# Tagged Questions

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### Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
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### Risk function of binomial random variable

Suppose $X\sim Binomial(100,\theta)$, True estimator($\delta$) $= X/100$; $$R(\theta ,\delta) = E_{\theta}\left[\left(\theta - \frac{X}{100}\right)^2\right] = \theta(1-\theta)/100$$ I am ...
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### Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $\theta$ and variance $a>0$. Let $Y$ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
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### 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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### Derivation of standard error of mean

I was going through this wikipedia article on standard error. I could not understand the crucial step here. It goes like this: This formula may be derived from what we know about the variance of a ...
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### 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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### Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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### normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
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### 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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### 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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### Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $\epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
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### Testing a hypothesys about a survey?

A survey of $61, 647$ people including questions about office relationships. Of the respondents, $26$% reported that bosses scream at employees. Use a $.05$ significance level to test the claim that ...
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### Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
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### Is difference between two sets of measurements significant?

Consider the following experimental setting: I have two machines $m_0$ and $m_1$ of which I would like to know which one performs better. For this I have set up an experiment to measure the time it ...
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### Biased MLE estimate of mean (expectation)

Please give an example of p.m.f. or p.d.f. , the maximum likely-hood estimate of whose mean (expectation) is a biased estimator . Thanks
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### Variance- covariance matrix

Consider $H$ denotes hat matrix and $e$ denotes residual. In the book Applied regression Analysis by Draper/Smith, it is written that : $\mathbb V(e_i)$ is given ...
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### Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
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### Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
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### How do I convert non-normal distribution to a normal distribution?

I have the below graph I drew using histogram in Excel but when I got my data and graphed it, certainly it is not a normal distribution. My assignment required me to gather some data and draw normal ...
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### Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
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### 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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### A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
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### Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
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### Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
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### “Approximation” of a maximum likelihood confidence set

I have some trouble constructing the "approximate" set, how can it be defined or calculated? Suppose that three characteristics in a large population can be observed according to the following ...
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### Alternatives to absolute error?

Let me explain my scenario in which I need to calculate absolute error. Lets say the X is the actual value. And X' is the value of X with some error 'e'. So X' = X + e'. Lets say i = 1 to 10000. I ...
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### 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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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 $... 1answer 51 views ### Finding UMVUE for Poisson distribution using Rao Blackwell Let${X_1,X_2,... ,X_n}$be a random sample from a Poisson distribution with parameter$\lambda$.Let$\gamma(\lambda)=P(X<=1)$. Find UMVUE for$\gamma(\lambda)$. This is my attempt: First I ... 0answers 17 views ### Draw and compare the likelihood using R The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ... 0answers 23 views ### What is the problem with this model parameter estimation algorithm? In a statistical model with parameters$\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$\theta$... 0answers 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: $$\frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = ... 1answer 33 views ### Question about Logistic Regression - 7 I am currently studying Logistic Regression. I am facing a problem with understand the sentence in the red circle below. I am trying to figure out what he/she means by the sentence. Please let me have ... 0answers 24 views ### Question about Logistic Regression - 6 I am studying Logistic Regression and I have come across to understating the paragraph below. I kind of can understand, but it makes me confused when I read the sentence in the red circle, "It also ... 1answer 27 views ### Question about Logistic Regression - 2 How should I tell the difference between those two formulas in the circles below. I am studying logistic regression and I have faced two different formulas from two different documents. I don't know ... 1answer 19 views ### Question about Logistic regression-1 As I am currently studying Logistic regression, but I am still new to this. I have encountered a sentence and a word below in red circles that I can only image but not with my own clear ... 2answers 36 views ### Question about Logistic Regression - 4 I am currently studying on logistic regression. So I have found a document on the Internet explaining about it. Somehow, it explains Bernoulli distribution in the beginning and I am having a problem ... 1answer 42 views ### Whether or not X_1,…,X_n are independent and exchangeable For some n = 1,2,..., let Y_1,...,Y_{n+1} denote iid real-valued random variables. Define X_j = Y_j$$Y_{j+1}$, $\hspace{10mm}$j=1,...,n a) Are $X_1$,$X_2$,...,$X_n$ independent? b) Are ...
Can somebody help me interpreting the red circled sentences in planer English? I understand "We view $y_i$ as a realization of a random variable $Y_i$ that can take the values of one and zero" but ...