Questions tagged [mathematical-statistics]

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

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379 views

Derivation of M-step for pLSA

I was looking at section 6 of these notes and trying to understand the derivation of the M-step at the top of page 10. I understood the derivation for the model without background, but I do not ...
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2answers
215 views

Proper similarity measure for clustering

I have problems in finding a proper similarity measure for clustering. I have around 3000 arrays of sets, where each set contains features of certain domain (e.g., number, color, days, alphabets, etc)....
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1answer
479 views

Variance explained for binomial proportions

In Rosenthal and Rubin (1979) ``A Note on Percent Variance Explained as A Measure of the Importance of Effects'', they give an example of where $r^2$ is deceptively low: Suppose half the patients ...
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1answer
2k views

For linear regression, what's the distribution of error term from Classical and Bayesian point of views?

I know that linear regression is based on the assumption that the errors are normally distributed (from both bayesian and classical views). I'm just trying to verify this assumption based on the final ...
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342 views

Integral of survival function

If X is a nonnegative random variable representing the life of a component having distribution function F, and S is the survival ...
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2answers
11k views

Mean residual life

If X is a nonnegative random variable representing the life of a component having distribution function F,the mean residual life ...
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6answers
15k views

A robust (non-parametric) measure like Coefficient of Variation -- IQR/median, or alternative?

For a given set of data, spread is often calculated either as the standard deviation or as the IQR (inter-quartile range). Whereas a standard deviation is ...
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2answers
239 views

Finding expectation of log

Does somebody have an idea to find the following expectation: $$E(X\log(a+bX)),$$ How can I proceed, can we go like this: $$=E(X)E[\log(a+bX)],$$if yes then what?
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Eigenvectors of a covariance matrix with only positive elements

If all the elements of a positive-definite covariance matrix are positive, how can I prove that the coefficients [elements] of the first principal component [first eigenvector] are all of the same ...
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0answers
74 views

Distribution of norms of vectors whose components have a known distribution? [closed]

I know very little of statistics, so forgive me if the question is badly posed. Suppose I have $n$ samples $x_1,\dotsc,x_n$ drawn from some distribution $f$ (let's say $n\approx 100$). Fix $1\leq k \...
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8answers
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Are bayesians slaves of the likelihood function?

In his book "All of Statistics", Prof. Larry Wasserman presents the following Example (11.10, page 188). Suppose that we have a density $f$ such that $f(x)=c\,g(x)$, where $g$ is a known (nonnegative, ...
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2answers
117 views

Independence in a sum relationship

I have a model of total reaction time T, which is a composite of a selection time S and a discrimination time D. So a person first finds something, this takes the tS. Then he discriminates and reports ...
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4answers
57k views

Expected value of a natural logarithm

I know $E(aX+b) = aE(X)+b$ with $a,b $ constants, so given $E(X)$, it's easy to solve. I also know that you can't apply that when its a nonlinear function, like in this case $E(1/X) \neq 1/E(X)$, and ...
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1answer
103 views

Common parameters for conditional likelihood

I am trying to understand the concept of conditional likelihood in the context of logistic regression. One paper I am reading defines $L(\theta; y|x) = f(y|x; \theta)$, then goes on to point out ...
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1answer
2k views

Frequentist statistics references for someone well versed in modern probability theory

Coming from a rigorous background in analysis and modern probability theory, I find Bayesian statistics straightforward and easy to understand, and frequentist statistics incredibly confusing and ...
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3answers
1k views

Some questions about statistical randomness

From Wikipedia's statistical randoness: Global randomness and local randomness are different. Most philosophical conceptions of randomness are global—because they are based on the idea that "in ...
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1answer
330 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,...
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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 ...
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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}{...
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1answer
743 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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120 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
592 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 ...
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1answer
517 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 ...
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1answer
4k views

Showing that the order statistic $X_{(n)}$ 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
93k 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, ...
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1answer
255 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 ...
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1answer
2k 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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7answers
849k 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
20k 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
18k 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 ...
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1answer
205 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
468 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 ...
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1answer
21k 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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4answers
17k 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
352 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 ...
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1answer
383 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 $...
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1answer
2k views

Is possible use HoG features to train a Neural Network?

I was wondering if it is possible use HoG features to train a Neural Network, I know that in the original paper by Dalal and Triggs they used the data generated to train a SVM. If not is possible or ...
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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
191 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 ...
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3answers
5k views

Is it true that in high dimensions, data is easier to separate linearly?

I have often seen the statement that linear separability is more easily achieved in high dimensions, but I don't see why. Is it an empirical fact? An heuristic? Plain nonsense?
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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
505 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 ...
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2answers
641 views

Does the principle of indifference apply to the Borel-Kolmogorov paradox?

Consider Jaynes' solution to the Bertrand paradox using the principle of indifference. Why doesn't a similar argument apply to the Borel-Kolmogorov paradox? Is there something wrong with arguing that ...
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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/...

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