sample odds ratio was 1.1667 with a 95% confidence interval of (0.5394, 2.5227). I am then ...
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The Law of Total Covariance on a Gaussian Process

Suppose that we have a Gaussian process with zero mean and covariance function $k$, $$ f(x) \sim \mathcal{GP}(0, K(x,x')) \tag{1} $$ It is usually assumed that there are a collection of training ...
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The division of Gaussian mixtures

In the study of probabilistic graphical models (PGMs), the loopy belief update propagation (LBUP) message passing algorithm requires the division of unnormalised probability distributions. If the ...
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Wilk's proof of convergence of LRT

In his book, Mathematical statistics, Wilks uses several times an argument that is a bit obscure to me (I'm referring to page 411 of the book). Basically, we have a sequence of roots of the maximum ...
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Neyman-Pearson $\alpha$-level test always exists for continuous distributions

The Neyman-Pearson lemma as in the classical book by Casella and Berger, gives to conditions for the existence of $\alpha$-level tests: The critical region must be of the form: $\{x:f(x|\theta_1) >...
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variance bounds of functionals

$X_1,\ldots,X_n$ are i.i.d standard random variables. $a_1,\ldots, a_n$ are constants such that $\min_i a_i > 0$ and $\max_i a_i < \infty$ $\hat c$ is given as the solution to the equation: ...
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Sample variance of distances of random points on [0, 1]

Let $X_1, \ldots, X_n$ be $n$ points chosen uniformly on [0, 1]. Let $Y_i = \displaystyle \min_{j \neq i} \{ |X_i - X_j| \}$ be the shortest distance between $X_i$ and any of the other points. Can we ...
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Finding the uniformly most powerful invariant test

As a "bonus" question (i.e. volontary), I got a question regarding invariance in my inference class I'm taking. So I'm trying to understand invariance, and uniformly most powerful invariant tests (...
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replace the worst $20\%$ in normal distribution- exercise

Here is an exercise, I tried to solve. It is a long since, I study statistics, so I would really appreciate if someone could check the answer/solution I give below. Exercise: Assume we are given a ...
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Time series determined by other time series

Intuitive Question Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots, X_t^k\}$. Is there a way to determine how much of each series is dependent on the others. Formal Question More ...
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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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Bayesian information criterion from measure theoretic point of view?

Bayesian information criterion (BIC) is well known and it is derived from the maximizing the posterior density function which is equivalent to solving the marginal likelihood integral. My question is: ...
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Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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t-distribution and Degrees of freedom

Why t- distribution have n-1 degrees of freedom? I know that it is used when population variance is not known but what determines n-1
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Why not use always a binomial exact test to compare two proportions instead of chi square?

I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that ...
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Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
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Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions $f_i:X\...
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389 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ \...
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Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
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Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
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consistency of Neyman Pearson lemma in the case simple vs simple test for exponential families

Basics, jump to section 2 for the question : I know that in the case of an exponential family with 1 parameter, meaning the distribution function of the sample variables can be written like : $$ f_X(...
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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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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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prove that $(n-1)(\frac{1}{n} \sum_{i=1}^{n}\hat{\sigma^2_{-i}}-\hat{\sigma^2})=-\frac{S^2}{n}$ in i.i.d sample(the factor n-1 in the jackknife bias)

let $x=(x_1,\cdots,x_n) $ are n independent samples from unknown distribution. The Jackknife Samples are selected by taking the original data vector and deleting one observation from the set. Thus, ...
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Find the UMVUE of $b^{\mu}$

Let $X_1,X_2,..X_n$ be a random sample from Cauchy$(\mu,1)$ population. Find the UMVUE of $b^{\mu}$ where $b$ is any positive real number. Now actually calculating sample mean won't work here because ...
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Minimal sufficient statistic criterion

(Note: This is not about Bayesian inference, but about classical inference) Let $\{P_\theta\}_{\theta\in \Theta}$ be a family of probability measures on $\mathbb{R}^n$ with density functions $f_\...
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Finding Best Unbiased Estimator of Uniform Distribution

Let $X_i$, $i=1,...,n$ be iid with $f(x,\theta) = \frac{1}{2\theta}$ for $-\theta<x<\theta$. Find the best unbiased estimator of $\theta$ if one exists. So I first tried $T(X)=X_{(n)}$, which ...
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Confidence interval for mean of a normal population

Consider a normal population with unknown mean $\mu$ and variance $\sigma^2=9$. To test $H_{0}:\mu=0$,against $H_{1}:\mu \ne 0$. A random sample of size 100 is taken. Based on this sample, the test ...
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Sufficient statistic for a function of the parameter

We know that if $T$ is a sufficient statistic for $\theta$ then $f(T)$ is a sufficient statistic for $f(\theta)$ if $f(.)$ is a one -one function. But,what if $f$ is not one one? For example, in case ...
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To find which of the following statistics is (are) sufficient but NOT complete

Let $X_1,X_2,X_3,....X_n$ be a random sample from a distribution with the probability density function $$f(x|\theta) = \begin{cases} \dfrac{x}{\theta^2}e^{\frac{-x}{\theta}}, & \text{if $x>0$;...
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Causal Inference A Primer Study Question

I am reading Pearl's Causal Inference book and attempted at solving study question 1.2.4. Here is the entire problem: In an attempt to estimate the effectiveness of a new drug, a randomized ...
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Simplifying Likelihood Ratio

(Answered here: https://stats.stackexchange.com/questions/372040/rejection-region-for-likelihood-ratio-test) I have a data set $((Y_1,x_1),(Y_2,x_2),...,(Y_n,x_n))$ where $Y_i$ is distributed as $N(\...
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Sampling error of correlation coefficient (Phi coefficient) for binary variables

Suppose I have two correlated binary variables (A and B) with known probabilities ($p_a$ and $p_b$) and correlation (Phi) coefficient in population - $\rho$ . Is there any analytic function for ...
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Equivalent defintions of minimal sufficient statistics

Wikipedia claims that the statistic $S(X)$ is minimal sufficient if and only if $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$ $\iff$ $S(x) = S(y)$. It is also claimed that this is a ...
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noise-free Gaussian Process likelyhood

I am learning Gaussian Process reading GPML. I am a bit confused with understanding the Bayesian analysis. Let consider the standard linear regression model with "Gaussian noise", i.e, $$ f(\textbf{x}...
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UMVUE of $\sqrt{a}/b$ for Gamma distribution

Suppose $(X_1,X_2,\ldots,X_n)\sim \operatorname{Gamma}(a,b)$, independent and identically distributed with pdf: $$f(x)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx},\quad x>0$$ Find the UMVUE of $\frac{\...
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Can a weakly consistent estimator beat a strongly consistent one?

Suppose we have two estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ of $\theta$, both with the same bias. If we have $$ \begin{align} &\hat{\theta}_1 \xrightarrow{a.s.}\ \theta \\ &\hat{\...
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Help understanding Casella & Berger's explanation of a sufficient statistic

This is from Casella and Berger's Statistical Inference: Definition: A statistic $T(\mathbf{X})$ is a sufficient statistic for $\theta$ if the conditional distribution of the sample $\mathbf{X}$ ...
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Proof of Cochran's theorem?

I'm trying to understand a proof of Cochran's theorem Let $X_1, X_2, \ldots, X_n$ be a random sample from an $N(0,1)$ distribution and let $x$ represent the vector of these observations. ...