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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How is the F-Stat in a regression in R calculated [duplicate]

I am running a regression and I'd like to be able to do the calculation to get to the F stat .3062. How is this .3062 calculated? Can you help? ...
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29 views

Evaluating integral to obtain marginal PDF related to Tikhonov Regularization

I am attempting to derive the marginal PDF for an application of the Gibbs Sampler. My joint PDF contains: $P(b,x) = \frac{1}{\sigma^{n}}\exp \left( -\frac{1}{2\sigma^2}\left\lVert ...
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44 views
+50

Likelihood and sufficient statistics

a)Find the maximum likelihood estimador for $a$ in the density $f(x;a)=\frac{2}{a^2}(a-x)I_{(0,a)}(x)$. b)Is it a sufficient statistics? I did $$\prod f(x;a)=\prod \frac{2}{a^2}(a-x)I_{(0,a)}(x)$$ ...
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36 views

Sufficient statistics, continuous distribution

Let $X_1,X_2,..,X_n$ be a random sample from the density $$f(x;\theta)=\theta x^{-2}I_{[\theta,\infty]}(x)$$ a)Is $T_1=min[X_1,...,X_n]$ a sufficient statistics? My doubt here is the right way to ...
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1answer
37 views

Sufficient Statistics, normal distribution

Let X be a single observation from $N(0,\theta)$. $(\theta=\sigma^2)$ a)Is X a sufficient statistics? b)Is |X| a sufficient statistics? What I did ...
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20 views

What are some advanced algorithms in bayesian networks? [on hold]

What are some advanced algorithms in bayesian networks? I am familiar with the conventional algorithms of network construction and inference in bayesian networks. What are some algorithms that provide ...
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2answers
27 views

Sampling distributions of sample means

Given a population from which we draw repeated samples of fixed size, say, 'n', my questions is what size is considerable? Is there any lower bound on sample size? Also how many such samples do we ...
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13 views

What is inference, training and testing in Undirected Graphical Model? [closed]

I have a Undirected Graphical Model (UGM) - $ \sum_i w_i\phi_i $ . What is the inference and training here? Suppose I have a train data and test data how do I train and test using this data and UGM? ...
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54 views

multiple regression advice on results

I am testing to see if there is a relationship between my dependent variable Y and any of 4 explanitory variables x1, x2, x3 or x4 First I started by doing simple linear regression and got these ...
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2answers
29 views

Calculating P- value

In ANOVA table , I got $F$-statistic $=330.285$ degrees of freedom (df) due to regression $= 6$ Error degrees of freedom $= 9$ But i don't know how to calculate $p-$value ? I have tried in ...
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1answer
48 views

Identify the distribution

Can anyone tell me which distribution has the density $$f(x)=\frac{x}{\sigma ^2}e^{-\frac{x^2}{\sigma ^2}}I(0,\infty)(x),\sigma >0$$
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27 views

Why are $M$-estimators NOT scale equivariant?

Consider the following location model. $$ x_i = \mu + u_i, (i = 1,\dots, n), $$ where $u_i$ are $i.i.d.$ with density function $f_0$. Hence, $x_i$ are $i.i.d.$ with density function $f_0(x-\mu)$. It ...
4
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1answer
103 views

Neyman-Pearson lemma

I have read the Neyman-Pearson lemma from the book Introduction to the Theory of Statistics . But I have not understood the lemma . Can anyone please explain me the lemma in plain words ? What ...
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0answers
4 views

Analogies between Gaussian Discriminant Analysis and Joint Probability Distribution

Can I say GDA is like a full joint probability distribution over all the feature random variables? I mean, if we are given some random variables, we try to inject some conditional independencies, and ...
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32 views

Bayes estimator, poisson distribution with exponential prior

If X ~ Poisson($\theta$) and $\pi(\theta)$~$exp(1)$(prior). Find the Bayes estimator for $P_\theta(X=0)$ with respect to quadratic loss $f(\theta|x)=\frac{e^{-n\theta}\theta^{\sum x_i}}{\prod x_i}$ ...
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0answers
23 views

Inferences with non-normal data

I have the data of index closing values that I later will use to run some regressions. When examining the data, I find heteroscedastic residuals and that the distribution is non-normal. In fact, it ...
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0answers
10 views

Do I get the HRSEs from my OLS or WLS regression?

I have a multiple regression linear model which I ran a simple OLS test on. I then performed the White test and found that it was heteroskedastic. Then I performed a Weighted Least Squares ...
1
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1answer
37 views

Time Series Comparison - Correlation and Regression Model

I am trying to see if and how the news for affects the financial markets. I have a time-series for both of them. Should I standardise the series? I have a monthly return on prices from the Dow and a ...
0
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1answer
37 views

Should I use the t-test or the Wilcoxon rank sum test given these qqplots and Shapiro-Wilk stats [duplicate]

I have 4 groups and I want to test if the pairwise difference in means are significantly different. There are 6 pairwise differences. The QQnorm plots of the 4 groups look like this: The ...
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0answers
10 views

EM to Variational EM in LDA

Why exactly, when learning hidden variables distribution in LDA(Latent Dirichlet Allocation), one cannot use to the EM (Expectation Maximization) algorithm and have to resort to a variationnal EM ...
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0answers
12 views

Finding the underlying pdf by sampling “buckets” of values

The situation I'm looking at a system where I can perform the following trial: $N$ samples are randomly taken from a population of $V$ different values (which can be treated as categorical), ...
2
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49 views

Combining two probabilistic predictions

I am solving a machine learning task in which I need to predict a label $\tau$ from input $\vec x$. The input $\vec x$ can be considered as two parts $\vec u$ and $\vec v$ ($\vec x$ can be thought of ...
2
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2answers
148 views

Likelihood Ratio for the Bivariate Normal distribution

For a random sample from a Bivariate Normal distribution with $\rho=\frac{1}{2}$ and equal variances, i.e. $\sigma^2_x=\sigma^2_y=\sigma^2$, I would like to derive the Likelihood Ratio Test for the ...
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0answers
8 views

Probabilistc Inference for Hybrid Models

I am looking for a library that can solve (calculate the MAP estimate) of the variables in a probable graphical model in which some variables are discrete and some are continuous. I understand that ...
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14 views

Process interpretation of confidence intervals?

Frick (1998) explains how statistical inference can be interpreted as making causal claims about processes, without resorting to random sampling and infinite populations. This view seems particularly ...
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1answer
65 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, ...
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10 views

How to calculate the parameter value for a test with Item Response Theory?

Given a set of responses to a test with multiple choice I wish to analyse it with Item Response Theory: wikipedia on IRT. I am planing to use the 3PL (3 parameters) in the Item response function. How ...
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3answers
39 views

Inference about parameter $\theta$ be same?

Let $\mathbf x$ be a sample point and $T(\mathbf x)$ be a statistic of $\mathbf x$. Similarly, let $\mathbf y$ be a sample point and $T(\mathbf y)$ be a statistic of $\mathbf y$. In the book ...
2
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1answer
63 views

Example of sample $X_1,X_2,\ldots,X_n$

In the book Statistical Inference by George Casella, it is written that An experimenter uses the information in a sample $X_1,X_2,\ldots,X_n$ to make inferences about an unknown parameter $\theta$. ...
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16 views

Inference problem, potential testing methods

For a person who hasn't studied statistics thoroughly, but instead is only at entry to median level, when facing a hypothesis test/inference problem, how does she/he know all the potentially possible ...
1
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1answer
30 views

Confidence interval for a function of the MLE

I am studying an old assignment in which I have calculated the MLE for a sample from an exponential distribution. It then gives the formula for the median of an exponential distribution ...
0
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1answer
50 views

Problem with generalized likelihood ratio test from samples from beta distribution

I was trying to resolve this exercise: This exercise is from the book "Statistical Inference, Second Edition" by Casella and Berger. Checking the solutions manual, I was understanding the solution ...
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28 views

Limiting Distributions and Weak Law of Large Numbers

I have that $Y_1, Y_2, ..., Y_n$ are i.i.d. Poisson random variables with mean 1, and that $U_n = \sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$. Given that I have a sequence $U_1, U_2, ..., U_n$, I'm ...
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13 views

Inference in bivariate continuous distributions

We have two nodes in different positions, which are represented by two random variables X,Y, with two prior bivariate continuous distributions, p_X , p_Y. f(X,Y,U,V) is a constraint on both ...
0
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1answer
18 views

Are daily updated data a sample or a population?

I want to do some tests based on all historical data of a product updated on a daily basis. These data are not supposed to have any time trend associated with them just be pulled on different days. I ...
0
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20 views

Indirect inference and p-value estimation

Let there be three continuous predictor variables X1, X2, and X3. Each variable is of length n. The correlation between the variables are rho12, rho13, and rho23. There is a binary outcome variable ...
0
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1answer
43 views

Does a lower t-stat suggest better evidence for rejection? [duplicate]

Say I have these two models: $y = \beta_0 +\beta_1x_1 + u$ $y = \beta_0 +\beta_1x_1 +\beta_2x_2 + u$ and the $p$ value for $H_0:\beta_1 = 0 $ with $\alpha = 10\%$ for both is less than 0.001, but ...
3
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73 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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0answers
10 views

Estimating conditional probability of bernoulli data

Assume I have $i=1,\dots,N$ fathers, each with $j=1,\dots,n_i>0$ sons. Now there is a binary event $A_{i,j}$ with outcomes 1 and 0 and the respective probabilities $p$ and $1-p$. Now I want to ...
0
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0answers
12 views

Choosing which distribution is most accurate

Let's say we have two samples from different normal distributions and we want to determine which distribution is most accurate relative to an ideal value. How would we evaluate which one is more ...
0
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2answers
26 views

Observed and Observable

What is observed and what is observable? I found this two word frequently in the context of random variable and realization of ...
3
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1answer
39 views

Contaminated mark-recapture: estimating set size from sampled subsets

Someone poured marked balls in my urn! Simplistically, I think this is a capture-recapture problem where, after drawing and marking balls from the urn, somebody added an unknown number (approx 25% of ...
5
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2answers
147 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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0answers
69 views

Elastic/ridge/lasso analysis, what then?

I'm getting really interested in the elastic net procedure for predictor shrinkage/selection. It seems very powerful. But from the scientific point of view I don't know well what to do once I got the ...
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34 views

Is a coin fair? [duplicate]

Is a coin fair ? In other words, does a coin come up $50$% heads ? If a coin is unfair, then it would not come up $50$% heads. My thoughts : let's first identify the population and the parameter ...
3
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1answer
45 views

Estimating Size of a Set based on two Overlapping Subsets

I've searched everywhere for a similar question and many things come close but are not the same. I'm looking for a way to estimate the size of a set if two partially overlapping subsets are known ...
0
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2answers
36 views

Test if sampled data are randomly sampled

Is there a way to test if data are (or at least seem) randomly sampled? In other words, is there a way to measure if my data are randomly sampled -- instead of coming from a complex survey sampling ...
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1answer
25 views

How can I infer the joint distribution of an observed and a latent variable?

I have a dataset of school children with three features: Age $x$ of the student answering the survey Year group $a$ of the student answering the survey Year group $b$ of the best friend of the ...
0
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1answer
52 views

$H_0=250g$ and $H_1\neq 250g$" [closed]

We have a sample of size $100$ with a standard deviation of $5g$ It was decided that if the sample mean is between $245g$ and $255g$ while the sample average is $250g$ if $\mu=250g$ or $\mu\neq250g$ ...
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19 views

A Hypothesis Testing Question and Verification

Suppose we have two p.d.f.'s: $f_0(x)=1,$ if $0\leq x\leq1$, and $f_0(x)=0$ otherwise $f_1 (x)=2x$ if $0\leq x\leq1$, and $f_1(x)=0$ otherwise We want to test the hypotheses ...