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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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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15 views

Risk function of binomial random variable

Suppose $X\sim Binomial(100,\theta)$, True estimator($\delta$) $= X/100$; $$R(\theta ,\delta) = E_{\theta}(\theta - (X/100))^2 = \theta(1-\theta)/100$$ I am copying this same from text book but ...
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2answers
19 views

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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29 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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58 views

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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33 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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17 views

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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14 views

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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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: $p(x,y) = \prod_{t=1}^{T} ...
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$Y_{(n)} = X_{(n)}/\mu$?

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$ I am sorry if the question looks so simple and I am nt ...
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22 views

Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|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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142 views

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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36 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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a complete sufficient statistic in geometric distribution

Exponential family has a very good property that could be used to conclude if a statistic is complete: $X_1,X_2,\ldots,X_n$ are from exponential family which has the form as: $$f(x\mid \theta ...
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26 views

How to prove a statistic is an ancillary statistic?

Though with the knowledge that an ancillary statistic is a statistic has distribution that is independent of the parameter, I feel like I still don't know very well for verifying a statistic is an ...
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11 views

95% Confidence interval of proportion test without calculator

I need some help regarding a calculation that I need to be able to do only with a basic calculator of a proportion test: A statistician is choosing a sample of 200 seeds. If 155 of these 200 are ...
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13 views

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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111 views

Weighing correlation by sample size

I'm a scholar in the humanities trying to not be a complete idiot about statistics. I have a problem relevant to some philological articles I'm writing. To avoid introducing the obscure technicalities ...
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37 views

4 points, how to know if it's growing over time?

I've an array of 4 points, which formula should I use to detect their growth ? ...
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30 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
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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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infer the initial state from draws

I went through binomial distribution and Chi-square test etc and got confused further. This question might be very basic and simple. I have three states (Combination of two colors, both has equal ...
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28 views

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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24 views

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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33 views

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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31 views

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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40 views

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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38 views

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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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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28 views

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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34 views

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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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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26 views

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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38 views

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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131 views

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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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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16 views

“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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27 views

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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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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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 ...
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27 views

Statistical inference

Ten balls are picked without replacement from a box of 50 balls. All turn out to be red. A argues that the box contains only red balls. B argues that it contains 25 balls each of red and and some ...
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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 ...
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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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Confidence Intervals with two variables

A random sample of $50$ men shows the following relation between annual income $Y$ (in dollars) and education $X$ (in years): estimator for $Y, (\hat{Y}) = 1200 + 800X $ Average income is $Y = ...