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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60 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 $[X_1], . . . , [X_n]$,...
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1answer
105 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 $\...
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165 views

In cross-validation, which is the AUC population parameter I really want to estimate?

In machine learning, AUC is usually used as a performance metric of an algorithm. As one is interested in the performance of the algorithm when applied to new cases beyond those used during the ...
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1answer
339 views

Connection between MLE (Maximum Likelihood Estimation) and introductory Inferential Statistics?

The first thing that one learns in statistics is to use the sample mean, $\hat{X}$, as an unbiased estimate of the population mean, $\mu$; and pretty much the same would be true for the variance, $S^2$...
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1answer
437 views

How to compute marginals in Sum-Product Networks?

This should be fairly easy, but for some reason i'm having hard time getting it to work and I've spent a long time trying to figure it out myself. In the last paragraph of page 4 of the original Sum-...
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1answer
197 views

Inference on and quasi variances in Bradley-Terry-models

Upfront I am not a statistician but a medical doctor. I have a working knowledge of statistical methods in my field, but this is my first time with pairwise comparisons and due to a lack of formal ...
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1answer
175 views

Unbiased estimator of distribution function in two-stage randomization design

Apologies for a long post. I believe the answer of my questions involve basic statistics though I am reading it in the context of two-stage randomization design. The questions appear in the ...
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2answers
188 views

What is statistic in statistics?

I am getting difficulty in understanding the definition of the statistic. From wikipedia, I come to understand that statistic is any 'information' (for example, range, mean, variance) of any sample ...
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1answer
220 views

Variance-Covariance Matrix for $l_1$ regularized binomial logistic regression

Given design matrix $X \in \mathbb{R}^{n \times p}$ and response vector $y \in \{ 0,1 \}^n$, I want to find the variance-covariance matrix of the coefficients $\hat{\beta}$ from an $l_1$-regularized ...
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1answer
462 views

Minimal sufficiency with indicator functions

The following theorem can be used to demonstrate that a statistic is minimal sufficient: Let $f(X|\theta)$ be the pmf or pdf of a sample X. Suppose $\exists$ a function $T(X)$ such that, for ...
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1answer
59 views

What is the name/relevant details of this exponential-family related structure?

Suppose that $X$ comes from an exponential family $$ p_\theta(x) = h(x)\exp(\theta x - A(\theta)), $$ and that, conditional on $X$, $Y$ also comes from an exponential family of the form $$ p_\eta(y\...
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2answers
103 views

Inference on random graph, with an application to mobile sensors

I've attended a course on Machine Learning and another one in Network Analysis, and I wonder if this two topics already intersect, in particular I'm interested in the following model: we have a ...
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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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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 ...
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191 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 ...
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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 ...
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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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700 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 ...
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6answers
2k views

Inference to the population when the survey response rate is only 30%

I have conducted a survey in which the questionnaires were sent out to 450 individuals, but only 30% of them answered the questionnaires. Is it still valid to interpret the usual inference analysis (...
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2answers
3k 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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3answers
2k views

Does R have post hoc tests robust to unequal sample sizes/population variances?

While reading Discovering Statistics Using R pp. 431-432, Dr. Field says that "There are a variety of tests designed to deal with these situations [multiple comparison procedures with unequal ...
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3answers
310 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
627 views

Is this alternative method to Metropolis-Hastings salvageable? What is it called?

For my application, I need to calculate an integral over a specific distribution. This distribution is obtained by Bayesian inference - the density at $\Theta$ is proportional to $P(\Theta)f(\Theta)$, ...
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2answers
637 views

Results of bootstrap reliable?

I am using the bootstrap algorithm to compute standard errors of the estimates of my normalmixEM output. I am not really sure if they are reliable? My code is (data here): ...
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1answer
227 views

Bayesian analysis of data

I have a big dataset in the form: $X_1, X_2, X_3, X_4, Y$. All the $X_i, i \in {1,...,4}$ come from different unknown distributions and $Y$ follows a bernoulli distribution, so it can take only values ...
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2answers
416 views

Why does continuous Bayesian analysis seem to give this contradictory result?

Let's say you have a process that generates data according to r = sin(t) + epsilon, where epsilon ~ N(0,V) is Gaussian noise. The unconditional variance of r is 0.5 + V. Let's say we're forecasting ...
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1answer
755 views

Sufficient statistics for Uniform $(-\theta,\theta)$

So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ? In other ...
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3answers
174 views

Statistics can't be a function of a parameter - but isn't the sample a function of the parameter?

I have a question that relates to this post: Can a statistic depend on a parameter? But on it, the discussion focuses much on the t-statistic given as an example by the question asker. My doubt in a ...
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2answers
168 views

Admissibility under the loss function

Suppose $X_1 , ..., X_n$ are random samples of exponential distribution with mean $\theta$. Determine $a$ and $b$ such that $a\sum_{i=1}^n X_i +b$ be admissible under the loss function $L(\theta,\...
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1answer
1k views

GLM Categorical Variable Level grouping / simplification

I am trying to find information regarding a technique which is commonly used in the insurance pricing industry. This relates to a GLM model where a categorical variable is used in the model. The ...
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3answers
158 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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3answers
187 views

How to measure student skill level based on a set of questions answered at different points in time and of different difficulties levels?

I am working on an e-learning system with a friend for our final year (Computer Science) project which is part of the under-graduate programs mandatory 'courses'. I have a question about making ...
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4answers
370 views

What if your randomly formed groups are clearly not similar?

What if, before you begin the data collection for an experiment, you randomly divide your subject pool into two (or more) groups. Before implementing the experimental manipulation you notice the ...
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3answers
119 views

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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1answer
45 views

Motivations for experiment design in statistical learning?

My interests in statistics centre around statistical learning, including Bayesian inference, inference in combinatorial spaces, Monte Carlo methods, Markov decision processes, modeling stochastic ...
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2answers
1k views

What is the Difference between Inductive Reasoning and Statistical Inference?

In my seminar work I used the following sentence: Overfitting stands out as the most important aspect of machine learning and statistics. Here, I want to replace "statistics" with either ...
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2answers
606 views

Can a posterior expectation be used as a approximate for the true (prior) expectation?

Let's say that the likelihood of observation $x$ given a random latent variable $z$ and a model parameter $\theta$ is defined as $p(x|\theta, z)$. As far as I know, if I want to obtain $p(x| \theta)$,...
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1answer
88 views

Inference for the Zeta distribution

I have a problem where I am aware that data is well-modeled by a Zeta distribution such as $P(X=x) = x^{-a}/\zeta(a)$, and would like to learn the Zeta distribution parameter $a$ from the data. More ...
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4answers
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Which book should I read to get started with machine learning, Elements of statistical learning or Pattern recognition in machine learning?

I want to learn machine learning. I found tons of material on the internet but couldn't decide which book to get started with.
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1answer
247 views

Estimating population size of a subgroup based on independent samples without replacement

Let a bag have 1000 balls, 100 red and 900 blue. Now, let us get ten independent samples (without replacement within each sample) of the original population (10% each). Proceed to count the number of ...
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1answer
15k views

How do I calculate a posterior distribution for a Poisson model with exponential prior distribution for the parameter?

Suppose: $N \sim {\rm Poisson}(\lambda)$ $\lambda$ is unknown, but we believe that it can be assumed $\sim \exp(1)$ If I want to calculate $N | X$, i.e., $P(model | data)$, I need to use the Bayes ...
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2answers
335 views

question about MSE mean square error

The following is taken literally from Wikipedia's mean squared error in the mean subheading: "Suppose we have a random sample of size $n$ from a population, $X_1$, ... ,$X_n$. Suppose the sample ...
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1answer
76 views

When testing multiple hypotheses, what does it mean when there are not enough extremes? [closed]

Suppose you are testing a large number of hypotheses, say a million. Unlike the usual situation where you have a lot of very small p-values, in this case all of your p-values are greater than 5%. ...
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1answer
38 views

Bootstrapping $R^2$ from a set of experimental data

My question is probably best asked through an example. Suppose we conduct some designed experiment measuring how three different fertilizers influence the stem lengths on sunflowers, and we repeat ...
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1answer
139 views

Interpreting the sampling distribution of $\bar{X}$ as the posterior distribution of $\mu$

Suppose we wanted to do inference on the mean $\mu$ of some population for which the variance is known. The conventional frequentist approach is as follows. Given a random sample $X_1, \dots, X_n$ ...
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2answers
104 views

Probability that $n$ trials will succeed given that $k$ succeeded [duplicate]

I'm not sure exactly how to ask this, or if there is such a thing. I'm new to statistics and have just studied confidence intervals and confidence levels of survey data, such as the confidence of ...
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2answers
307 views

Unknown process outputs binary results, how to prove that this process is (or not) a Bernoulli trial

I have an unknown process that produces binary results. I am trying to determine if this process is a Bernoulli trial. From wikipedia: In the theory of probability and statistics, a Bernoulli ...
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1answer
686 views

Cramer-Rao Lower Bound

Let $X_1,..,X_n$ be an iid sample of $N(0,\sigma^2)$. Find an unbiased estimator of $\sigma^2$ and its lower bound. I found that $$\hat{\sigma}^2 = \sum_{i=1}^{n} X_i^2$$ is an unbiased estimator ...
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1answer
2k views

Kalman filter with input control noise?

assume we have a standard Kalman filter with input controls, following wikipedia notation (http://en.wikipedia.org/wiki/Kalman_filter) where the latent state is $x_{t}$ and the observation is $z_{t}$, ...
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3answers
167 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 ...