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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17k 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
7
votes
2answers
159 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
679 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
6k views

Maximum Likelihood Estimator for Multinomial.

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 ...
6
votes
1answer
159 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
120 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
485 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
575 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 ...
5
votes
2answers
154 views

domino's 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 ...
4
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3answers
381 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
2answers
61 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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1answer
455 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
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3answers
80 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
1answer
160 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
131 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
1answer
56 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
864 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
37 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
63 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
152 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
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1answer
125 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
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2answers
78 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
22 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 ...
3
votes
2answers
108 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
66 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
5answers
184 views

Why the natural log is there in MLE?

Why do we use natural log for MLE?
3
votes
1answer
8k 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 ...
3
votes
2answers
144 views

why is argmin $\|w\|^2$ equivalent to $\operatorname{argmax} 1/\|w\|$

I was wondering why the maximization of $1/\|w\|$ is equivalent to minimizing the squared norm of $w$. Shouldn't it be equivalent to just minimizing the norm of $w$? This is a very basic optimization ...
3
votes
2answers
735 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 ...
3
votes
2answers
72 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 ...
3
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2answers
429 views

Why does the standard deviation change from confidence intervals to hypothesis tests?

When considering two-sample data that involves a difference of proportions, both a confidence interval and a hypothesis test can be done. The standard deviation used for a difference of proportions ...
3
votes
1answer
44 views

calculate expectation of MLE

It's a question about whether $\hat{\theta _{MLE}}$ is an unbiased estimator of $\theta$. n independent pairs $(X_{1},Y_{1}), (X_{2},Y_{2}),....(X_{n},Y_{n}), n\geq 3$, where $Y_{i}=\theta ...
3
votes
1answer
37 views

Determine sample size according to some unknown distribution with given error rate and confidence

Assume $x\in\mathbb{N}$ obey some unknown distribution, and I can sequentially and independently acquire infinite samples of $x$. Now, given an error rate $\epsilon$ and confidence $1-\delta$, can I ...
3
votes
1answer
844 views

Rolling standard deviations

I am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: ...
3
votes
1answer
1k 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!
3
votes
2answers
104 views

distribution function

Suppose that $X$ and $Y$ are two random variables such that: $$E \left(\frac{a}{a+X} \right)=E\left(\frac{a}{a+Y}\right)< \infty \qquad\forall a > \pi.$$ Can we conclude that $X$, $Y$ have the ...
3
votes
1answer
49 views

When does the variance of a consistent estimator go to zero?

I came across the following statement (marked as true) in multiple-choice section of an old exam: The variance of a consistent estimator goes to zero with the growing sample size. As far as I ...
3
votes
1answer
71 views

Showing distribution has a $\chi^2$ distribution with df = n

Let $X_1,X_2,....,X_n$ denote independent identically distributed random variables such that $X_1$ has density $p_1(x;\theta)$ where $\hspace{15mm}p(x;\theta) ...
3
votes
1answer
583 views

How do I 'reverse engineer' the standard deviation?

My problem is fairly concrete and direct. My company loves to do major business decisions based on many reports available on the media. These reports relates how our products are fairing in ...
3
votes
1answer
49 views

Survey: prove 10% of population “Don't know”?

I have a simple question that I expect is a standard situation, but I can't seem to find the right answer (maybe because it's too simple). I have a survey with the following answers: (What is your ...
3
votes
0answers
34 views

Pareto distribution confidence interval

$X$ is distributed by Pareto with $$f_X (x) = \frac{\alpha k^{\alpha}}{x^ {\alpha +1}},\alpha,k>0,x>k.$$ Derive a 95% confidence interval for $k $. My friend said I gotta do this $$Pr ...
3
votes
1answer
43 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
3
votes
0answers
56 views

t-distribution and Degrees of freedom

Why t- distribution have n-1 degrees of freedom? I know that it is used when population variance is not known but what determines n-1
3
votes
0answers
52 views

showing that $E[(\hat\theta -\theta)^2] \lt Var(\bar X)=\dfrac{1}{n}$. [closed]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
3
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0answers
41 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 ...
3
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0answers
40 views

What is the appropriate statistical test to see if a quantity has been distributed differently into discrete bins?

Say I have $10^6$ balls, $3$ bins $A,B,C$, and $2$ machines $X$ and $Y$ that distribute the balls into the bins according to an internal set of rules (i.e. a probability distribution). If I run both ...
3
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0answers
40 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
3
votes
0answers
76 views

Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions ...
3
votes
0answers
131 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
3
votes
1answer
73 views

How is the “cooking” done in surveys

In my country there's an official center undertaking surveys of voting intention every 4 months. However, they provide only "direct" voting intention, and the statistics obtained are usually pretty ...