The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
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58 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
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61 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
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16 views

The distribution of the ith order statistic for discrete random variables

Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$. We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim ...
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25 views

Normalizing multiple different features from unknown distributions

I'm doing some "exploratory" data analysis over a large set of classes/proteins, with a few hundred different features (I.E. Continuous variables) extracted from the data. The features are calculated ...
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54 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
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57 views

Estimating a sub-population characteristic based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ...
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44 views

Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
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84 views

hypothesis testing practice question

This is a practice question for a final exam. Two types of cordless weed-trimmer batteries are tested. One group of 5 batteries averaged $1.4$ hours, while the other group consisting of 8 batteries, ...
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40 views

degrees of freedom for a chi squared goodness-of-fit test

For a statistics project, I gave out a 20 question multiple choice quiz with each question containing five answers. I would like to run some hypothesis tests on the data by using a Chi squared ...
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88 views

Probability distribution for a digit of a number

If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
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65 views

How do I must do it?

For a sample of size n from a random variable with density function $$f_{\theta}(x)=\frac{2x}{\theta^2}, x>0$$ find the confidence interval for $\theta$ of average length consistently lower level ...
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24 views

Inference the time a process would take depending on the number of threads

I have several datasets consisting on: Number of threads n Start process time t1 Stop process time ...
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124 views

Is there a statistical hypothesis test that uses the mode?

Is there a statistical hypothesis test that considers the mode rather than the mean or median?
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7 views

Function of a complete statistic is complete

Quick question: If a have a statistic $T$ complete, is $a^T$ complete ($a$ constant)? In general, if a statistic is complete, then is every one-to-one function of it complete? Thanks.
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18 views

Confidence bounds given random verification

[Edits made for clarification and brevity.] I'm working on an idea for a fault detection algorithm, and I've boiled it down (I think) to the following problem. A box contains 10 balls. The balls can ...
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18 views

Why is the marginalized inverse-Wishart distribution not equal to the inverse-gamma distribution?

Given that the inverse-gamma distribution is the one-dimensional version of the inverse-Wishart distribution, why will (philosophically speaking) an inverse-Wishart distribution that originally has ...
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22 views

Estimate distance between approximated posterior and true posterior

I'm working on a paper about using graphical models to do some prediction tasks with known observations. Since the model is complicated, finding the maximum a posteriori on the true posterior ...
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25 views

Adjusting regression for small sample bias

I have a set of data points $\{x_i\}$. These data points are grouped so that (say) $i\in\{1,2,3\}$ is group $A$, $i\in\{4,5,6,7\}$ is group $B$, etc. I would like to test the null hypothesis of no ...
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37 views

Find the maximum of an integral function with respect to another function

I'm facing this statistical data analysis problem, where I have to maximize a certain statistic in order to find the optimal filtering function. I'm a little bit out of practice with the mathematics ...
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44 views

Classification problem: admissible rule is a Bayes rule for some prior $\pi$

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$. Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in ...
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31 views

Comparison of two error distributions to determine “goodness of fit”

I am a physicist who is a few years out of doing his last course in statistics, so I am hoping to get some advice when comparing some data I recently generated. The context is as follows. I have two ...
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38 views

Hypotesis test: $X_i | \theta \sim Exp(\theta)$ (Likelihood Ratio Test)

Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$ P.S: ...
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42 views

Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
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94 views

Central Limit Theorem Clarification

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
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54 views

Hypothesis testing with poisson distribution

At a nuclear plant great care is taken to measure the employees health.These are the number of visits made by each of the 10 employees to the doctor during a calender year. 3,6,5,7,4,2,3,5,1,4 ...
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231 views

Exponential integral approximation

I have an equation that contain exponential integral of the form: $$ \begin{equation} E_k\left(\frac{a+b ~x}{c}\right) \end{equation} $$ Where $k\geq 0$ ($k=0,1,2,...$), $a$, $b$, and $c$ are ...
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51 views

Properties of almost sure convergence

If, $\Sigma$ is the population covariance matrix and $S$ is the sample covariance matrix, $p$ is the number of variables, $\frac{p}{n} \rightarrow c$ as $n \rightarrow 0$, $\frac{1}{p}|S|_{F}^{2} ...
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71 views

Nondimensionalize an equation with logs

I have an equation with logs in it. Specifically the equation takes the form: $log(Y)=\alpha+\beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3$ Where $X_1$ through $X_3$ are variables. The trouble is that Y ...
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20 views

How to test that a a time series follows a Markov process?

I am searching for a statistical test that tells wether a finite-alphabet time series is a Markov process of a given order or not.
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46 views

Interval overlap maximisation problem

Consider a line of equally spaced sensors and a disturbance which travels unidirectionally along the line at a fixed speed such that the disturbance takes time $\tau$ to travel between adjacent ...
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25 views

Point Estimator.

When the point estimator under consideration has a pdf , the $P[T=\tau(\theta)]=0 $ where $\tau(.)$ is some function of parameter $\theta$ T is an estimator of $\tau(\theta)$ But i did many ...
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65 views

Calculation of conditional joint probability given certain conditionals for data which aren't independent

The context of this problem is the estimation of the distribution of a parameter $v$ given sets of data $A$ and $B$, where $A$ and $B$ are not independent. Suppose I know $P(v | A)$ and $P(v | B)$. ...
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25 views

Choosing an appropriate part of an unreliable dataset

I have a dataset of ~2000 entries (for example, model of car). For each car model I know the weight of the car, and the power output. I don't know the price or age, which is likely to affect the ...
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17 views

minimum distance estimation approach in inference

I am having a bit of problem with a very basic concept in the minimum distance estimation approach in statistical inference. I've read a paper which uses, for a parametric model family of discrete ...
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32 views

Run test statistics

I'm currently implementing the Wald-Wolfowitz runs test on a variable $\epsilon \in \{0;1\} $. According to the original paper, the number of runs, $V_n$ converges to the normal distribution with: ...
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31 views

Confidence interval for $k$

I have exams tomorrow and In preparing for it, I came across this question and I need help. Suppose I have a random sample $ X_1, ..., X_n$ from a population with a density function $$f_{x}(x) = ...
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61 views

Survival Analysis

I have some survival times which are exponentially distributed for two groups G1 (treatment) and G2 (control). The data are censored with a censoring distribution given by h(c), so I only observe: 1) ...
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678 views

Beta Distribution Sufficient Statistic

So I have this homework problem that I am struggling a little bit with coming to a solid answer on. The problem goes like this: Suppose X~Beta($\theta,\theta), (\theta>0)$, and let $\{X_1, X_2 , ...
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35 views

Test the hypothesis that $B_1=0$ at the $5 \%$ significance level

I am told to test the hypothesis and this is what I did: $H_{0}:\beta_{1}=0$ $H_{a}:\beta_{1}\not=0$ So then I have $$t^{*}=\dfrac{\hat{\beta_{1}}-\beta_{1}}{\dfrac{s_{\beta_1}}{\sqrt n}}$$ ...
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104 views

Need help deriving plug-in (functional of distribution) estimator

I need help with homework exercise, have no idea how to approach it. Assume we have i.i.d. observations $x_1,\ldots,x_n$ of a continuous random variable $X$, taking values in $\mathbb R^+$. Define ...
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141 views

Classification: Why k-Nearest Neighbor method is more appropriate for a Mixture of Gaussians?

I'm reading a book named "The Elements of Statistical Learning" in which it states 2 scenarios when we are trying to predict the class label: Scenario 1: The training data in each class were ...
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35 views

What is a good way to deal with finite trains from random variables?

EDIT: Disregard this question if it is formulated too confusing. This link provides you with the updated version of the same question What is the most powerful test for process discrimination based ...
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128 views

What statistical hypothesis test to use for comparing results of two equations?

Given a function $f\left(x\right)$, I have two formulas to compute the coefficients of the same harmonic series approximation to $f\left(x\right)$. Call the results of each formula $^1c_k$ and $^2c_k$ ...
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48 views

Estimate the size of a set given random sub sets.

Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible. I'm not going to quibble ...
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Conceptual doubt between observed values and samples

first of all, I know almost nothing about statistics (and I do no like it or I've never found a good reference). Apparently it is usual to write capital letters to denote random variables and small ...
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24 views

Which hypothesis test to use

Two identical machines are used to make a special coin. We want to know if they have the same variability. A random sample is taken from each machine : $$ \begin{matrix} MachineA & 135 & ...
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16 views

Poisson distribution confidence intervals and hypothesis

I think I have A and B correct but I have troubles with parts C and D. A) What is the p value if we suppose the following : finding golden apples in a tree follows a Poisson P(2) with $\lambda = 2$ ? ...
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16 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
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Is Expectation Propagation (EP) affected by the prior?

I understand EP by reading Minka's thesis: http://research.microsoft.com/en-us/um/people/minka/papers/ep/minka-ep-uai.pdf I'm trying to apply it to solve a Bayesian inference problem. However, I'm ...