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 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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12 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
63 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
33 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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64 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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56 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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12 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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7 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 ...
2
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60 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
90 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
61 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
65 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
75 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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0answers
27 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
47 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 ...
2
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1answer
50 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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29 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 ...
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56 views

A doubt on the definition of p-value

The p-value is the probability, under the assumption of the null hypothesis $H_0$, of obtaining a result equal to or more extreme than what was observed at given data. This means, if I were to ...
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1answer
40 views

whether Y(employees injured) variation is due to X1(job function) or X2(population)

Here is the actual question- There are 1000 employees in a firm, and the firm has four departments namely D1, D2, D3 and D4 with 100, 200, 300, 400 employees respectively. Now, each employee is ...
3
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0answers
26 views

Is there a test/technique/method for comparing principal components decompositions between samples?

Is there any methodical way to compare the directions, magnitudes, etc of PCA results for different samples? I'm leaving the nature of the test deliberately vague because I'd like to hear all the ...
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1answer
43 views

If my normality test is non-significant, am I safe to use the t-test?

I took a 30 unit sample from a population. The sample distribution resulted to be normal. Can I state that the population distribution is normal too? If so, with what level of confidence?
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1answer
52 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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1answer
45 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
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2answers
108 views

Normalization to non-degenerate distribution

I am reading de Haan's Extreme Value Theory (2006). In the discussion of distribution of sample maximum, he said "in order to obtain a non-degenerate limit distribution, a normalization is necessary". ...
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1answer
40 views

Show that a statistic is ancillary

Let $X_{i} \sim U(0, \theta) $ and $X=(X_1,\dots,X_n)$. Show that $$ \frac{X_{(1)}}{X_{(n)}}$$ Is ancillary for theta I coulxnt find a way of doing it that looks convenient. Any idea? P.s: ...
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27 views

Testing association between exposure and disease

In a particular example from the book Epidemiologic Research by Kleinbaum [example 15.1], I have three problems. Consider the data in table 01. These data pertain to a follow-up study concerning the ...
2
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2answers
65 views

Significant difference and correlations

Is it true that when there's no significant difference between groups then there will be correlations between groups? My situation is as follows: I have a sample that was measured using two ...
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1answer
47 views

How do you measure the accuracy of an inference hypothesis/procedure?

Take inference to mean reasoning/predicting the value of a hidden/laten variable $Z$ given some evidence/data $X$. For example, maybe you are trying to find out if your patient has Cancer (Z = 1 if he ...
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17 views

Two-tailed test [duplicate]

In one-tailed test , we give our decision at $\alpha$ level of significant. But in two-tailed test , why do we give our decision at $2\alpha$ level of significant? Why do we not give the decision of ...
3
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1answer
64 views

Estimating total number of people from an observed sample

The well known "German tank problem" shows how to answer the question: "If I have tanks which have an increasing serial number, and I see a sample of tanks and record their serial numbers, what is the ...
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1answer
20 views

Do I need to care about constants in Expectation Propagation

I am trying to approximate a certain factor in my graph. Following Tom Minka's tutorial what I have to do is as follows: $$ \prod_{i=1}^3 q_{w_i}(\pi_2)\approx \int ...
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86 views

Basic problem in Bayesian inference

I have questions with the following Bayesian inference problem I found in the book by Bertsekas & Tsitsiklis (Introduction to Probability 2nd ed.). Problem is as follows (P.445, Problem 2): ...
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1answer
41 views

Getting all zero correlations,$\rho_{ij}=\frac{\mathbb cov(e_i,e_j)}{(V(e_i)V(e_j))^{1/2}}$

Consider the general regression model $$Y=X\beta+\epsilon$$ where, $Y$ is an $(n\times 1)$ vector of observations, $X$ is an $(n\times p)$ matrix of known form, $\beta$ is a $(p\times 1)$ vector ...
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131 views

Estimating abundance using non-normal count data

I have sample counts of $n=20$ or $n=7$ taken from right-skewed and zero-inflated populations. The challenge in each case is to use the sample to estimate the total count in that population. Each of ...
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1answer
44 views

What is max-sum / max-product variant of loopy BP computing?

In (Nowazin and Lampert, Structured Learning and Prediction in Computer Vision, p. 29.), they say that in the max-sum variant of loopy belief propagation, the "variable max-beliefs are no longer ...
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3answers
475 views

Do the pdf and the pmf and the cdf contain the same information?

Do the pdf and the pmf and the cdf contain the same information? For me the pdf gives the whole probability to a certain point(basically the area under the probability). The pmf give the probability ...
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4answers
199 views

Can a trend stationary series be modeled with ARIMA?

I have a question / confusion about stationary series required for modeling with ARIMA(X). I am thinking of this more in terms of inference (effect of an intervention), but would like to know if ...
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1answer
42 views

Maximum likelihood estimator for variance in two linear models

I am learning MLE's at my inference class and this is a problem I came accross. Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha ...
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1answer
103 views

Asymptotically unbiased estimator using MLE

I am learning Maximum likelihood estimators for a inference class. And this is a problem I came across. Let $X_1,X_2,X_3,\ldots, X_n$ be a random sample with p.m.f $$p(X)=\theta(1-\theta)^x; ...
2
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0answers
54 views

UMVUE for normal distribution $\sigma$

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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17 views

Maximum Likelihood estimators in reation to linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
0
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2answers
29 views

Pivotal to estimate lambda of a exponential

I am studying interval estimation by the method of pivotal quantities. Let $X_1,X_2,...,X_n$ be a random sample from a p.d.f $f(x;\lambda)=\lambda e^{-\lambda x}, x>0,\lambda >0$. I have to ...
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26 views

Constructing Counterfactuals and Estimating Prevalence

I'm a social scientist working on a research project where I try to estimate the prevalence of lying in responding to a certain sensitive question. The way I estimate it is to rely on a ...
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33 views

Updating a Dirichlet distribution with partial data

I've got some categorical data where each observation has multiple attributes, and I want to make a probabilistic model of this using Dirichlet distributions. For example, in the two dimensional case ...
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30 views

Expectation maximization with variant length at observing data

Imagine one loaded dice. Based on EM algorithm, how could we compute how much it loaded if we introduced: Variant length on each rolling attempt (look at first and second attempt below 1st one has 6 ...
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1answer
104 views

Testing significance of a random effect glmmADMB model

Below is the output from a model of novel object test scores fit with the nbinom1 (quasi-Poisson) option in glmmADMB. I used ...
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26 views

Superiority test versus non-inferiority test

What is your best (or the best quote that you have) plain language explanation of the difference between tests for superiority and tests for non-inferiority? I think a test of superiority a simply a ...
4
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2answers
89 views

How to statistically test upper bound

Suppose a theory claims that a random variable $R$ (of unknown distribution $F$) must satisfy a certian upper bound $R < c$ (where $c$ is known constant). Suppose I perform a set of measurements ...
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30 views

Model averaging effect sizes of Gamma family GLMs

I'm trying to get some model averaged effect sizes from a set of candidate models, all of them assuming a Gamma error distribution, according to the theory given by the book from Burnham and Anderson ...
2
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1answer
65 views

What the dimension of an exponential family tell us about that family?

In Wikipedia it is stated that: A vector exponential family is said to be curved if the dimension of $$ {\boldsymbol \theta} = \left (\theta_1, \theta_2, \ldots, \theta_d \right )^T$$ is ...