Inference, in a statistical context, refers to drawing conclusions from data containing an element of randomness introduced by e.g. measurement error, sampling variation, or assignment of experimental treatments. A common inferential paradigm is drawing conclusions about population parameters from ...

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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 ...
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34 views

nonparametric method to calculate the probability how alike two samples are

I have two samples with each couple of hunderd observations. I want to calculate a probabilty how much they look alike. I'm aware of tests like kolmogorov smirnov but I don't think I need this. I ...
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25 views

Greedy subtree selection in Nested Hierarchical Dirichlet Processes

I'm implementing the Nested Hierarchical Dirichlet Process as described in this paper by Paisly et. al, 2014: http://arxiv.org/abs/1210.6738 My question is about the variational objective in Equation ...
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27 views

Is there any method to quantify parameter estimation uncertainty of method of moments fitting technique?

If I want to fit a distribution (let's say we can be certain about the type) to observations using maximum-likelihood method, I have many options to express the parameter estimation uncertainty due to ...
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45 views

Copulas with Regression

Copulas are joint distribution of uniform marginal distributions. Traditionally I have seen examples of fitting a Copula to the data and then simulating from the data. I haven't seen much on Copula ...
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184 views

Why doesn't the standard deviation represent a normal distribution?

Why doesn't the standard deviation of a sufficiently large sample represent a normal distribution that we can make inferences from? Let me list my thought process, so hopefully someone can highlight ...
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21 views

How to find three probabilities with two different values or ratings?

I would like to know how to find three probabilities of two values.... Specifically...I want to know the three soccer venues (HOME DRAW AWAY) proabilities with two ratings... Example: I have two ...
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25 views

How to infer correlations from correlations

I have a question regarding correlation inference. Consider, I have two sets of variables X and Y. For an x element of X I know the correlation to an unknown variable z. I also have the covariance ...
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1answer
37 views

Inference about the true intercept of the model and the OLS being BLUE

Consider the following population regression model: $$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$ where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = ...
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1answer
34 views

Poisson confidence interval using the pivotal method

I am trying to build a confidence interval for the Poisson distribution using the pivotal method. I have the theory down but I am struggling to come up with $h(Y, \lambda)$, the probability ...
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64 views

Expectation-Maximization with dependent latent variables

Deriving the equations for a Expectation Maximization over my model, I end up with a posterior for the latent variables (E-step) that prevents me from going on. Generative model I assume my data is ...
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1answer
31 views

Backward message passing in variational Bayesian inference

I have come across in a research paper that, I do understand the logic. But the paper has't mentioned about the way of updating $\eta_{t}$. When I asked from the authors they said when we equate ...
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1answer
39 views

Bayesian inference

Assume two demographics $[F,M]$ and each person has a choice of attending only one of four different lectures $[A,B,C,D]$ all occurring at the same time so they can only attend one. The following ...
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23 views

Conjugate prior equivalent prior sample size with respect to the mean

In Cowles's book ([Applied Bayesian Statistics - With R and OpenBUGS Examples–(http://www.springer.com/statistics/statistical+theory+and+methods/book/978-1-4614-5695-7)), page 108, there is a ...
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69 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, ...
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19 views

How Can I Build A Regression Model With Collinear Data?

Hello there my fellow Cross Validated members; I’m here today to brainstorm a little bit with all of you out there, to flesh out our collectively acquired data analytic skills, and to try and find new ...
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5 views

Multiplying distributions with different conditioning

I saw this expression in a UBC machine learning class lecture, and I'd like to understand how the math works. Suppose we're trying to predict a class label $y$ given some data $x$. There are prior ...
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16 views

Variance of precision in conjugate prior

How can I calculate the variance of the precision in a normal distribution, knowing I used a conjugate prior?
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105 views

What is the distribution of the conditional mean E(Y|X) in a multiple regression?

Suppose the model is $$ Y = b_0 + b_1X_1 + b_2X_2 + b_3D + b_4X_1D + e \\ e \sim\mathcal N(0, \sigma^2) $$ Where $D$ is a categorical variable. $$ E(Y|X_1, X_2, D=1) \sim\mathcal ?? \\ E(Y|X_1, ...
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30 views

asymptotic distribution of joint random variables

I am trying to understand the asymptotic distribution of the following expression under normality $$ {\hat \sigma \hat S - \sigma S} $$ Where $\sigma$ and $S$ are the population standard deviation ...
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16 views

Compare two diagnostic Likelihood Ratios

I want to compare the LR+ and LR- in substratas according to age, gender, duration of symtoms. LR+ is the proportion of true positives/proportion of false positives so what I need is to compare ratios ...
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52 views

What is relationship between Fisher Information and Variance in natural exponential Family?

I know that $Var(\hat\theta)\geq 1/I(\theta)$ where $I(\theta)$ is Fisher information. Let take an example of natural exponential family with density $f(x)=\lambda\exp(-\lambda x)$. In this case we ...
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19 views

Score Confidence Interval for Difference of Proportions

In Agresti et al. (2008) "Simultaneous confidence intervals for comparing binomial parameter", it is suggested to use a studentized range distribution with a score statistic. However, exactly how you ...
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13 views

Is data fusion suitable for this application?

I have a situation where the delay vehicles experience at an intersection can be obtained through 3 or 4 types of sensors. Each sensor provides its own type of data with their own unique ...
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1answer
39 views

What criterion are used to accept/reject hypotheses in ridge regression?

On what basis might one accept/reject a hypothesis when running ridge regression? For example, if I have five predictor variables as part of a ridge regression... what criterion would I use to accept ...
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12 views

Why are chordal graphs special for inference in the context of Probabilistic graphical model?

I was trying to make a list of the reasons of why chordal graphs are important or interesting in the context of inference and probabilistic graphical models. Some of the reasons I have so far are: ...
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16 views

adaptive surveying for a maximal income that depends on a parameter

I have a product that I would like to price for the highest income. The income $I$ from this product will depend on the asking price $c$: $$ I(c) = N \cdot E(c)$$ where $E(c)$ is the expected ...
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22 views

Sampling: the % of homeowners who own at least 2 TVs

It is planned to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than ...
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104 views

Observed Fisher information under a transformation

From "In All Likelihood: Statistical Modeling and Inference Using Likelihood" by Y. Pawitan, the likelihood of a re-parameterization $\theta\mapsto g(\theta)=\psi$ is defined as $$ ...
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1answer
42 views

Finding the unbiased variance estimator in high dimensional spaces

The problem comes from linear regression. Assume the regression function is linear, i.e. $$ f(X) = \beta_0+\sum_{j=1}^pX_j\beta_j $$ .Given a set of training data $(x_1, y_1),\ldots,(x_N,y_N)$,we try ...
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195 views

Invalid inference when observations are not independent

I learned in elementary statistics that, with a general linear model, for inferences to be valid, observations must be independent. When clustering occurs, independence may no longer hold leading to ...
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120 views

Example of an inconsistent Maximum likelihood estimator

I'm reading a comment to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the power of a likelihood ratio or ...
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35 views

Inference methods for multimodality and label switching

Imagine that there are three professions in the world $a,b,c$ (astronauts, doctors and statisticians) and that the Gross Domestic Product (GDP) of a city can be modeled as a linear regression of its ...
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44 views

Sufficient and complete statistic

Let $ X_1, ... , X_n $ be i.i.d random variables with pdf given by $$f(x;\theta) = \exp(-(x-\theta))I_{(\theta, \infty)}(x)$$ It is asked to find a sufficient statistics for $ \theta $ and to verify ...
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25 views

Robust multivariate Wald test for significance in proportional odds model

I am using the rms package (Harrell) to estimate a proportional odds model to determine the association between an ordinal outcome (frequency of pain) and the ...
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1answer
71 views

LR test on marginal effect

Say I have the following regression model: $$\text{Wage}_i = constant + α·\text{YearsOfEduc}_i + β·\text{Age}_i + γ·\text{CompletedHighSchool}_i + \mbox{δ·$\text{NumOfSiblings}_i$} + ...
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1answer
43 views

Wald test on marginal effect

Say I have the following regression equation: $$Wage_i = YearsOfEduc_i + Age_i + NumOfSiblings_i + u_i$$ How would I go about peforming a wald test of the hypothesis that for an individual with ...
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110 views

Should we address multiple comparisons adjustments when using confidence intervals?

Suppose we have a multiple comparisons scenario like such as post hoc inference on pairwise statistics, or a multiple regression, where we are making a total of $m$ comparisons. Suppose also, that we ...
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70 views

Unbiased estimator with minimum variance for $1/\theta$

Let$ X_1, ...,X_n$ be a random sample feom a distribution $Geometric(\theta)$ for $0<\theta<1$. I.e, $$p_{\theta}(x)=\theta(1-\theta)^{x-1} I_{\{1,2,...\}}(x)$$ Find the unbiased estimator ...
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15 views

Detecting parameter influence

I have a data set consisting of a system's responses to various test configurations. Every test configuration corresponds to a different parameter set. These parameters can have either continuous ...
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9 views

Combining weighted evidence based probabilities?

I'm trying to identify people by determining if a data sample matches a set of existing samples (assume DNA if it helps). In addition to the samples I have a function which gives a probability that ...
3
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68 views

Inference of Pearson's rho from distribution perturbation

I would like to infer the correlation between random variables $Q$ and $R$, however, I have access only to the distribution of $Q$ and the distribution of $P=Q+R$. We can see how Pearson's $\rho$ ...
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2answers
106 views

Maximum likelihood estimator for $\theta$ and $E[X]$

Let $X_1,..., X_n $ be a random sample of a variable with PDF: $$f(x|\theta)=\frac{\theta}{x^2} I_{(\theta, \infty)}(x), \theta >0$$ Find the maximum likelihood estimator for $\theta$ and $ E[X]$ ...
2
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2answers
125 views

How can I calculate t-score without knowing true population mean?

I am studying now t-scores. As far as I understand, t-scores are used when we don't know true population parameters (such as: standard deviation and population mean) and cant use z-scores. Here is ...
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1answer
80 views

Bayesian inference of a clinical trial for clinicians

I am a clinician who is more adept than average at interpreting clinical trials in a frequentist manner. At this point, interpreting a trial as a frequentist has kind of become a procedure: check ...
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2answers
86 views

Inferring prior distribution

Suppose that we take a sample ($X_1, X_2, ... X_n$) from a distribution where we assume that $X_i $~$ Bin(n_i, p_i)$ and $n_i$ is known for every $i$. We also assume that $p_i$'s are independent and ...
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31 views

Proof for Sufficient and complete statistic (Shao)

Please can you help me, with this question: Let $X$ be a random variable with a distribution $P_\theta$ in $\{P_\theta : \theta \in \Theta\}$, $f_\theta$ be the p.d.f of $P_\theta$ w.r.t a measure ...
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1answer
57 views

Random Effect Model

One factor random effect model: $$y_{ij}=\mu+\tau_{i}+\epsilon_{ij}\quad i=1,2,\ldots,a; j=1,2,\ldots,n$$ where, $y_{ij}$ is the $j$th observation of $i$th treatment effect $\mu$ is the overall ...
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1answer
58 views

Unbiased estimator for $P(X_1=1)$

If $ X_1, ... ,X_n$ are IID binomial with parameters $ n$ and $p, $ find an unbiased estimator for $$G(p)=P(X_1=1)=np(1-p)^{n-1}\, .$$ I need to find this estimator so I can apply Lehmann-Scheffé ...
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42 views

Computing the F-ratio under null hypothesis

Statistical model for a Completely Randomized Design: $$y_{ij}=\mu+\tau_{i}+\epsilon_{ij}\quad i=1,2,\ldots,a; j=1,2,\ldots,n$$ where, $y_{ij}$ is the $j$th observation of $i$th treatment effect ...