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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Question on sufficient complete statistics proof and estimators of zero

I am trying to prove theorem 7.3.23 in Casella and Burger. Theorem: Let T be a complete sufficient statistic for a parameter $\theta$, and let $\phi(T)$ be any estimator based only on T. Then $\phi(...
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18 views

Casella and Berger Likelihood Ratio Tests statistic vs Wasserman LRT

It seems like there is a discrepancy between these two authors on what a LRT is. Casella and Berger state on pg. 375. That the LRT statistic is: $\lambda(x)=\frac{L(\hat{\theta}_0|x)}{L(\hat{\theta}|...
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31 views

expected value of the average cubed

I can not resolve an issue of the book Mathematical Statistics of Shao, is as follows: If $E|X_{1}|^3$ is finite, get $E(\bar{X}^3)$ and $Cov(\bar{X},S^2)$ If $E|X_{1}|^4$ is finite, get $Var(S^{2})$...
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24 views

How to calculate survey bias due to preference for first answer?

I was recently given the results to a survey in which participants chose answers to questions they would be likely to randomly answer, and in which the survey population is known to have a preference ...
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44 views

Calculate calculate $E(\frac{1}{\bar X})$, Var$(\frac{1}{\bar X})$

Consider n i.i.d. observations from a Poisson (λ) distribution. Suppose $\bar X =\frac1n\sum_{i=1}^n X_i.$ How do I calculate Var$(\frac{1}{\bar X})$ or even $E(\frac{1}{\bar X})$ for that matter. ...
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17 views

Multilinear loss in Uniform-Exponential model

Let a prior $\pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta))$ and $f(x\mid\theta)=\theta e^{-\theta x}$. Taking the multilinear loss $$...
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43 views

Finding the Cramer-Rao Lower Bound

Given the probability density function $$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$ and $0$ otherwise, find the Cramer-Rao Lower Bound for $\...
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Additive model ANOVA with interaction. How to obtain gammas from the graph?

I try to complete this exercise of statistic that said that I need to compute the values of alpha, beta and gammas from the graph. Graph: I allready compute the values algebricaly, and this are: <...
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13 views

Distribution of Subvector Sums

Suppose $X_1,\dots,X_N \sim_{iid} \mathcal{N}(0,1)$ are iid normal, and let $K=N/2$. Let $S$ denote the collection of all subsets of $\left\{1,\dots,N\right\}$ with $K$ elements. For any $s\in S$ ...
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Can I bound $P[R > x + \epsilon]$ independently of R?

I have this probability distribution: $P[\Theta < \varphi] = \frac{\varphi}{\pi}$ for $\phi \in [0,\pi]$. Now I have $n$ samples of $D = R\Theta$ i.i.d. ($R>0$) and I want to estimate $R$ as $\...
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31 views

Bayesian inference for sum of random variables

Assume that we have a random variable $Z = X + Y$ for $X$ and $Y$ independent. Then if w use two independent data-sets $D_1$ and $D_2$ to try and approximate the distribution of $Z$, i.e. $$p(Z|D_1,...
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17 views

Relation between Bayesian analysis and Bayesian hierarchical analysis?

I have been studying a Bayesian hierarchical model. In that model all I am dealing is with the estimation of parameters. In Bayesian analysis, loosely speaking, we update our prior knowledge (in light ...
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42 views

Confidence interval for exponential - is it the shortest possible?

The confidence interval for an exponential distribution is said to be: $$\frac{2n\bar{x}}{\chi^2_{1-\alpha /2,2n}}<\frac{1}{\lambda}<\frac{2n\bar{x}}{\chi^2_{\alpha /2,2n}}$$ In general we aim ...
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21 views

Sampling from Gaussian Process Posterior

Anyone know of a Python package that both fits a Gaussian Process to data, and also lets you sample paths from the posterior? I'm interested in sampling the colorful lines on right (b) of the ...
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32 views

Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space

Suppose that we are given an exponential distribution model with a pdf $f(x,\theta) = \theta^{-1}\exp(-x/\theta)$ with an iid sample $X_1, \ldots, X_n$, and we would like to test hypothesis $H_0 : \...
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39 views

Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-degenerate random variable.

Let $X_1,X_2,...X_n$ be iid with the same chi-square distribution with one degree of freedom. Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-...
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74 views

Theoretical distribution of a random variable

Martin has $n$ words, and he wants to make a computer program that chooses for him $k$ words (and shows them to him), where $k \le n$, for as many times as he clicks a button until all of the words ...
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64 views

Does a Markov process have memory zero?

I have the following question: The Markov process with two states and a transition matrix $$P =\begin{pmatrix} 0.3 & 0.7 \\ 0.3 & 0.7 \end{pmatrix}$$ has memory zero. Is it true? My ...
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30 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy.

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
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37 views

testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...
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38 views

Rank of a random matix

This arises in Time-Series modelling. Suppose $Y_i \sim N_p(0,\Sigma_i)$ and they are not necessarily independent (but assuming $\Sigma_i$ to be p.d.). Then for any ${a}\neq 0 \:\:\:\:$ $Y_i'a\neq0$ ...
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22 views

Conditioning multivariate Gaussian on a function of coordinates

I have a pretty general question and I would really appreciate if you give me any hints or point me towards some relevant literature. Suppose $X$ is an $n$-dimensional Gaussian vector. What is the ...
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20 views

Penalty function of multi-peak fit?

The question I have is about the answer from here by @Silvia: http://mathematica.stackexchange.com/questions/26336/how-to-perform-a-multi-peak-fitting I can only understand some of the code but the ...
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14 views

Denominator in Maximum Posterior Estimation - How to Interpret?

Suppose we're given a sequence $x_1,\ldots,x_n$ of realizations of i.i.d. $\mathcal{N}(\mu,\sigma^2)$ random variables and we want to apply maximum posterior estimation to estimate the parameters $\mu$...
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23 views

Covariance estimation and Graphical Modelling

I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions. Suppose, $X_i \overset{iid}{\sim} N_p(\mu,\Sigma)$, $i=1(1)n$...
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62 views

Likelihood ratio test for a normal distribution with unknown mean

Suppose $X_1,X_2,…,X_n$ is a random sample from a normal population with mean $μ$ and variance 16. Let sample size=16. Find the likelihood ratio test for $H_0:μ=10 $ against the simple alternative ...
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22 views

Likelihood ratio test using random number generation

Let $X_1,X_2,...,X_{10}$ be a random sample from the density $\theta_1 x^{(\theta_1-1)}I_{(0,1)}(x)$ and let $Y_1,Y_2,...,Y_{20}$ be a random sample from the density $\theta_ 2y^{(\theta_2-1)}I_{(0,1)...
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Prove $\lim_{n\to\infty}\Bbb P[\max_{k\le n}x_k \le\sqrt{2\log n-\log(2\log n)-log 2\pi +2x}]=\exp({-e^{-x}})$

Prove that $\lim_{n\to\infty}\Bbb P[\max_{k\le n}x_k \le\sqrt{2\log n-\log(2\log n)-log 2\pi +2x}]=\exp({-e^{-x}})$, where $x_1, x_2$, etc. are independent with common density $(2\pi)^{-1/2}e^{-x^2/2}$...
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23 views

How to compare standard deviations of different sample sizes

I am interested in estimating the value of an unknown, random variable, X. X changes over time. So, X = X(t) = Xt At specific time intervals, I estimate the value of Xt using 7 different methods. Some ...
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20 views

What to do about Missing values for Multivariate regression analysis

I am required to perform multivariate analysis on all countries in the World bank database regarding digital divide. I am confused becasue when I look at factors such as School enrollment, there are ...
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45 views

UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$
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53 views

Sampling substrings of a beaded necklace to determine the necklace composition

I have a necklace composed of 100 beads, where each bead is one of 13 colors. If I am only able to look at one 4 bead sub-sequence at a time (connected, as they would be on the necklace) , how many ...
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14 views

comparing 2 datasets which have different distributions

I'm currently analysing two datasets. They report the same information, but in different ways. I am looking to draw comparisons between the way items fail in each of the datasets. In the first ...
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36 views

Why isn't the estimator of the square of a parameter the square of the estimator of the parameter?

Let $X_1,...,X_n$ be a sample from a distribution having as a p.d.f: $f(x) = \frac1{\theta} e^{-x/\theta}, x,\theta > 0$ and $0$ elsewhere. The maximum likelihood estimator of $\theta$ is $\bar{X}...
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38 views

Why is $nS_X ^2/\sigma ^2$ $\chi ^2(n-1)$, while the other is $\chi^2(n)$?

Suppose $X_1,...,X_n$ is a random sample from a distribution having $N(\mu, \sigma^2)$. What is the conceptual difference between: $$ \frac1{n} \sum_{i=1}^n (X_i - \bar{X})^2$$ and $$ \frac1{n} \...
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28 views

Creating a minimal sufficient statistic with Likelihood function

To find a minimal sufficient statistic you can take the likelihood ratio and find a function $T$ so that the ratio does not depend on the parameter $\theta$ , as page 18 here http://sites.stat.psu.edu/...
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Positive semi-definite in Linear model

Suppose $Y_{n \times 1} \sim N(X\beta,\sigma^2V)$ where $V_{n\times n}$ is invertible and $X_{n\times p}$ is of rank $p$ and $\beta_{p \times 1}$ is unknown and to be estimated by $Y$ and $X$. Which ...
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158 views

Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$. Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: \theta_1 &...
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31 views

How is the Variance of this estimator equal to $\theta$?

Currently going through solutions of a worksheet and I don't understand the jump between two lines of working. "$\hat{\theta}_1$ and $\hat{\theta}_2$ are independent unbiased estimators for an ...
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73 views

How to measure the stability of datas

The background: I have a server handling $n$ kinds of requests, denoted by $k_1, ..., k_n$, at a certain time, many requests has been processed, the average time it takes to process $k_i$ is $t_i$, ...
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42 views

Estimating Prevalence of Disease, Defects, or Spam With Screening Tests

Background. A screening test is a relatively quick and easy or inexpensive preliminary test that gives preliminary warning of undesirable condition $D$, such as disease in a patient, defect in a ...
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78 views

Rao-Blackwell theorem and conditional distribution

Let $X_1,..,X_n$ random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$ i) Find a unbiased estimator of $\lambda$ based only ...
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49 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
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15 views

Statistical Modeling with the combination of two models

I'm having a modeling problem now. Assume we have discrete random variable Y and continuous random variables X and Z. First, we assume a logistic regression between Y and Z.(Assumption One) Also, we ...
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57 views

Asymptotically normal but biased estimator

This is the problem 2.11 from Lehman book "Theory of point estimation" 2-nd edition. Construct a sequence $\{\delta_{n}\}$ of estimators of $g(\theta)$, satisfying $$ \sqrt{n}[\delta_{n} - g(\theta)]...
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37 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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How can we have $T_n \xrightarrow{\mathbb P_\vartheta} \vartheta$ if $T_n$ are defined on different spaces?

Here is how I understand the standard parametric model in statistical inference: We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. ...
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45 views

Conditional Probability/Expectation in the EM algorithm

I'm doing a study in which I measure data under a random censoring process. The observed data which may be interpreted as the lifetime of a subject, is denoted by $t$, with the censoring variable $c$ (...
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49 views

calculating $E(X_{(i)}| \sum_{i=1}^5 X_i)$

suppose 5.5,3.5, 2.5,4.5,2 be a random sample from of gamma distribution with parameters of $ \beta,\alpha=2$. if $Z_{(i)}$ be i-th order statistic a random sample of size 5 from $\Gamma(2,1)$, how ...
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49 views

Limiting Distributions and the 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 ...