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Questions tagged [inference]

Drawing conclusions about population parameters from sample data. See https://en.wikipedia.org/wiki/Inference and https://en.wikipedia.org/wiki/Statistical_inference

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271 views

Dealing with dependent data in a Bayesian model

Background: Consider a series of dependent data points, $$ y_1,y_2,y_3,\cdots,y_N. $$ In cases where the dependence is well described by an exponentially decaying auto-correlation function, it is ...
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129 views

Finding MLE and MSE of $\theta$ where $f_X(x\mid\theta)=\theta x^{−2} I_{x\geq\theta}(x)$

Consider i.i.d random variables $X_1$, $X_2$, . . . , $X_n$ having pdf $$f_X(x\mid\theta) = \begin{cases} \theta x^{−2} & x\geq\theta \\ 0 & x\lt\theta \end{cases}$$ where $\theta \...
6
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1answer
260 views

Sufficient statistics for $\mu_1 - \mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known,...
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134 views

Multiple maximum likelihood estimates for discrete parameter

Suppose I have a bivariate likelihood function, $L(\theta ,\lambda |\mathbf{x})$, where $\theta$ can take on continuous values, but $\lambda$ can only take 'count' values $(0,1,2,...)$, and $\mathbf{x}...
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91 views

Question on Inference - Catching Cheating Students

In their paper "Catching cheating students", Levitt and Lin propose a simple reduced-form method to identify cheating of students in exams. The strategy works as follows: For each possible pair of ...
5
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192 views

Effects of model selection and misspecification testing on inference: Probabilistic Reduction approach (Aris Spanos)

This question is about pre-test bias, inference after model selection and data snooping within the Probabilistic Reduction (PR) methodology by Aris Spanos (which is related to the Error Statistics ...
5
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244 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
5
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1answer
141 views

How to do inference over two steps in a graphical model simultaneously?

I have observed data $D$ about a physical object described by $M$. I would like to determine the posterior distribution of $M$ given $D$, or $p(M|D)$. Now I can't infer this directly because unknown ...
5
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701 views

Variance of marginal posterior distribution

Suppose $Y_1,\dots,Y_n\mid\mu,\sigma^2 \sim \text{ iid } N(\mu,\sigma^2)$ and suppose the priors $\mu \mid \sigma^2 \sim N(\mu_0, \sigma^2 / \kappa_0)$ and $1/\sigma^2 \sim \text{gamma}(\nu_0/2, \nu_0 ...
4
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38 views

How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
4
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27 views

Before using CV-selected Regression model for Inference, shouldn't model performance be evaluated on unused test set?

I just came across a biokinesiology paper that used some Machine Learning methods, but I think there is a flaw in their methodology. The authors had data on stroke patients and used Lasso regression ...
4
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42 views

If a statistic can be written as a function of a minimal sufficient statistic almost everywhere, is it minimal sufficient?

I know that if $T(X) = f(W(X))$ for one-to-one $f$, where $W(X)$ is minimal sufficient, then $T(X)$ is also minimal sufficient. But my textbook does not include "almost everywhere" or "almost surely" ...
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86 views

Interesting application of E-M algorithm

Suppose the following dataset: [3 4 3 4 6 12 12 7 8 9] [2 5 3 4 12 2 2 10 7 6] [3 4 3 4 5 11 10 7 8 9] These numbers are totally random. So this dataset, depicts ...
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55 views

Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals (CIs) for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fitted parameters and after that to ...
4
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314 views

Optimal Stopping for Bernoulli One-Armed Bandit with a Fixed, Known Payout

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. Suppose I have two arms $k = {1, ...
3
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37 views

Who invented train/validation/Test method and when?

I can't seem to find here or in other places the earliest source for this method. it seems the holdout method was separately proposed by Highleyman in 1962, and cross validation was separately ...
3
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24 views

How to Use Chi-Squared Test for Inference about Three-way independence

If I recall correctly, three random variables X, Y, and Z are three-way independent iff these two statements are met: P(X∩Y∩Z) = P(X)P(Y)P(Z) X, Y, and Z are all pairwise independent of each other. ...
3
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29 views

How much of a problem is inference after model selection when few models are manually compared?

tl;dr: I found a better model than the one I first thought of while inspecting the data and performed a few steps of variable selection/model fine-tuning. I assume that this is a (mild) case of ...
3
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1answer
649 views

use and misuse of Winsorization

I am doing research on Winsorization (and trimming), which has been broadly applied in many fields, but I think many researchers didn't do it in a "rigorous" way. Or maybe even worse, they misuse it. ...
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151 views

Existence of UMVUE of $\theta$ for sample from $\small{\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}}$?

Suppose $X_1,X_2,\ldots,X_n$ is a random sample drawn from a distribution with pdf $$f_{\theta}(x)=\small\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}\quad,\,\theta>1$$ Does there ...
3
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124 views

Finding the UMVUE of $\theta^2$ where $f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of ${\theta^2}$ I ...
3
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1answer
40 views

Survival model for an epidemic — can the observations be treated as independent?

I've been thinking about ways to tackle an epidemic modelling problem I've been working on, and I've come up against a conceptual difficulty over the way survival analysis works. Here's a really ...
3
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56 views

Fit data based on generating function

Suppose I have iid data generated from a discrete random variable $X_i \sim D(\lambda)$, and I would like to infer the parameter $\lambda$. Unfortunately, I do not know the likelihood function for $D$,...
3
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77 views

What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?

My textbook gives the following theorem for exponential families: Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(...
3
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72 views

confusion about the link between residuals, error terms, sample size and CLT in ANOVA

I feel a little confused about the assumption of the ANOVA and what it ensures mathematically the errors have to be iid and normally distributed N(0,1). independance of observation. Is it not a ...
3
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0answers
64 views

Bayesian inference of a coin's bias when we don't directly observe the flips

Consider a coin with bias $p$. We generate a random sample $x_1, \dots, x_n \sim \text{Bernoulli}(p)$, but we do not observe results of these coin tosses. Instead, for each $x_i$, we observe a set of ...
3
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51 views

Show that there is no efficient estimator for the variance of a normal distribution using properties of the exponential family

I want to prove the statement in the title using the following statement from Wikipedia: it was proved that efficient estimation is possible only in an exponential family, and only for the natural ...
3
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88 views

Can an asymptotically efficient estimator be biased?

In "Theory of point estimation" by Lehmann and Casella (1998) there is the following definition: It is also said that So terms of the asymptotically normal sequence of estimators can be ...
3
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83 views

Dealing with multiple cases per subject

I want to analyze a table with surgery operations data. For some patients, about 10%, multiple operations have been made, so there may be multiple rows for a patient in the table. My goal is to ...
3
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0answers
101 views

Mean of a Pearson Type VI distribution

I have a question that gives me the density of a Pearson type VI distribution and then says to state the range of parameter(s) for which the expression for the mean is valid. In your calculations, ...
3
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0answers
93 views

Likelihood with random censoring

Suppose to observe a random sample from a r.v. $Y_i=\min(T_i,C_i)$ where $T$ and $C$ are iid absolutely continuous distribution. I would like to inference about a parameter of $T$ (for example, $\...
3
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117 views

What type of drawn sample is more informative about $\theta$?

We have two sampling methods: 1. $X_1,\ldots,X_n$ from a Bernoulli$(\theta)$ distribution; 2. $Y_1,\ldots,Y_n$ from a Geom($\theta$) distribution. Which is more informative about $\theta$ and ...
3
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0answers
94 views

Does MCMC method can be used to calculate the mean and variance of the distribution of random variable functions?

I am not professional in Probability & Statisticsin, in order to clearly describe my problem, please be patient of the long introduction.THANKS! Background of my question Assume I have several ...
3
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0answers
239 views

Residual Bootstrap for Lasso Inference

I am a bit confused why the standard residual bootstrap fails to make consistent inference for lasso. Especially, it is claimed that the standard residual bootstrap fails to reproduced the signs of ...
3
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0answers
133 views

Model training: difference between inference, marginalisation, and estimation

I am working through the lecture slides of Carl Rasmussen's Probablistic Machine Learning course. In slide 6 of the first lecture it lists a number of ways that one can learn the parameters (A, C) and ...
3
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469 views

Under what conditions will a Bayesian posterior fail to converge to a point mass?

Let's say you have a Bayesian model: $$\theta' \sim g(\theta|\mu) $$ $$ y \sim p(y|\theta')$$ And we have some data ($n$ data points) $\mathbf{y}_n$, which we will use to perform inference on $\...
3
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2answers
68 views

Looking for resources (papers, books) that explain the impact that non-random sampling has in test statistics

The majority (if not all) of test statistics assume random sampling. Consequently, probability values obtained in t-tests, ANOVAs, regression, HLM, etc., are intrinsically linked to the assumption of ...
3
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0answers
413 views

Inference for quasibinomial GLM with LASSO penalty using selectiveInference package

I would like to carry out inference on a binomial LASSO model, but take into account the fact that my data are overdispersed and use the quasibinomial family instead. R package ...
3
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1answer
253 views

How to make Bayesian-style inference for a Poisson process?

I am working on a fleet management software recently. Normally, the arrival of merchant request is a Poisson process. That is to say, on average we have a new merchant request every 10 minutes, but ...
3
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0answers
146 views

Check that a statistic is complete

I have a question regarding completeness of a statistic. So the problem is: $n$ numbers are chosen randomly and independently between $a$ and $b$ ($0 < a < b$) but the information about $a$ and ...
3
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1answer
58 views

Inference using Gibbs sampling

Suppose there is a one-dimensional normal distribution $\mathcal{N}(\mu, \sigma)$ for which we want to infer the joint distribution of the parameters using Gibbs sampling. Let $D$ be the data, ...
3
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0answers
62 views

Finding the limiting distribution of sample covariance $\hat{\sigma}_{XY}$?

How can I find the limiting / asymptotic distribution of $\sqrt{n}(\hat{\sigma}_{XY}-\sigma_{XY})$, provided $\sigma_{XY}=E[(X-\mu_X)(Y-\mu_Y)]$ and $\hat{\sigma}_{XY} = n^{-1}\sum(X_i-\bar{X})(...
3
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0answers
373 views

How to check the assumptions behind an inference procedure, in the case of very large data sets

On this site it has been confirmed multiple times that, contrary to what is often heard, hypothesis tests don't have any issues with large sample sizes. As a matter of fact, the probability of Type 1 ...
3
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1answer
102 views

How do I perform Bayesian Updating for a function of multiple parameters, each with its own distribution?

I have a variable that is a recursive function involving other variables with known distributions (see problem below). Let $b(t+1) = b(t) + C \sqrt{b(t)}$ where I know $C \sim N(1.82, .0298)$ and ...
3
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1answer
103 views

Can an independent t-test be used on paired data when the pairing is unknown?

Suppose the effectiveness of a training course is examined, and performance of each individual in a group is taken both before and after, and the differences are compared in a paired $t$-test. Would ...
3
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1answer
56 views

Parameter Inference when Model is a bad fit to the data.

I am working with gamma-ray data from the Fermi Satellite. The data has been binned into energy dependent maps of the sky -- e.g. three dimensions (energy, latitude, longitude) and is extremely high ...
3
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0answers
341 views

Finding the uniformly most powerful test

Let $X_1,X_2,...,X_n$ denote a random sample from density, $$f(x;\theta)={1\over 2\theta}, \quad 0<x<2\theta.$$ Find the uniformly most powerful test for testing $H_0:\theta \le \theta_0$ vs ...
3
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0answers
75 views

Given a single sample X from $N(0, \theta)$ is $|X|$ a sufficient statistic for $\theta$?

My first idea on how to proceed was to treat $|X|$ as piecewise where $X=$ $X$ for $X \in [0, \infty)$ and $-X$ for $X \in (-\infty, 0)$, then use the conditional probability definition for a ...
3
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0answers
71 views

Confidence and credible interval: cases

I am having difficulties in understanding these two approaches. Let's say given the data I compute both confidence and credible interval, then what is the intuition/interpretation of having: Big CI ...
3
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0answers
534 views

In a simple hypothesis vs composite hypothesis test, are the p-values the same?

In a simple hypothesis test, we have something like $H_0 = 5$ and $H_A = 10$ while in a composite hypothesis test we have something like $H_0 = 5$ and $H_A >5$. Since the p-value is defined to be ...