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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1answer
2k 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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719 views

Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$

Suppose $X_1, X_2, . . . , X_n$ are i.i.d Poisson ($\theta$) random variables, where $\theta\in(0,\infty)$. Give the UMVUE of $\theta e^{-\theta}$ I found a similar problem here. I have that the ...
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182 views

How to go about selecting an algorithm for approximate Bayesian inference

I am wondering if there are any good rules of thumb for how to go about selecting an approximate inference algorithm for a problem/model (specifically when exact inference is intractable)? When you ...
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1answer
755 views

Simple linear regression with a random predictor

We understand a SLR model as $$y_i = \alpha + \beta x_i + \varepsilon_i$$ with $\varepsilon_i$ i.i.d with equal variance. Suppose I have two instruments measuring a common entity, say, density of ...
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2answers
198 views

Resampling within a survey to account for missing data

Suppose I have survey responses that look like this: ...
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4answers
317 views

Does sampling from a large dataset lead to correct inferences?

Say we have some population, and we obtain a "representative" random sample of that population, $(y_i, x_i)_{i = 1}^n$, where $n$ is very large (millions) and $x_i = (x_{i1}, x_{i2}, ... x_{ip})'$ is ...
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816 views

Intuitive explanation of desirable properties (Unbiasedness, Consistency, Efficiency) of statistical estimators?

From literature I understand that the desirable properties of statistical estimators are Unbiasedness - we want the estimator to give the correct parameter value theta, on an average, irrespective of ...
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2answers
143 views

Drawing conclusions of several inferences with the same data in one study

In a sample size of 100, we identified the existence of two attributes A and B. Our goal is to assess whether there is any ...
6
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1answer
485 views

What is the problem with overdifferencing a long memory time series?

Suppose I have a long memory time series and instead of using fractional differentiation I take a first difference. What kind of problems am I going to run into? Is there any advantage to doing the ...
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134 views

Time series forecasting using statistical tools

I'm building a system which needs to poll some feed of articles in a smart way. When polling, I can only know the number of new articles (could be $0$ - no new articles). I don't have the info when ...
6
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1answer
5k 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 $...
6
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1answer
124 views

How to measure uncertainty of a parameter when false positives exist?

The main goal of my research is to measure the percentage of brown dwarf stars in the Pleiades star forming cluster that are actually double stars (i.e. the brown dwarf star has a companion brown ...
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1answer
4k views

How to write a poker player using Bayes networks

This is my first question on stackexchange and also my first time implementing a Bayesian network so I will apologize ahead of time for any novice mistakes I make. The goal of my project is to ...
6
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2answers
2k views

Standardized residuals vs. regular residuals

I've got an easy question concerning residual analysis. So when I compute a QQ-Plot with standardized residuals $\widehat{d}$ on the y-axis and I observe normal distributed standardized residuals, why ...
6
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1answer
140 views

Calculating the integral of a PDF inside a closed contour of constant density

I'm working with some two-dimensional probability distributions which have emerged from Bayesian inference work I'm doing. These PDFs are stored on regularly spaced Cartesian grids. I feel like it ...
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1answer
180 views

Lesser-known but powerful probabilistic inference algorithms

What are the lesser-known but powerful probabilistic inference algorithms? Most references about probabilistic graphical models describe popular inference methods like Variable Elimination and ...
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0answers
126 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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128 views

Explanatory variables may bias predictions

I' m asking this question out of sheer curiosity, my teacher was not able to explain it. If I'm using logistic regression with categorical variables they are coded like {1,2,3}. I guess it wouldn't ...
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3answers
120 views

Likelihood function when $X\sim U(0,\theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(0,\theta)$. Derive the likelihood function given the sample $x_1, ..., x_n$. Answer The likelihood function is: \begin{...
5
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1answer
231 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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2answers
650 views

Differences between prior distribution and prior predictive distribution?

While studying Bayesian statistics, somehow I am facing a problem to understand the differences between prior distribution and prior predictive distribution. Prior distribution is sort of fine to ...
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1answer
3k 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
167 views

Definition of Statistic

I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the ...
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1answer
1k views

Bayesian vs. frequentist estimation

I don't really understand the connection between bayesian to "normal" frequentist estimation. Suppose we want to estimate the expected value of a population given a sample. In frequentist statisics ...
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3answers
992 views

Variational autoencoder: Why reconstruction term is same to square loss?

In variational autoencoder (see paper), page 5, the loss function for neural networks is defined as: $L(\theta;\phi;x^{i})\backsimeq 0.5*\sum_{j=1}^J(1 + 2\log\sigma^i_j-(\mu^i)^2) - (\sigma^i)^2) + \...
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1answer
5k views

Asymptotically Normally Distributed

When we say an estimator is consistent, we mean "as the sample size increases, sampling distribution of the estimator converges to the true parameter value." But when we say "an estimator is ...
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1answer
2k 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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2answers
255 views

What can we say about the likelihood function, besides using it in maximum likelihood estimation?

I found that often in literature that likelihood values are often used to compare different estimation method for the same model. And I got the impression that is the only way likelihood values are ...
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1answer
149 views

Fiducial Inference in Machine Learning

I was looking at the Fiducial Inference page on wikipedia, which is an alternative to the traditional Frequentist and Bayesian standpoints. Although it was out of favour in mainstream statistics for ...
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2answers
502 views

MRF MAP inference for non-submodular pairwise terms

I have a multilabel MRF MAP inference problem (a labeling problem). The graph has relatively few nodes, about a thousand or so. The pairwise term is (very) not submodular (it does not satisfy the ...
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1answer
210 views

Means of groups A and B differ significantly. I want to classify values into A or B

My data look like this. The variable on the $x$ axis is height in inches. The variable on the $y$ axis is whether someone hovers when urinating at a public toilet. Each of the 103 points is a female ...
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3answers
101 views

Why is sufficient statistics/data reduction normally taught in Statistics?

In most upper-level classes on statistical inference, data reduction and sufficient statistics are normally taught, but without too much motivation. I understand sufficient statistics are important ...
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159 views

Simple linear regression: If Y and X are both normal, what's the exact null distribution of the parameters?

Suppose $Y \sim{N(a,b)}$, $X \sim{N(c,d)}$, and $Y$ is independent of $X$. After sampling 25 observations from both $Y$ and $X$, I run the following regression model: $Y=\beta_{0}+\beta_{1}X + \...
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1answer
704 views

Find UMVUE of $\theta$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

As a slight modification of my previous problem: Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\...
5
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1answer
359 views

What is a pivotal statistic?

I'm currently reading "Computer Age Statistical Inference" by Efron and Hastie. In section 2.1, they talk about some of the mechanisms that frequentist inference uses to circumvent the defect of ...
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2answers
577 views

Coverage probability of credible intervals if we take Bayesian model literally

Let's say I have a Bayesian model with a proper prior $\pi$, likelihood $L$, data distribution $p(x|\theta)$ (assume $\theta$ is a scalar) and the vector of sample values $x$: $$p(\theta|x) = \frac{\...
5
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1answer
411 views

In what situations would one use Approximate Bayesian Computation instead of Bayesian inference?

I'm not sure why one would use ABC/Likelihood-free inference methods instead of standard Bayesian inference methods. Is this fundamentally a conceptual problem of mine? Are there any concrete ...
5
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1answer
497 views

How to perform a sensitivity analysis in Bayesian statistics?

Bayesian inference is drawn from the posterior distribution or - in case we are interested in forecasting - from the predictive posterior distribution. However, these values are heavily affected by ...
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2answers
230 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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3answers
174 views

Bayesian inferential target

In frequentist, i.e., sampling-based statistics, we envision a target population to which inference is made. Notwithstanding the fact that our so-called random samples from this population are ...
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1answer
2k views

Assessing variable importance in generalized additive models (GAM)

In a linear model, it's easy to assess the importance of each explanatory variable. If the assumptions of the model are met, given two explanatory variables $x_1$ and $x_2$, both with a regression ...
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2answers
219 views

A Question on Elementary Statistical Inference

A box contains $5$ white and $2$ black balls. A coin with unknown $P(Head)=p$ is tossed once. If it lands HEADS then a white ball is added, else a black ball is added to the box. Then a ball is ...
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1answer
196 views

Hosmer-Lemeshow recommendations

During lectures I came across following statement: If you want Hosmer-Lemeshow test to be valid, number of expected events ($E_1g$) should be >5 in most of $g$ groups Then after few lectures, ...
5
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1answer
451 views

complete sufficient statistic exercise

I have to find complete sufficient statistic of the following pdf $$f(x|\theta)=\frac{\theta}{(1+x)^{(1+\theta)}},\quad 0<x<\infty,\theta>0.$$ My Attempt: The joint density $$f(\mathbf x|\...
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2answers
705 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 ...
5
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1answer
89 views

Is it invalid to use a single sample to estimate more than one proportion?

Lets say that I have a jar of white, black and red marbles (1000 total) but I don't know the quantity of each colour. I now take a sample of 10 marbles from the jar which contains 3 x white, 4 x ...
5
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1answer
923 views

Does Loopy BP give the same solutions as a Gibbs sampler?

The literature in MCMC and LBP never refer to the fact that the two methods look (on expectation) exactly the same. To illustrate, first consider a simple Ising model, that is, a graphical model ...
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1answer
59 views

Approximate the critical region such that the size of the test tends to $\alpha$

Consider this question, Suppose $X_1, X_2, . . . , X_n$ is a random sample from an exponential distribution with mean $\lambda$. Assume that the observed data is available on $[X_1], . . . , [X_n]$,...
5
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1answer
103 views

Showing MSE of $\bar{X}\mathbf1_{\bar{X}>0}$ is less than that of $\bar X$ when sampling from $\mathcal N(\theta,1)$ population

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from a $\mathcal{N}(\theta,1)$ population where $\theta>0$. I am trying to compare the estimators $T=\bar{X}\mathbf1_{\bar{X}>0}$ and $\...