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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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UMVUE of q/p when X∼bin(n,p) [closed]

Finding UMVUE of q/p is similar to (1/p) - 1. How to proceed
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Are $\mu_{\hat{p}}$ and $\sigma_{\hat{p}}$ considered parameters or statistics?

Is $\mu_{\hat{p}}$ (the mean of the sampling distribution of $\hat{p}$) and $\sigma_{\hat{p}}$ (the standard deviation of the sampling distribution of $\hat{p}$) considered parameters or statistics, ...
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Logic behind hypothesis testing and one-tailed tests

Suppose I have the following: Let's consider a hypothetical experiment to determine whether James Bond can tell the difference between a shaken and a stirred martini. Suppose we gave James Bond a ...
Darren's user avatar
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Measuring departure between the posterior predictive distribution and the true data generating distribution

Suppose, I am trying to measure the departure between the posterior predictive distribution and the data generating distribution. So, in this case, assume that there is a single observation $$X \sim N\...
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Average Rank versus Ranked Average in Parameter Estimation

I have the following problem: In a cricket tournament, the eleven batsmen of a team play 100 matches before the final. The runs scored by each are available. Determine the average rank of the batsmen ...
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Unequal observation Two-Way ANOVA

Righto, I have some data on fish. I want to compare the weight of male and female (F = 1) fish across different ponds (Ponds 1 to 4) so using a two-way ANOVA. I want to do this in Excel, using a Two-...
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Understanding equicontinuity in asymptotic normality with nonsmooth objective functions

I am working on the normality of extremum estimators with nonsmooth objective functions. Assume that my objective function is $Q_n(\theta)$, where $n$ is the sample size. I denote by $\hat \theta_n$ ...
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Help developing intuition behind sufficient statistics (Casella & Berger)

Migrated to Cross Validated I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272): A sufficient statistic captures all of the ...
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Creating an Estimator for the Dimension of Bernoulli-distributed Vectors from Observed pairwise Dot Products

I have I individuals defined by vectors $P_i \sim \mathcal{B}(1,1/2)^d$ iid. We can note $\overline{P}_i = \langle P_i, \textbf{1} \rangle$ the proportion of 1's in individual i; $c_{ij} = \langle P_i,...
yann kerzreho's user avatar
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Fisher's information for a function that consist in many indicator functions

I have the following pdf: $$ f(x) = \theta I_{(-\frac{1}{2},0]}+ I_{(0,\frac{1}{2}]}+(1-\theta) I_{(\frac{1}{2},1]} $$ I've tried the following \begin{align} I(\theta) &=-E[\frac{d^2}{d\...
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Intuition for bounds of Adaptive Conformal Inference

I have been reading the paper by E. Candès and Gibbs about Adaptive Conformal Inference (here is the original papel). The main idea is to update the miscoverage level $\alpha_t$ as $ \begin{cases} \...
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The sufficient statistic and unbiased estimator of normal variance

Suppose we have a normal distribution with mean $\theta_1$ and variance $\theta_2$. I know that $\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is an unbaised estimator of $\theta_2$ and has a variance $2\...
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The idea/intuition behind replacing elements in bootstrapping(Statistics)

I have read several posts on this, none of them directly deals with this. I don't understand the idea/intuition behind replacing elements when one bootstraps(Statistics). As in given a data set that ...
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Maximum Likelihood Estimation for Poisson Mean with Given Observations

You have a sample of $n$ i.i.d. realizations of the random variable $X$ distributed as a Poisson with parameter $\lambda$. It is known that: $n_1$ values are greater than or equal to $2$; $n_2$ ...
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How to calculate risk of choosing option0 vs option1

I have an expirement where every day a person has to pick between $image0$ and $image1$, where based on your selection you can win $1$, loose $1$, or $0$. I am given the following probabilities: $r0$: ...
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the sum of $O_p$ --$ O_p(s^2\frac{\log d}{n}+s\sqrt{\frac{\log d}{n}}) $

I read papers in the area of inference for high-dimensional graphical models and these papers always state the convergence rate of the estimator. Using $O_p$ is a good choice. Maybe I made some ...
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How to express Variational Auto-Encoder, ELBO with Random Variables?

In the context of VAE, variational inference and Bayesian statistics more broadly the equations typically involve densities that are somewhat loosely defined. Often the notation refers to a ...
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Probability that one estimator is larger than another

Imagine having $n$ independent and identically distributed (i.i.d.) observations from a variable $X$, which in the population follows a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$. For $\sigma^...
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How to compare real-time physiological data between pre-defined time blocks

I have a set of physiological data for approx. 20 patients measured per second over approximately 8 to 10 minutes. The data is grouped into the following blocks of time: calibration, test, first ...
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P-value for testing a median to be $M \geq M_0$

I'm trying to solve a question from an introductory textbook on statistics. I am to use the Sign Test and determine if there is significant evidence that the median of a dataset $M$ is "at least&...
fatCat9999's user avatar
5 votes
2 answers
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What is the frequentist's Bayesian prior for a coin with unknown bias

A "coin" has a fixed unknown bias $0\le p\le1$ for heads, and out of $n\ge0$ tosses it yielded $0\le h\le n$ heads. Note that this occurs with probability $P(h\;|\;p,n)=\binom{n}{h}p^h(1-p)^{...
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Estimate the correlation coefficient of a two-dimentional normal distribution $(X,Y)$, given some samples of $(|X|, |Y|)$

I have two random variables $X,Y$, whose joint distribution is a two-dimensional normal distribution, and the expectations of both $X,Y$ are zero. Let $\rho={\rm cov}(X,Y)$ be their correlation ...
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When to use t-distribution?

Let the dataset $6, 12, 12, 9, 7, 16, 10$ be given. Each data point is considered to be an outcome of a random variable $X_i$, where the $X_i$'s are assumed to be independent and Poisson distributed ...
Need_MathHelp's user avatar
4 votes
2 answers
257 views

Given the maximum likelihood function- estimate the value of the parameter

Lets say I have the pdf and maximum likelihood function: $ f_X(x) = \begin{cases} \frac{\alpha \beta^\alpha}{x^{\alpha+1}}, & x > \beta, \\ 0, & x \leq \beta. \end{cases} $ $ \begin{...
Need_MathHelp's user avatar
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2 answers
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What exactly is P-value and what is its relation with the significance level?

The formal definitions I have seen differ but the one I thought I understood was: "the probability of an observed or more extreme result assuming that the null hypothesis is true". Let's say ...
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Expected Prediction Error from The Elements of Statistical Learning

I am working self-studying the Elements of Statistical Learning and have a question regarding how equation $(2.28)$ is derived. Similar question here but without a satisfactory answer: How to ...
InvestingScientist's user avatar
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Evaulating conditional pdf efficiently given marginal pdfs

Let $\mathbf{A}$ and $\mathbf{B}$ be two random variables with joint distribution $p(\mathbf{A},\mathbf{B})$. The joint prior is defined in relation to a data model, given by $$\mathbf{y} = f(\mathbf{...
Zero's user avatar
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when and how can you convert any summation into an integral?

I was wonder through my self-study of physics if there is a consistent way of converting sum to integrals, because it seems all tricks for me. For instance, the simplest case is the riemann sum but ...
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Confidence interval for difference between means

I have the following information about data: $$\sum_{i=1}^{12}x_i=2114; \;\sum_{i=1}^{12}y_i=2144;\;\sum_{i=1}^{12}x_i^2=373160;\;\sum_{i=1}^{12}y_i^2=383666$$ And I want to calculate the confidence ...
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Collaborative Planning, Forecasting, and Replenishment (CPFR) model

I'm trying to understand better the CPFR model but I can't find anywhere a numerical example of this. I'm looking for a numerical example with solution for Collaborative Planning, Forecasting, and ...
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Measuring Robustness in Variational Bayesian Inference and Nonlinear Filtering

I am interested in how to properly pose/measure robustness, in a qualitative or potentially quantitative manner, when inferring a probability density function (pdf) either by Bayes' rule or a ...
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How to understand likelihood function bayesian

$\mathcal{N}(W^T \cdot X, \beta^{-1})$ This is the likelihood distribution for Bayesian linear regression, right? So, the thing is, if I'm doing batch mode Bayesian regression, then: Weights (W): Size:...
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Bayesian linear regression about finding the likelihood

Pick a single data point $(x,t)$ and calculate and plot the likelihood for this single data point across all $w$ in your parameter space $(w_0 \times w_1)$ (for a single data point it is a univariate ...
User's user avatar
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Can a Student's t-test be used to identify whether or not a single observed data point should be judged as an outlier from a null distribution?

FYI: This question may be more thematically appropriate for Cross Validated's site, but, from previous experience, there is considerably less traffic there compared to Math, and I still feel that this ...
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1 vote
1 answer
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Confidence interval with shortest length for location parameter of uniform distribution $U (\theta, \theta+1)$ based on pivot $X_{(1)} - \theta$

$X_i$ is iid from a uniform distribution on $(\theta, \theta + 1)$, $\theta \in R$. Show that $X_{(1)} − \theta$ is a pivot. Showing that the pivot pdf does not depend on theta, i solved by first ...
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Problem about critical value for multiple comparisons

Consider a one-way layout of fixed effect model: $$y_{ij}=\eta+\tau_i+\epsilon_{ij}, \epsilon_{ij} \sim N(0,\sigma^2),i=1,...,k,\ j=1,...,n,$$ $\tau_i$ is the main effect, and we want to test $H_0:\...
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Asymptotic Distribution and Describe Sources of Increasing Power in an hypothesis testing problem

I am currently dealing with the following problem in a past exam (with no solution): Suppose $S$ follows the Poisson distribution with mean $2\lambda>0$, here $\lambda$ is a parameter. Another two ...
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Bayes Credible set for Gamma distribution

Given iid observations of x that follows $beta(\theta, 1)$ pdf and we assume $\theta$ has a $gamma(r,\lambda)$ prior pdf. The likelihood function of the beta is given as $$L(\theta|x_i) = \theta^n e^{-...
Maale Faustus's user avatar
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What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves?

!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
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1 answer
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Choosing Null Hypothesis

A company producing light bulbs is interested in the average lifetime (in hours) of their bulbs. It is assumed that the lifetime (in hours) of a light bulb follows an exponential distribution. The ...
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1 answer
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How to use argmax in Bayesian posterior estimate?

I have some yield function $f_\beta(x)$. I want to find the value $x$ that maximizes the yield. However, my function is parameterized by a parameter $\beta$. For simplicity let's assume there is only ...
Willem's user avatar
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Find minimal sufficient statistic of this random sample with cursed support

Suppose $X_1,X_2,...,X_n$ is a i.i.d random sample with probability mass function $p(x_i,\theta)$ where $x_i \in \{\theta,\theta+1,\theta+2,...\}$ and $\theta \in \mathbb{R}$. I claim that minimal ...
ArshakParsa 's user avatar
1 vote
1 answer
60 views

Calculate the posterior distribution

How can I solve the letter (a)? Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from $1$ to $N$. You see a cable car at random; it is numbered $203$. You ...
Siqueira's user avatar
2 votes
1 answer
39 views

Bayesian Inference Intractability

When looking at Bayesian posteriors $$ p(z \mid x) = \frac{p(x \mid z)p(z)}{\int p(x \mid z')p(z')dz'} $$ The denominator commonly intractable. I understand this is due to the possibility of high ...
Lehmann's user avatar
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1 answer
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UMVUE of $\mathbb{E}[X^2]=\lambda^2 + \lambda$ where $X\sim\mathrm{Pois}(\lambda)$.

This is the same question as this: UMVUE of $E[X^2]$ where $X_i$ is Poisson $(\lambda)$. Here, I restate the problem for completeness: Let $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} \mathrm{Pois}...
pbb's user avatar
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1 answer
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Are sample mean and variance unique estimators?

I get that the sample mean and sample variance are unbiased and consistent estimators. But i'm wondering if they are unique estimators that are unbiased and consistent. i.e. for any statistic A, if A ...
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1 vote
0 answers
129 views

Exponential Family with Complete Sufficient Statistic

Suppose that $X$ is in an exponential family taking values in $\sigma$-finite space $(\mathcal{X}, F_{\mathcal{X}}, \nu)$ probability density function $f_{\theta}(x)=h(x) \exp \{\eta(\theta)^T T(x)-\...
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Basu's theorem and completeness

Recently, I was reading up on the Basu's theorem and what i gathered of it was that if a statistic $T$ is complete and minimal sufficient then it is free from Ancillary statistics. My question is why ...
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Econometrics Question - Causal effect in a non-randomized trial

I am trying to establish a specification of a binary choice model (logit/probit) that dictates treatment assignment. The context (and subsequent cross-sectional data) is related to a government that ...
econstudent's user avatar
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How to Calculate Probabilities for a Causal Graph?

I was reading a paper on causal inference. In this paper I came up with this causal graph In this causal graph, if we ignore E (Environment variable), and consider all variables are binary for ...
Swakshar Deb's user avatar

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