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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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
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98 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 ...
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311 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
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471 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 ...
4
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200 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 ...
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45 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 ...
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125 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
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79 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. ...
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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", ...
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4k 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
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153 views

Why the natural log is there in MLE?

Why do we use natural log for MLE?
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75 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 ...
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225 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 ...
3
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2answers
62 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
290 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
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2k 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 ...
3
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102 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
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82 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 ...
3
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168 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
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1answer
46 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
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0answers
59 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
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1answer
66 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 ...
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62 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
3
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1answer
131 views

finding the maximum likelihood estimator: conditional generalised linear model

Find the maximum likelihood estimator of $\alpha$,$\beta$ and $\lambda$ given the model ...
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63 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? ...
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120 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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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 ...
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2answers
55 views

Does a proportion have to be a rational number?

Does a proportion have to be a rational number? For example, Assume we have a square with side $2$ units. We are throwing a circle of radius $1$ unit over the square. Let $X$ be the area of the ...
2
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2answers
65 views

Distribution of the sample variance

This is my first post to this great website :) It seems like an excellent place to learn. I have a question however that is bothering me as I cannot figure it out through my textbook. The sample ...
2
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1answer
121 views

Basu's Theorem's application

I can't solve a problem, for which I am suppose to use Basu's theorem. Suppose that $X$ and $Y$ are independent Exponential random variables with common parameter $\lambda$. I have to show that $X + ...
2
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2answers
172 views

Test for Bernoulli distribution

I'm thinking about the following problem. Assume that one is given a sequence of binary data $(x_i)_i\in\{0,1\}^n$ and that wants to determine whether or not this sequence has been generated by a ...
2
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2answers
169 views

expectation value 3

Suppose that $X$ is a non-negative random variable and there exist constants $A,B$ such that $$\forall t > 0\colon P(X>\frac{1}{t})<Bt $$ and $$\forall t > 0\colon E(\sin(tX))<At$$ I ...
2
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1answer
52 views

Expectation of the MLE $e^{-\frac{1}{\overline{X}}}$

I am having a bit of a problem with examining the properties of a maximum likelihood estimator. I feel like I am missing something simple, but I have been unable to find someone doing an example quite ...
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102 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
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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 ...
2
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4answers
49 views

The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
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2answers
81 views

connection between PCA and linear regression

Is there a formal link between linear regression and PCA? The goal of PCA is to decompose a matrix into a linear combination of variables that contain most of the information in the matrix. Suppose ...
2
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1answer
68 views

Differentiation for least squares method?

Is there any reason that we use mathematical differentiation of least squares method for regression analysis? The theory say we use differentiation supposing the sum of errors is 0. I I don't really ...
2
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2answers
43 views

Show that as $d$ goes to $\infty$, a standardized version of $X$ has the STD Normal Dist

I am currently stuck on this problem and I would greatly appreciate some help. The problem is as follows: Let $X$ have a chi-square with $d$ degrees of freedom. Show that a standardized version of ...
2
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1answer
141 views

Standardizing Sum of Uniform Distributions

Let X be a Unif(−6,6) variable, that is, X is Uniformly distributed over the interval (−6,6). Let $X_1,X_2,...,X_n$ be independent Unif(−6,6) variables. Let $Y=\sum_{i=1}^nX_i$ Find the mgf $M_Y(t)$ ...
2
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1answer
203 views

Maximum Likelihood Estimator

Let the random variable $X$ have a uniform density given by $$ f(x;\mu,\sigma)=\frac{1}{2\sqrt 3\sigma}I_{[\mu-\sqrt 3\sigma,\mu+\sqrt 3\sigma]}(x) $$ where $-\infty\lt\mu\lt\infty$ and $\sigma\gt ...
2
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1answer
102 views

Fun with Powerball history. What probability theories apply?

I wanted to play around with probability theories and see if I find any statistical inferences from the Powerball lottery history going back to 11/1/1997. What theories could I apply besides simple ...
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2answers
68 views

Show that $(1-\frac{1}{n})^{\sum_{i=1}^{n} X_i}$ is an unbiased estimator of $\tau(\theta)=e^{-\lambda}$

Let $S=\sum_{i=1}^{n} X_i$ than show that $T(X_1,\cdots,X_n)=(1-\frac{1}{n})^{\sum_{i=1}^{n} X_i}$ is an unbiased estimator of $\tau(\theta)=e^{-\lambda}$ where $X_1,\cdots,X_n$ are IID from ...
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2answers
42 views

Showing independence of random variables

When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are ...
2
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1answer
37 views

Let $\{x_1, \ldots, x_n \}$ be a sample. Why can one subtract $L$ from every sample value $x_i$ to get equivalent results?

Suppose we have a sample $\{x_1, \ldots ,x_n\}$ obtained from i.i.d RV's $X_i\sim N$ (normal). Then in books on statistics, depending on the values obtained, they subtract a fixed quantity $L$ from ...
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92 views

Rolling standard deviations

I am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: ...
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1answer
124 views

Finding the solution of Sufficient Statistics for this function

Suppose $X_1,X_2,\ldots,X_n$ be a random sample of distribution with probability density function $$f(x\mid\theta) = \theta x^{\theta-1},\quad 0\lt x \lt 1,\quad 0\lt \theta \lt \infty$$ how can i ...
2
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2answers
241 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 ...
2
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1answer
61 views

Monte-Carlo simulation

Let $X:=X_1,..,X_n$ iid $\sim P$-distributed where $P$ is a known distribution and $T_n(X_1,..,X_n)$ is a test-statistic with unknown distribution $P(T_n(X)\leq t)$. Furthermore I am interested in the ...
2
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1answer
221 views

Proof that $\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}$ are independent

Let $X_i\sim N(\mu,\sigma^2)$ ; where$[i=1,2,\ldots,n]$ $Z_i\sim N(0,1)$ ; where$[i=1,2,\ldots,n]$ Proof that $\bar Z=\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^{n}(Z_i-\bar ...