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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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_ ...
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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 ...
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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. ...
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18 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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42 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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49 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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12 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 ...
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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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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 ...
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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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100 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: ...
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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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43 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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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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57 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 ...
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48 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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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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56 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} - ...
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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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39 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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34 views

Fisher Expected Information for a Gaussian Process model

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am ...
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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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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 ...
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37 views

Do higher order sample moments converge to the distributional mean?

The Methods of moments estimation is based on the law of large numbers, which says that the sample means of i.i.d. random variables from any distribution converge to the distributional mean as the ...
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48 views

Hazard function for proportional odds model

The Cox proportional hazards model for survival data with covariate ${\bf z}$ is defined through the hazard function $h(t,{\bf z})$ by $$ h(t,{\bf z}) = h_0(t)~\cdot\theta~~,~~~\theta = \theta(\beta, ...
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29 views

Questions on the theory of Lasso

The linear model ${\bf Y}={X\beta}+\epsilon$, where ${\bf Y}$ is a $n\times 1$ vector, and ${\bf X}$ is $n\times p$ matrix. $n\lt p$ and $rank({\bf X})=n$. $\epsilon\sim N(0, \sigma^2)$. How to prove ...
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inferential statistics methods for population?

Is that true to use inferential statistics methods when we study whole population? I mean for example is that true to use hypothesis test when whole population are under study? Suppose I am studying ...
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$Z$ is Normal$(\sigma,1)$, find UMVUE of $P(Z\leq 0)$.

Given i.i.d. samples $X_1,...,X_n$ from Normal$(\sigma,1)$, find the UMVUE of $g(\sigma)=P_\sigma(Z\leq 0)$. I tried to use Lehman-Scheffe theorem. We now that $\sum_1^nX_i$ is sufficient and ...
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29 views

How can I tell whether sample size is inadequate or not ?

I am given sample size of 15322 students and our research topic is to find out a relationship between students academic performance and participation in sports team. The question asks " do you think ...
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40 views

how to find relationships in dataset with multiple variables

I have a large project data set ,which includes numeric values like dollar amounts, and non numeric quantities like country codes, purpose codes etc I want to find relationships between the variables. ...
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Showing that moment estimates are asymptotically bi-variate normal.

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let ...
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How to parameterize some emprical data

I would like to describe a bunch of data that I have collected as a function of two variables. The data is phytoplankton absorption in my local area that has changed in concentration. The data looks ...
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$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
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How to represent data graphically in which a respondents can choose more than one group?

I am analyzing a survey in which one of the questions was (What is the industry segment of your company?) and the respondents can choose more than one category. I have used a pie chart to represent ...
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196 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
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114 views

Statistical sample with age ranges. How to extrapolate it using the real age distribution over population.

I have a data set consisting in the classification of the numbers of suicides by age range. I want to figure out if there is or not association between the number of suicides and the age range. But, ...
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89 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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36 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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How to account for both between-subjects and within-subjects covariables?

I have a data set I'm trying to analyze and I can't figure out how to include the two different kinds of covariables in my analysis. Without the covariables, the analysis isn't too complicated. It's ...
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Outlier Contained in Prediction Interval (Tme series Forecasting Problem)

In my stats class today, the professor was showing us some output from MINITAB on a prediction interval that was calculated (from time series data using standard linear regression). For one of the ...
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Uniformly Most Powerful

Let X1;X2; : : : ;X10 denote a random sample of size 10 from a population which has an exponential distribution with parameter ; > 0, i.e. with pdf f(x) =   e 􀀀x if x 0; 0 otherwise. (a) Find ...
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Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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48 views

Comparing models to smoothed data

I am attempting to fit a model to a noisy data set. I am performing this modeling in two stages - first, smoothing it out by fitting an analytic mixture model to it, and second, fitting my final model ...
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Showing mutual contiguity

The problem: Let $P_n$ and $Q_n$ be the distribution of the mean of a sample of size $n$ from the $N(0, 1)$ and the $N(\theta_n, 1)$ distribution, respectively. Show that $P_n$ and $Q_n$ are ...
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81 views

Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
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Fitting of the Lévy triple

Given a Lévy process and its triplet $(\mu,\Sigma,\nu)$ i.e. the triplet such that for each $t\ge 0$ $ X(t) = bt + W_A(t) + \int_{|x|<1} x \tilde N (t, dx) + \int_{|x|\ge 1} x N(t,dx)$ where ...
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138 views

Iterative Mean, Covariance Algorithm Convergence

The problem is to show that the following iterations converge to the vector $\mu$ and the matrix $\Sigma$. We have data in the form of nx1 vectors $\mathbf{Q}_k$, $1 \leq k \leq N$ where ...
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Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...