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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12
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3answers
31k views

Why the sum of residuals equals 0 when we do a sample regression by OLS?

That's my question, I have looking round online and people post a formula by they don't explain the formula. Could anyone please give me a hand with that ? cheers
11
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1answer
100 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
8
votes
1answer
10k views

Maximum Likelihood Estimator of parameters of multinomial distribution

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
8
votes
1answer
140 views

Expected value and variance of ratio of two sums of two sets of random variables

Let $X_1,X_2,\ldots,X_n$ be iid $\operatorname{Gamma}(\alpha,\beta)$ random variables. Suppose that, conditionally on $X_1,X_2,\ldots,X_n$, the random variables $Y_1,Y_2,\ldots,Y_n$ are independent ...
7
votes
2answers
193 views

What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
6
votes
1answer
777 views

How to quantify the differencen between 2/4 and 20/40?

Assume I have two methods to do prediction. The first method makes 4 predictions and 2 out of 4 are correct. The second method makes 40 predictions and 20 out of 40 are correct. The prediction ...
6
votes
1answer
2k views

Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
6
votes
3answers
77 views

distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
6
votes
1answer
62 views

Limit of median of uniform distribution

Let $X_1,X_2,\ldots$ be a random sample from the uniform distribution on the interval $(0,1)$. Assuming that $n$ is odd, find the pdf of the sample median (say $M_n$). Does the pdf of the r.v. ...
6
votes
1answer
252 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 ...
6
votes
1answer
138 views

Asymptotic efficiency of maximum likelihood estimate

Let us consider a simple statistical model $\{f_{\theta}\}$ where $\theta\in U$, an open subset of $\mathbb{R}$. Let $X_1,\dots,X_n$ be sample drawn from $f_{\theta}$. I know, under some regularity ...
6
votes
2answers
642 views

Taylor series approximation statistics

how can I show the following: Let $X_1, X_2,\ldots, X_n$ be i.i.d Poisson with mean $\lambda$. Let $Y = |\{i: X_i =0\}|$. Then $\lambda$ is estimated by $$\eta = - \log(Y/n)$$ Use Taylor series to ...
6
votes
1answer
637 views

Trying to understand the basics of bayesian inference

This paper gives a somewhat gentle introduction to Bayesian inference: http://www.miketipping.com/papers/met-mlbayes.pdf I got to section 2.3 without much problems but got stuck in understanding that ...
6
votes
0answers
120 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
6
votes
0answers
39 views

Find a function such that follows to normal in distribution

Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each ...
5
votes
2answers
245 views

Domino's Advertising Pizza Claim

I just got a Dominos promotional flier through the post and one of the graphics advertising 'create your own pizza' lists the various toppings and claims there are 'more combinations than people in ...
5
votes
1answer
344 views

How to prove SSE and SSR are independent

Consider $Y=X\beta+\varepsilon$, where $X$ is n by p, $\beta$ is p by 1 and $\varepsilon$ is n by 1 with covariance matrix = var($\varepsilon$)=$\sigma^2 I$. Give expression for the regression and ...
5
votes
1answer
38 views

Find $\sum_{k=0}^{\infty}(1-1/n)^{2k}\frac{e^{-n\theta}(n\theta)^{k}}{k!}$ (the variance of $(1-1/n)^{X_1+\cdots+X_n}$)

Given a random sample $X_1,\ldots,X_n$ from Poisson distribution with an unknown parameter $\theta>0$.$T:=(1-1/n)^{X_1+\cdots+X_n}$. Find $\operatorname{var}(T)$. My work: I find $T$ is a UMVUE ...
5
votes
0answers
33 views

single variable is significant but overall test is not

I do a multiple regression with 3 independent variables $X_1$, $X_2$ and $X_3$. The correlation between $Y$ and $X_1$, $Y$ and $X_2$, and $Y$ and $X_3$, are each large and statistically significant. ...
4
votes
3answers
479 views

Does “Big Data” Have a Ramsey Theory Problem?

I'm erring on the side of conservatism asking here rather than MO, as it is possible this is a complex question. "Big Data" is the Silicon Valley term for the issues surrounding the huge amounts of ...
4
votes
1answer
13k views

How to calculate the covariance matrix

I tried searching a lot on the net and got the following sources: Source One Source Two The first source seems to be incorrect cause when I calculate it using matlab it comes to be different from ...
4
votes
2answers
84 views

Bivariate distribution with normal conditions

Define the joint pdf of $(X,Y)$ as: $$f(x,y)\propto \exp(-1/2[Ax^2y^2+x^2+y^2-2Bxy-2Cx-Dy]),$$ where $A,B,C,D$ are constants. Show that the distribution of $X\mid Y=y$ is normal with mean ...
4
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2answers
1k views

Vague Gamma prior?

I'm looking at a MCMC algorithm where the author takes a Gamma(shape = 0.001, rate = 0.001) prior distribution, which they refer to as a vague prior. For all my searching, I am struggling to see how ...
4
votes
1answer
772 views

computing the bias and standard error of a uniform distribution with unknown upper limit?

Let $X_1, \ldots, X_n \sim \mathrm{Uniform}(0,T)$ and $T^\wedge = \max\{X_1, \ldots, X_n\}$, which is the estimator of $T$. What is the bias and se of this estimator? If $n=1$, then the calculating ...
4
votes
3answers
121 views

Question about English sentences in statistics?

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 ...
4
votes
2answers
60 views

I am running a series of experiments that I expect to have similar outcomes. What is the best method to measure statistical significance?

Following on from this comment on an answer to my previous question, I'd like to know two things: what the best statistical test I can use to measure significance on the experiments I'm running? ...
4
votes
1answer
174 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 ...
4
votes
3answers
136 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...
4
votes
2answers
54 views

Help: SPSS and Data Interpretation of Voters. Republican vs. Democrats (1993 election)(Almost finished)

Hello everyone, I am Julieta this time I get stuck in the following exercise. It is a statistical analysis of pools, the statement is quite long I will try to keep it short and put some links. Note: ...
4
votes
1answer
76 views

MLE for the PDF $f_\theta(x)=\theta x$ on $0\leq x\leq\sqrt{2/\theta}$: where is the mistake?

Consider $f_X(x;\theta)=\theta\cdot x$, $x\leq\sqrt{\frac{2}{\theta}}$. Find the maximum likelihood for the estimator $\hat{\theta}$ of $\theta$. By definition, the MLE of ...
4
votes
1answer
58 views

Difficulty to compute an integral

Have somebody ideas to evaluate the following integral ? $$J_n=\int_{-\infty}^{+\infty} \left(\frac{\pi^2}{4}-\arctan(x)^2\right)^n\,dx$$ I'm trying this because I have shown that the empiric ...
4
votes
1answer
2k views

Normalization for Chi square test

The formula for the Chi-Square test statistic is the following: $\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}$ where O - is observed data, and E - is expected. I'm curious why it depends on ...
4
votes
1answer
90 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
4
votes
1answer
120 views

Can a class test scores with a bimodal distribution provide statistical evidence for cheating?

I know the normal distribution can represent many things in nature. Most items are normally distributed. I recently watched a video of a professor who claims that biomodal distributions provide ...
4
votes
1answer
60 views

Same Expected Value but different variances. Is $E[U(X)] \ge E[U(Y)]$?

Let $U: \mathbb R -> \mathbb R$ be a concave function, and let $X$ be a random variable with a normal distribution, expected value $\mu$, and standard deviation $\sigma$. Let $\lambda \gt 1$, and ...
4
votes
1answer
123 views

How to use Laplace method to get the asymptotic expansion of multiple integral

I meet difficulty when I try to get the asymptotic behaviour of multiple integral as x tends to plus infinity. And $-1<$p$<1$ $$\int_x^{+\infty}\int_x^{+\infty}e^{-{\frac{1}{2\sigma^2(1-p^2)}\ \ ...
4
votes
1answer
186 views

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $\sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for an ...
4
votes
1answer
140 views

Philosophy of Statistics (Likelihood Function)

Last week during statistics class, my professor asked us a few basic questions about statistics. We could answer most of them except these three questions that we could not provide him good answers. ...
4
votes
2answers
532 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $$f(x;\theta) = \begin{cases} 1 & \text{if } \theta-1/2<x< \theta+1/2 \\ 0 & \text{otherwise} \end{cases}$$ Let $Y_1=\min \lbrace X_1,\ldots,X_n ...
4
votes
2answers
81 views

I have a bunch of sets. Some sets contain bad values. I know which sets have them, but not which values are bad.

My company sends email on behalf of many other companies. Hotmail tells us when we start sending spammy messages, but they only say "some of the emails this giant batch of messages had spammy stuff", ...
4
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0answers
96 views

Divergence based robust inference

The term 'divergence' means a function $D$ which takes two probability distributions $g,f$ as input and puts out a non-negative real number $D(g,f)$. I have learnt that the inference based on ...
4
votes
0answers
32 views

Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
4
votes
0answers
79 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
votes
2answers
8k views

Is it possible to calculate the mean and standard deviation from a median and quartiles?

Any advice would helpful. I understand that the reporting of median and quartiles for small samples is an indication of skewed data. If such is correct, then is it useless to try to work out the mean ...
3
votes
2answers
133 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
3
votes
2answers
106 views

Why is there a difference between a population variance and a sample variance

Sorry if this answer is simple but I was wondering why is there a difference between a population variance and a sample variance? I understand The variance is calculated as: $$\text{Var} = ...
3
votes
1answer
41 views

Likelihood function for a distribution with both discrete and continuous components

Suppose $X_1, X_2, \ldots, X_n$ are $IID$ normal RVs with mean $\mu$ and variance $1$. However, we observe only $Y_i$'s where $Y_i = \max (0, X_i)$. I would like to know how to write likelihood ...
3
votes
2answers
43 views

MSE of an estimator as sum of bias and variance

I am reading that how the MSE of an estimator $\hat{\theta}$ of $\theta$ can be expressed as $E(\hat{\theta} - \theta)^2$. Then this can be further simplified to $ (E[\hat{\theta}] - \theta)^2 + ...
3
votes
2answers
72 views

Reasoning for confidence interval

Suppose $$X_1,\dots,X_{20} \sim f_X(x;\beta)$$ where $$f_X(x;\beta) = \frac{1}{\beta} e^{-\frac{x}{\beta}},\quad x>0;\beta>0$$ It can shown that ("details omitted") $$P(0.52 \bar{X} \leq \beta ...
3
votes
2answers
81 views

Distribution of the Objective Value and the Variables in an Optimization Program

For random variables $X$ and $Y$, where $X\sim f(X;\theta)$ ($X$ is drawn from some distribution with pdf $f$ which is parametrized by $\theta$ ), $Y=g(X)$; we know that we can find the pdf of $Y$ if ...