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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Multiple Linear Regression Zero Conditional Mean Assumption

Greene [1] and Wooldridge [2] emphasize that in the standard multiple linear regression model $${\bf y}=X{\bf b}+{\bf e}$$ a key assumption is that $$E[{\bf e}|X]=E[{\bf e}].$$ Or, in other words, $X$...
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Complete statistics for $f_X(x) = e^{-(x - \mu)} I_{\mu, \infty}(x)$

I am studying parametric statistical inference. One of the self study I have to find a sufficient, minimal and complete statistic for the $\mu$ parameter of the following p.d.f. $$ f_X(x \mid \mu) = e^...
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How does graphical model of a GP look like?

I'm trying to understand the difference between GP and Markov process. I couldn't find answers on the internet. I figured that graphical models can tell the difference, hence my question.
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When should one do class rebalancing? [duplicate]

Does anybody know a source when class rebalancing should be considered? Say one has a very small dataset. About 70 observations. When would class rebalancing make sense? When the 0/1 ratio is 70/30, ...
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Two-tailed tests… I'm just not convinced. What's the point?

The following excerpt is from the entry, What are the differences between one-tailed and two-tailed tests?, on UCLA's statistics help site. ... consider the consequences of missing an effect in the ...
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compare 2 different sized multivariate samples

I have a two data tables like below. The dataset1 represents failed candidates. The dataset2 represents the successful candidates. I want to know, by applying some inference statistics which var (...
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22 views

Modeling count data to poisson distribution when expected is zero

I am working on a project where we saw an event occur in 5 of 39 patients. We don't expect this event to occur at all (though we know they occur very rarely, we don't expect to see any in this ...
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1answer
121 views

Two-sided UMP test for exponential densities?

I'm struggling with a problem from Lehmann & Romano's book *Testing Statistical Hypothesis." Suppose $X_i$ is a random sample from $$f(x) = \frac{1}{b}e^{-(x-a)/b}\mathbf{1}_{x>a}$$ The ...
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Predicting the values of a function given a non-symmetric loss function [on hold]

The problem is the following : I know the values of the function $f : \mathbb{N} \to \mathbb{N}$ only for $n \leq N_0$ i.e I have access to $(f(1),f(2), \dots, f(N_0)) $. Question. Predict the ...
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1answer
111 views

Poisson Binomial Distribution - confidence intervals

I'm working on a project which involves multiple trials for which the probability of success is not the same across trials. Given the unequal probabilities per trial, I'm using the Poisson Binomial ...
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302 views

Confidence intervals and Central Limit Theorem with only one sample

I know that to construct confidence intervals, standard errors must be calculated, a process which in turn makes use of the CLT (but I am not clear how). I also understand that, very generally, the ...
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40 views

Bayes formula alternate expression using alpha

I know that Bayes theorem is: Posterior = Likelihood * Prior / Evidence However, I am confused about the above notation in the picture. How do we get to the above three notation? How does ...
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Estimate probability of selecting the same card more than once

Suppose I have a deck of N=1,000 cards where each card is a unique number from 1 to 1,000. Draw 1: draw n=10 cards at random. Put them back Draw 2: draw n=10 cards at random. Put them back ... ...
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How to correctly state a multiple-comparison hypothesis pair

I need to compare multiple treatments over a predefined set of benchmark instances. However, I'm facing some difficulties on how to correctly state my hypothesis pair. I want to verify if there are ...
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What is the difference between conditioning on regressors vs. treating them as fixed?

Sometimes we assume that regressors are fixed, i.e. they are non-stochastic. I think that means all our predictors, parameter estimates etc. are unconditional then, right? Might I even go so far that ...
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1answer
409 views

UMVUE for pareto distribution

Let $X_1,\ldots,X_n$ random sample with $f(x;\theta,a)=\frac{\theta}{a}(\frac{a}{x})^{(\theta+1)}I_{(a,\infty)}(x),a>0,\theta>0$. Find the UMVUE for $\theta$ when $a$ is fixed. My attempt $$f(x;...
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240 views

Sequential Inference And Evidence (Jaynes 2003): Is it valid? Is it used?

Exploring the work of ET Jaynes, Probability Theory (11th Printing 2013) has led to consideration of the technique he identifies as Sequential Inference (p. 96); where the evidence, in decibels, ...
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How is MAP 'not invariant to reparametrization'? [duplicate]

I was watching a lecture on coursera on 'Bayesian Methods on Machine Learning' and I came across a statement that: MAP(Maximum a posteriori) is not invariant to reparametrization. I didn't quite ...
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Random forest “out-of-bag” ensemble

I am using the R package RandomForestSRC for random forest applications. In the manual for the main function (rfsrc) they mention a setting called ...
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1answer
1k views

What is the problem in the Neyman-Scott problem?

Let $Y_{ij} \stackrel{d}{=} N(\mu_i, \sigma^2)$ for $i\in \{1,\ldots,n\}, j\in\{1,2\}$. Also assume $Y_{i1}$ independant of $Y_{i2}$. The parameter of interest is $\sigma^2$. Setting up the ...
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Understanding Product of individual PDF for Joint PDF

Let's say that we make multiple noisy observation from a sensor node where $h$ is the parameter we want to deduce and $v$ is the noise. $$y[k] = h + v , k=[0,1,..n] $$ Question: The PDF for each ...
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2answers
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Coefficient estimate of multiple interaction terms in regression model

I am trying to estimate coefficient of a regression model with two interaction terms. I would appreciate any help. I will try to recreate my model then ask the question. ...
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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 ...
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1answer
259 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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Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are “no events”

I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
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55 views

Verifying whether $X$ is a complete statistic

The pmf of $X$ is as follows: $X = -1 \rightarrow p(x)= \theta$ $X = 0 \rightarrow p(x)= \theta^2$ $X = 1 \rightarrow p(x)= 1-\theta-\theta^2$ I know that to show whether $X$ is complete it is ...
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Statistical differences between two sets of predicted values [closed]

I have a few distinct groups and each group has units within them. Each unit has time series data to which I fit an exponential model and predict the time when it hits a certain threshold. I have such ...
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3answers
301 views

Does order of events matter in Bayesian update?

I'm wondering whether the order of events can lead to different Bayesian update. For example, consider a coin-tossing problem with unknown $p$, the probability of Head. Initially, $p$ is known to ...
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1answer
30 views

How to exclude events with low data (eg. threshold, outliers)

I have this data set and I want to filter only "Event" with a good conversion rate. We can say that good are those that have a higher than average conversion (but maybe you have better ideas). ...
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How do I find the expected values and covariance matrix of the order statistics of iid random variables sampled from the standard normal distribution?

Recently I was trying to learn more about Normality tests and came to know about Shapiro-Wilk test for Normality. I understood most part of it but one thing I didn't understand is that how do I find ...
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1answer
151 views

Bayesian Network: Calculate probabillity of child node given all probabillity tables

I have a Question about Bayesian Networks. I have a network with many parent nodes and one child node. I have the probabilities for the parents and for the child. The child node is binary, so there ...
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82 views

Bayesian update for Beta distribution

I'm wondering how to find a posterior of a beta distribution when the "new information" is not an outcome of a binomial trial. Let $p$ be the probability of Head of a (biased) coin toss. As usual in ...
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How to convert cohen's h to a percent difference in groups in R?

I'm trying to calculate the minimum detectable effect in an experiment after n samples. I'm able to use the pwr package like this to compute the minimum ...
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1answer
33 views

Rate of convergence of gradient descent inference in likelihood maximization

I am reading this classic paper on convergence properties of EM for Gaussian Mixture Models. In section 5, the authors compare EM with a gradient based inference approach. The gradient approach ...
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1answer
19 views

Hypothesis test for the difference of two means, should I consider annualized or monthly returns?

I have 10 years monthly returns. I calculated annualized return multiplying the mean return over the period for 12. Then I calculated the excess returns as difference between the annualized mean ...
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Using hold-out method for validation set: How to choose a DL model with model selection?

After >170 deep learning experiments were I did a (almost) full factorial design with >15 factors. I cannot measure performance with cross validation because that would require to much training of ...
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49 views

How to perform joint inference on multivariate normal variables?

Suppose I have the following model: $$\begin{aligned} \text C &\sim \mathcal N \left(\mu, \delta^2\right) \\ \forall i: \text L_i | \text C = c &\sim \mathcal N \left(c, \lambda_i^2 \right) \\...
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1answer
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Conducting “inference” on Titanic data set (and other non-random/“population-encompassing” data sets alike)

Presume I'm given a data set like Titanic, where the data on all the passengers is available (hence "population-encompassing" in the title). Then, by inertia, I proceed to conduct statistical ...
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Implications of current debate on statistical significance

In the past few years, various scholars have raised a detrimental problem of scientific hypothesis testing, dubbed "researcher degree of freedom," meaning that scientists have numerous choices to make ...
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Recommended textbooks for student majoring in applied statistics [duplicate]

I am currently a second year science student double majoring in biochemistry and applied statistics. The stats course im doing this semester (Statistical Theory) is focused on joint probability ...
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1answer
202 views

Understanding how the determinant of the multidimensional normal likelihood can overrule the prior probability

I am doing Bayesian inference. I have a normal prior probability distribution of some theoretical parameter $\theta$ and I am trying to update my knowledge of $\theta$ using some data $D$ and a model $...
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1answer
23 views

Interpreting predictive models in the presence of omitted variables

Suppose the best predictive model from a set of possible models is a univariable model, due to lots of moderate correlations with other variables for example. However, if I use this model for ...
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1answer
45 views

Time series explaining the trend

I'm very new to time series analysis and I've been tasked with trying to make sense of some data and was hoping you smart folks out there could provide some guidance. I have some data relating to ...
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1answer
27 views
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1answer
42 views

Statistics help! Reporting ANOVA results!

I am new to statistics and I need some help in understanding how to report the data of some tests I am running on R, I hope this is the right place! I have a dataset: ...
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1answer
103 views

Why is the log derivative estimator considered of large variance?

It's mentioned in the paper Variational Bayesian Inference with Stochastic Search that, the variance of the following approximation may be very large, but I didn't quite understand why this is so. It ...
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2answers
4k views

Should we address multiple comparisons adjustments when using confidence intervals?

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

Expectation of exponential family distributions

Is there a closed form of the following marginal (one dimensional data) $\pi(\theta|y) = \mathbb{E}_{x \sim \pi_R(x|y)} \pi(\theta|x)$, where both $\pi, \pi_R$ are exponential family distributions?
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1answer
224 views

Unbiased estimator of $\lambda(1 - e^\lambda)$ when $x_1,\ldots,x_n$ are i.i.d Poisson($\lambda$)

Suppose $x_1, x_2, x_3,\ldots, x_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. I was trying to find an unbiased estimator for $\lambda(1 - e^\lambda)$, but I could not ...
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3answers
132 views

Maximum likelihood estimator of $n$ when $X \sim \mathrm{Bin}(n,p)$

Given a random variable $X\sim Bin(n,p)$, where $p$ is known $p\in (0,1)$ , $n$ is an unknown positive integer and $x\in\{0,1,2,....n\}$, what is the maximum likelihood estimator of $n$? I ...