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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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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32 views

Proof that data-processing selection process is statistically independent of the sample?

I hope to replace a proof of my own, in a paper explaining that the "no free lunch" theorems for optimization actually address sampling and statistics, with a reference to an existing result on ...
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14 views

factor models and using cross sectional regression?

I have been doing some reading on factor models. In the literature it mentions that when creating a portfolio that maximises particular attributes it may lead to unwanted bias to other factors. I ...
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21 views

Finding the sample size

My main question is this: Assume I want to conduct a plebiscite of an entire town about whether or not the citizens want to have their water fluoridated or not, I want to conduct a survey first. If ...
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13 views

Distributive Law on Sum Product

I am reading a tutorial on Conditional Randome Fileds, Here is the link: http://people.cs.umass.edu/~mccallum/papers/crf-tutorial.pdf in the equation 1.24 it defines: $p(x,y) = \prod_{t=1}^{T} ...
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13 views

Reconstructing message from clippings

It's a question arising from genetics. Suppose we have a message - a list of 100 bits. Now we run a process which cuts off either one or a pair of bits (let's say with equal probability). As a result ...
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22 views

Probability Distribution in Cumulative Follow-Up Study

Data layout for a cumulative type of follow-up study is : $$\text{table 01. Data layout for a cumulative follow-up study}$$ $$ \begin{array}{l|cc|l} & \text{Exposed}(E) & ...
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20 views

Statistics on a column of values

I have a class project that I am working on as an engineering student but the course is probability and statistics so I don't really know how to make sense or use of what I am learning so I thought of ...
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10 views

Can I use a z-transformation instead of Welch-t-test

I am not a big mathematician and at the moment I am just refreshing some of my statistical knowledge. So, when doing a students t-test with two independent samples, a precondition is, that both ...
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28 views

UMVUE using complete and sufficient statistic

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 ...
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21 views

Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
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23 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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30 views

Parameter estimation of Lorenz system (nonlinear dynamical system)

My problem is as follows. I have to estimate parameters of Lorenz system using given data. Lorenz system is described by following system of ODEs: $$ \frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = ...
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17 views

Show there is no UMP test for $N(\mu,\sigma^2)$

Here's a testing problem. $X_1, \cdots, X_n, indep, \sim N(\mu, \sigma^2)$ where $\mu, \sigma^2$ are unknown. $H_0: \mu \le \mu_0$ vs $H_1: \mu > \mu_0$ where $\mu_0$ is specified. I ...
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7 views

Matrix Completion feature vectors

In the noisy matrix completion problem, with enough number of revealed entries say $|E| = nr^2 log(n) $, can we have a bound on the error in the singular vectors of a sub matrix. For, example say ...
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24 views

Question about Logistic Regression - 6

I am studying Logistic Regression and I have come across to understating the paragraph below. I kind of can understand, but it makes me confused when I read the sentence in the red circle, "It also ...
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16 views

Backwards quadratic programming to infer Q matrix

Consider the standard QP problem: $\arg\min \frac{1}{2}x^TQx +c^Tx$ Say I know the optimal $x$ for a large number of solutions to this problem with various (known) $c$, and identical (unknown) $Q$. ...
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16 views

Normal distributions with a 95% confidence interval

A factory produces items with a mean weight of 3kg. A sample of 100 items has mean weight 3.041kg. Assuming the weight of the items is N[a, 0.04] find a 95% confidence interval for a. I understand ...
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34 views

Proving a statistic is sufficient and NOT complete

Let X be a single observation from the distribution , $P(X=x)= \begin{cases} \theta, & \text{if $x$ =-1} \\ (1-\theta)^2\theta^x, & \text{if $x =0,1,2,3,...; 0<\theta<1$} \\ ...
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52 views

Sufficient statistic for normal distribution

Let X from a Normal distribution $(\theta,1)$. a) Find a sufficient statistic for $\theta$. b)Is $S_n^2$ sufficient for $\theta$ My answer for part a) The joint p.d.f= $1 \over (2\pi)^{n/2}$$e^{{-1 ...
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19 views

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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33 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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22 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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11 views

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 ...
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22 views

General procedure Vs pooled variance

When we compare means of two group. With independent sample, we have two method, but i don't know when use choose which one and when choose the other. Please help me classify them Two method are: ...
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34 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
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19 views

Cross correlation computations

What are useful ways/formula for calculating sample cross correlations (i.e. correlation factors between individual components of two different random variables). Say I have two sample matrices, $X$ ...
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18 views

Calculate probability of total population from sample

I have a population of infinity size. Each Unit is Boolean(True/False). I take a random sample of size N of which M elements are True(and N-M are False). I'd like to know what the probability is that ...
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25 views

How to find MLE function?

I have two vectors of known values x and y. And the relationship between them is y=sin($\theta$$\cdot$x)+$\epsilon$, $\epsilon$~N(0,1) . The question is how i find the MLE function for $\theta$?
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19 views

Stationary point of Unnormalized and Normalized KL-divergence minimization

I have encountered a problem, basically related to the http://arxiv.org/abs/1206.6679 . If I want to minimize the normalized KL-divergence KL(Q||P) with Q a multivariate Gaussian distribution. but in ...
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20 views

Jeffery's prior - help

I'm studying Bayesian inference and looking at prior choices. Currently I have looked at Laplace's uniform prior choice and now I am trying to understand Jeffrey's prior. I am having trouble ...
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25 views

Multivariate Normal Variables

Sorry if this is a bit hard to read. Not entirely sure how to use the MathJax formatting on the site... The random variables $X_1, X_2, X_3, X_4$ and $X_5$ are jointly multivariate normal. Their ...
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18 views

Point estimation for two different group

Let θ be the difference in the mean precipitation from the two groups. Estimate θ. Estimate the standard error of the estimate and produce a 95 percent confidence interval. About mean, I use ...
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21 views

Unbiased estimators of $\theta^3$

Let $X_1, ..., X_n$ iid $∼$ Bin($1,\theta$) $0 < \theta < 1$ and $n >= 3$ I'm trying to find two unbiased estimators of $\theta^{3}$, one that is a function of only $X_1, X_2, X_3$ and ...
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26 views

Proof of the independence of the sample mean and sample variance using characteristic function

This question is related to the one asked here Joint characteristic function of $\bar{X}$ and $\left\{ X_{1}-\bar{X},X_{2}-\bar{X},\cdots,X_{n}-\bar{X}\right\}$ is ...
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17 views

Showing that the sample variance for an SRS is a biased estimator of the population variance?

EDIT: I suspect I may be going about this all wrong, so maybe disregard this. So, I can get this far on my own: $E(\hat{\sigma}^2) = E(\frac{1}{n - 1}\sum_{i = 1}^n (X_i - \mu)^2) = \frac{1}{n - ...
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25 views

Computation of conditional probabilities

I am aware this question might not be well formulated, but it is not very clear for me neither, so if anyone could help me explicit it... I observe many examples e_i. For each example, I compute two ...
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9 views

Predict values of some numerical vectors by using other numerical vectors with all these vectors in the same vector set

I need to solve a problem about predicting values of some numerical vectors by using other numerical vectors with all these vectors in the same vector set, which is generated by one or more black box ...
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68 views

Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
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23 views

Interpretation of Cook's Distance

The Cook's Distance $D_i$ of the data $(Y_i, x_i)$, $i=1,2, \dots, n$ is defined as: $$ D_i := \frac{||\hat{Y}-\hat{Y}_{(-i)}||^2}{p{\hat{\sigma}}^2}$$ where $p$ is the number of parameters, the 'hat' ...
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8 views

Statistic (Linear normal model): $X_{hi} \sim N(\alpha_h+ \beta t_{hi}, \sigma^2)$. How to calculate $C_{95}(\beta)$ and $t$-test for $\beta = 2.5$?

Statistic (Linear normal model): $X_{hi} \sim N(\alpha_h+ \beta t_{hi}, \sigma^2)$. How to calculate $C_{95}(\beta)$ and $t$-test for $\beta = 2.5$? I am in the statistical model $X_{hi} \sim ...
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18 views

Estimation of a Random Variable , Probability

A machine consist of two engines ,which stop running independent of each other.In one week the probability of the engines stop running is $p_1$ for the first engine and $p_2$ for the second one.So for ...
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20 views

Efficient algorithm for point estimation of a dependent random variable

Suppose $X$ is a normal-distributed random variable and $f$ is a known smooth function (possibly quite complicated, with many oscillations). Let $p(y)$ be the pdf of the dependent random variable $Y = ...
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19 views

$p$-dimensional confidence set for normal linear model

this is a question about a confidence set in $p$-dimensional space for a normal linear model. If we define the cuboid $C:= \prod_{j=1}^p C_j(\frac{\alpha}{p})$, where $$C_j(\alpha) = [\hat{\beta_j} ...
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10 views

Loss specific inference in graphical models

As far as I have seen, in graphical models, the inference (for training or parameter estimation) is done via maximizing likelihood. While in many applications people need loss specific optimization of ...
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58 views

Find the MLE of the parameters

I need some help with the following problem: Let $X$ and $Y$ be independent exponential random variables with $$f_X (x| \lambda)={1 \over \lambda} e^{-x/\lambda},x>0,$$ $$f_Y (y| \mu)={1 \over ...
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27 views

What fundamental assumption is taken here?

In biology, the following is often used for collecting data from a population: if we have data from $n$ people, over $m$ years, then we have $nm$ years of data for one person (to simplify). To give a ...
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10 views

Tukey's half space depth

For multivariate distributions, Tukey's half space median is defined as "the maximizer" of the quantity: inf{P(a'(X-*x*)>=0|||a||=1}, where X~a multivariate distribution F and the infimum is taken ...
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26 views

Multiple comparisons Bonferroni

I want to use Bonferroni method to do multiple comparisons, but i would like to know how the statistic is being proved. From what i understood the statistic of Bonferroni is $$ t=\dfrac{(X_i - X_j )} ...
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28 views

Hypothesis test on variance of normal sample

This question is a follow-up to this discussion: Distribution of likelihood ratio in a test on the unknown variance of a normal sample Could someone audit the reasoning below? I am trying to derive ...