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Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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1answer
225 views

Eighth order moment

I read Nonlinear Dimensionality Reduction by Lee and Verleysen [Google Books] and came across the following theorem (p. 8): Let $\mathbf{y}$ be a $D$-dimensional vector $[y_1, \ldots, y_d, \ldots,...
7
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3answers
806 views

Sufficiency in Lehmann Scheffe

We are wondering what sufficiency in the Lehmann Scheffe Theorem is needed for. Our reasoning was: If an unbiased estimator is uncorrelated with all unbiased estimators of 0, it is UMVUE If the ...
7
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1answer
5k views

In calculating the F-measure with precision and recall, why is the harmonic mean used?

The article for F-measure in Wikipedia says: The traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall: $F_1=2\times\frac{precision \times recall}{...
4
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1answer
688 views

With simple random sampling, how to approximate variance of R=avg(Y)/avg(X)?

Recenly I am reading "Mathematical statistics and data analysis" written by Rice myself. At page 207, theorem A said: With simple random sampling, approimxate variance of $R=\frac{\overline{Y}}{\...
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0answers
118 views

Comparing two opposite ranked sets of data

Looking for the best formula to compare two sets of numbers Set #1 - Rank: Smaller numbers indicate higher "rank" (Ex: 1 is the best, 1,000,000 is the worst) Set #2 - Number of Items: The more items ...
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2answers
453 views

Why is the amount of variance explained by my 1st PC so close to the average pairwise correlation?

What is the relationship between the first principal component(s) and the average correlation in the correlation matrix? For example, in an empirical application I observe that the average ...
1
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1answer
444 views

Simple regression proof using general formula

I want to derive the least square estimation of the coefficients for $y=\beta_0+\beta_1*x_1+\varepsilon$ using the general formula. Can someone walk me through how to get from B to D in the image ...
3
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1answer
3k views

How to show order statistic is sufficient

I have some trouble showing sufficiency for largest order statistic ${x}_{n}$. This is from Casella's text, problem 1.6.3. Let ${p}_{\theta}$ be a density function. ${p}_{\theta}(x)=c({\theta})f(x)$ ...
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10answers
63k views

Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
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1answer
2k views

Estimating misclassification rate from summary classification values

I'm trying to work out the misclassification rate in a published study, where the results do not include this value. However, I have the sensitivity, ...
2
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1answer
233 views

Representation within a RKHS framework

Given a p.s.d kernel $Q$, can minimization/maximization of $Tr(X^TQX)$ over X be represented within a reproducing kernel Hilbert space (RKHS) framework? If there is a primary concern with the trace ...
0
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1answer
1k views

What test will tell me a normalized percentage of the data?

I'm working with baseball statistics and I have a list of the players' batting percentages. I want a formula that will tell me what percentage of the data is represented from a player's score ...
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6answers
656k views

What's the difference between variance and standard deviation?

I was wondering what the difference between the variance and the standard deviation is. If you calculate the two values, it is clear that you get the standard deviation out of the variance, but what ...
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2answers
15k views

What are the degrees of freedom of a distribution?

I am dealing right now with a lot of distributions, e.g., $F$, $t$, $\chi^2$. I was wondering why do these degrees of freedom signify for distributions such as the $F(m,n)$ distribution?
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3answers
11k views

Proof that moment generating functions uniquely determine probability distributions

Wackerly et al's text states this theorem "Let $m_x(t)$ and $m_y(t)$ denote the moment-generating functions of random variables X and Y, respectively. If both moment-generating functions exist and $...
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1answer
60 views

Repeating questions based on giving correct answers

I am making a web-app that is supposed to help people memorize Japanese kana symbols using flashcard. How it works at the moment is that on each new flashcard, user is presented with one symbol which ...
6
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1answer
197 views

Laplacian-Beltrami approximation based on an empirical sample

Given a probability measure $\nu$ on a subset $M \subseteq \mathbb{R}^N$ we construct the corresponding operator $$L^tf(x)=f(x)\int_{M} e^{-\frac{||x-y||^2}{4t}}d\nu(y)-\int_{M}f(y)e^{-\frac{||x-y||^...
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3answers
445 views

Mathematics base for data mining and artificial intelligence algorithms

Could you give me some clarification about data mining and artificial intelligence algorithms? What mathematics base they used for? Could you give me starting point, in mathematics, to understand ...
13
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1answer
14k views

Expected value and variance of trace function

For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^...
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3answers
12k views

What is so cool about de Finetti's representation theorem?

From Theory of Statistics by Mark J. Schervish (page 12): Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their ...
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2answers
3k views

Equivalent events

Define the term "equivalent events". If $M$ is the event that the number rolled from a die is a prime number, which event can be equivalent to $M$?
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2answers
1k views

need help understanding Dirichlet (coursera's PGM class week 7 - Bayesian prediction)

I'm trying to work through Coursera's probabilistic graphical models class (week 7: Baeysian prediction) and a have several questions. In the Dirichlet distribution, I'm having difficulty trying to ...
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0answers
282 views

Derivation of prediction intervals for a normally distributed population with unknown population standard deviation

I have via the ISO standard 16269 found the solution to a problem that I've been working on. Based on a couple of independent samples from a normally distributed population, I would like to determine ...
2
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1answer
343 views

Joint distribution of mid p-value and p-value

I have a question about the joint distribution of the mid p-value and p-value. We know that, for right tailed test with discrete test statistic $X$ with distribution $F$, the p-value is defined as $...
4
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2answers
2k views

Find residual sum of squares

Let $Y_1, Y_2,Y_3$ be uncorrelated random variables with common variance $\sigma^2>0$ such that $$ \begin{aligned} E[Y_1]&=\beta_1+\beta_2, \\ E[Y_2]&=2\beta_1 \\ E[Y_3]&=\beta_1-\...
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2answers
182 views

Distribution of a centered standardized sample

Foreword : this is not homework, but a real problem : in a Bayesian model comparison context, I am trying to work out the correct prior density of mixed-model parameters which are, for computational ...
13
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3answers
1k views

Why does this excerpt say that unbiased estimation of standard deviation usually isn't relevant?

I was reading on the computation of the unbiased estimation of standard deviation and the source I read stated (...) except in some important situations, the task has little relevance to ...
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0answers
442 views

Error propagation from fit parameters

I have two distinct data samples($A$ and $B$), and to each one a gaussian is fitted. I then evaluate the product $S = \sigma_A * \sigma_B$ ($\sigma_A$ and $\sigma_B$ and their errors are obtained ...
28
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3answers
2k views

Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s

I'm working with two independent normal distributions $X$ and $Y$, with means $\mu_x$ and $\mu_y$ and variances $\sigma^2_x$ and $\sigma^2_y$. I'm interested in the distribution of their ratio $Z=X/...
6
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2answers
1k views

What are U-type statistics?

In an article, I recently came across the mention of first and second order U-type statistics without further detail. Does anyone know what U-type statistics are? References will be highly ...
3
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0answers
238 views

Proving the convergence of KDE algorithms when the samples are non-i.i.d

I am currently working on convergence proof for a new method for non-parametric importance sampling, and I need some help... My method uses an MCMC algorithm to generate a set of dependent $M$ ...
0
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0answers
645 views

Variance covariance matrix for the inverse of a matrix

Suppose we have a matrix $\mathbf{A}=\begin{bmatrix}a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$ and know its variance covariance $\left(4\times4\right)$ matrix. Then how the variance ...
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0answers
1k views

Taylor Series and Multivariate Delta Method

I asked this question on https://math.stackexchange.com/ but did not get any answer. Sorry for cross posting. I'm trying to understand delta method for matrices and vectors to find the variance-...
4
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2answers
2k views

Probability of two values being equal in a sample drawn from a continuous distribution

I am reading about the Kolmogrov-Smirnov tests from the book Probability and Statistics by DeGroot and Schervish. In the initial few lines on this topic, the authors state the following:- Suppose ...
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1answer
748 views

Derivation of equations in kriging

I have some confusion regarding some derivations of the equations for kriging in the wiki article. $\newcommand{\Var}{\rm Var}$ It says that kriging error is given by: \begin{align} \Sigma_k^2(x_0) &...
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2answers
3k views

Which distributions have closed-form solutions for maximum likelihood estimation?

Which distributions have closed-form solutions for the maximum likelihood estimates of the parameters from a sample of independent observations?
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4answers
21k views

What's the difference between mathematical statistics and statistics?

What's the difference of mathematical statistics and statistics? I've read this: Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all ...
4
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1answer
177 views

Likelihood for Poisson data

In my book, it says: Independent random variables $X_1, X_2, \dots, X_n$ are modeled by a Poisson distribution with mean $\lambda > 0$. The likelihood for $\lambda$ based on data $\mathbf{x}...
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2answers
724 views

What is the purpose of dividing covariance by multiple of S.D. of X and S.D. of Y for computation of sample correlation by K.P. formula? [closed]

The r formula divides the covariance by S.D. of X and S.D. of Y. Does it help in estimating a correct estimate of the sample correlation coefficient ?
1
vote
1answer
192 views

Sum of uniform distributions

Let’s say $X_1$ ~ $uni(0,1)$ $X_2$ ~ $uni(0,1)$ $X_3$ ~ $uni(0,1)$ And $Y=0.1X_1+0.3X_2+0.6X_3$ What’s the $F(Y)$ (i.e., CDF)?
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1answer
775 views

Finding a comparable Control group for a treatment group?

I have a treatment group of size 30 (30 schools in California) that used a math supplemental software. In a simple analysis, I'd like to compare students' average Math growth between our treatment ...
3
votes
1answer
228 views

Under what conditions can a PLS regression model be expressed by single linear equation?

I am confused by two, yet inconsistent for me, facts: Since the PLS regression is expressed by matrices of scores and loadings as $$X=TP^T+E\\Y=UQ^T+F$$ how it can be translated into linear equation ...
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0answers
554 views

What are the characteristics for a problem to be statistical?

Today, I read a post Computing and Sustainability: What Can Be Done?. And I found that the author of this post can easily find statistical problems in other fields, such as computer science. Since as ...
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0answers
186 views

Rao-Cramér inequality [closed]

Suppose $p({\bf x},\theta), \theta \in {\bf \Theta} \subset {\bf \mathbb R^k}$ is a (regular) family of distributions. The score function is defined as $s({\bf x},\theta)=\frac{\partial}{\partial \...
4
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4answers
280 views

Is there any problem which can be solved only by fuzzy theory but not by statistics?

When statistics is solving all the problems, why does one study the fuzzy theory? Is there any problem which can be solved only by fuzzy theory but not by statistics? Please kindly answer with ...
10
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3answers
682 views

Information theoretic central limit theorem

The simplest form of the information theoretic CLT is the following: Let $X_1, X_2,\dots$ be iid with mean $0$ and variance $1$. Let $f_n$ be the density of the normalized sum $\frac{\sum_{i=1}^n X_i}...
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2answers
2k views

Does efficiency imply unbiased and consistency?

If I can prove that for an estimator $\hat{k}( \theta)$ I can write: $$\frac{\partial l(X_1, \dots , X_n)}{\partial \theta} = a(n, \theta)(\hat{\theta} - \theta)$$ Am i sure that the estimator is ...
3
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1answer
125 views

About sample mean and the mean of various samples

What I would like to know it is the precise use of both statistics -- the sample mean and the mean of various samples. I'm getting a bit confused with the notation in my textbook and sometimes I ...