Questions tagged [mathematical-statistics]

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

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Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional sufficient ...
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71 views

Can someone help solving the 4th order exponential equation subject to the constraint equations?

Let $$f(x)=\exp\left(\sum_{i=0}^4\theta_ix^i\right).$$ Given the following constraints: $\int_{-\infty}^\infty f(x)dx=1$ $\int_{-\infty}^\infty xf(x)dx=0$ $\int_{-\infty}^\infty x^2f(x)dx=1$ $\...
3
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1answer
170 views

Does stable distribution belong to exponential family?

According to Hougaard (1986), positive stable distribution on $\mathbb{R}^+$ belongs to exponential family, how about the case the support of stable distribution being less than zero? The purpose of ...
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275 views

MLE estimate of normal distribution

I am quoting this from Greene's econometrics book: The occasional statement that the properties of the MLE are only optimal in large samples is not true, however. It can be shown that when sampling ...
3
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1k views

Issue with the proof of PCA

I found a very nice PCA proof over here PCA_proof and I'm trying to understand it (I don't know what Langrange multipliers are so I'm trying my best). From the second page of the previous link, ...
3
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301 views

Mulitvariate normal truncated conditional expectation

There is three-variable multivariate normal distribution. Denote 3 variables with $X_1$, $X_2$, $X_3$. Let $\mu_i$ be means, and $\sigma_i^2$ variances of respective variables, and let $\Sigma$ be ...
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141 views

Draw balls from two different binomial distributions. What $r^2$ do you expect to find?

We have $N$ buckets in which we will put some balls. Before that, the buckets are split into two groups, group $A$ and group $B$. The number of balls that we will put in each bucket is drawn from a ...
3
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1answer
46 views

Finding for constant in a standard deviation problem

Let $X_1, \ldots, X_n$ be sample observations. Show that for any constant $c$, $$ (n-1)s^2\le\sum(X_i-c)^2 $$ where $s$ is the standard deviation of the observations. My professor gave this to ...
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625 views

Why does this multi-response Guassian LASSO not give a sparse solution?

I tried the glmnet package to learn multi-response Gaussian family. I have looked at the coefficients of the final model. The result is odd. All the features have ...
3
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128 views

Weibull Mixture question

Is it possible that a mixture of Weibull RVs is also Weibull distributed, and if yes, what are the necessary conditions?
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1answer
95 views
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340 views

Bounding the expectation of the difference between empirical vs generalization error

Let the (defect) difference between empirical and generalization error be: $$D[f_S] = I_S[f_S] - I[f_S]$$ where the empirical risk is: $$I_S[f_S] = \frac{1}{n}\sum^n_{i=1} V(f_S,z_i)$$ where $V(f,...
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127 views

Distribution of a binary matrix times a Bernoulli vector

Suppose we have the vector $\mathbf{Y} = (Y_{1},\ldots,Y_{n})$ where $Y_{i} \sim \textrm{Bernoulli}(p_{i})$ independently. For the applications I have in mind, $n$ will typically be several thousand, ...
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82 views

Finding the minimax estimator

Suppose $x$ comes from a Bernoulli distribution $f(x, \theta) = \theta^{x}(1-\theta)^{1-x}$ where $0 \leq \theta \leq 1$. Now suppose $y$ comes from the same distribution. An estimate for the ...
3
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1answer
283 views

ranking based on lower confidence interval

I have a database of bridge scores from a local bridge club that effectively contains for this question, three fields: name, date...
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3k views

How to calculate 'weighted' pooled/composite statistics for two given groups given group statistics

I have two sets of independent samples from the same distribution. For each, I have calculated sample weighted mean (u1, u2) sample weighted std deviation (d1, d2) sample weighted std error (e1, e2) ...
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96 views

Optimal Kernel Size for Lanczos resampling

The kernel for Lanczos resampling is defined by $$K(u) = \frac{a \sin(\pi u) \sin(\pi u/a)}{\pi^2u^2}.$$ How does one go about finding a value of $a$ to minimize the mean squared error, $\text{MSE} ...
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305 views

Variance of a difference in marginal proportions in a three-way contingency table

Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the ...
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659 views

How to prove that a t-distribution can be written as a ratio distribution?

If $X \sim N(0,1)$ and $Y \sim \chi^2(n),$ then it's "known" that $Z = X/\sqrt{Y/n}$ is $t$ distributed. Is there anywhere a proof for this? That in the end one can see the $t$ distribution?
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206 views

Proof of Theorem 7.3 in book of Probabilistic Graphical Models by Daphne Koller

I'm studying graphical models myself and reading contents about bayesian networks. When I am reading in page 371, section 8.1.4 Linear-Gaussian models, in Pattern Recognition and Machine Learning, I ...
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101 views

Is the following property for positive random variables fulfilled in general?

[I have cross-posted this from math.stackexchange: https://math.stackexchange.com/questions/476466/is-the-following-property-for-positive-random-variables-fulfilled-in-general ] Suppose we have a ...
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97 views

Maximum risk and sparse estimation

On Larry Wasserman's blog, he talks about the "Steep price of sparsity" here: http://normaldeviate.wordpress.com/2013/07/27/the-steep-price-of-sparsity/ In it, he points out that a sparse estimation ...
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127 views

Coefficient of variation to compare running time of two algorithms

I have developed two new sorting techniques using C language. I need to compare the performance of the both sorting techniques, to see which one is better than another. To do this, I used different ...
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520 views

Hypothesis test for the presence of a Gaussian signal in i.i.d additive Gaussian noise

Suppose there exists a sequence of $n$ numbers with two possible instantiations: The sequence contains all zeros; $n-1$ of the numbers are zeros, and one is a zero-mean Gaussian random variable $...
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348 views

Measuring parameter sensitivity and variability (standard-error) in k-fold cross-validation

I mainly use k-fold cross-validation for parameter tuning and model selection for prediction problems. Now, is there a standard or if not a less-known way to measure the sensitivity of the parameters ...
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108 views

Implications of lower-bounded total variation distance on hypothesis testing

Let $\{X_i\}_n$ be a sequence of $n$ random variables independently and identically drawn from either $P$ or $Q$. Thus the sequence $\{X_i\}_n$ has a product distribution, which is either $P^n$ or $Q^...
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163 views

Unbiased estimate of the semi-partial correlation

Is the sample semi-partial correlation a biased estimate of the population semi-partial correlation? If it is biased, what is an unbiased estimator of the population semi-partial correlation? Are ...
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1k views

Finding Expected Value of a discrete uniform random variable

I am stuck on a problem for my Statistical theory class. So the problem goes like this: Let X be the discrete uniform random variable, namely, X has the pmf: $f(x)=\frac{1}{\theta}, x=1,2,...,\...
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1k views

Unbiased hypothesis tests

Is there some textbook or expository account showing that the definition of "unbiased test" bears the same sort of relation to "unbiased estimator" that interval estimation generally bears to ...
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91 views

Is there any way to define a distance metric given a Hidden Markov Model?

Let's say I've gotten a HMM that describes user search strings for my e-commerce website. Let's also say that I've just received a search string from a customer that doesn't have any search results. ...
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69 views

Distribution of norms of vectors whose components have a known distribution?

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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470 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 ...
3
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260 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 $M$ ...
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710 views

Estimating Alpha parameter in a stable distribution

I am trying to estimate the alpha parameter of a supposed $\alpha$-stable distributed set of data. I have tried from the Hill estimator to more advanced fitting method, but they are or too ...
3
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75 views

Best way to find non-randomness regions in these or similar count data?

Let say I have data in a shape: [0,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,....] - so mainly zeros.... However I know how long is my 'signal' and how many counts are they. Is it possible ...
3
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1answer
188 views

Equivalent definitions of a sufficient statistic

I'm trying to understand the definition of a sufficient statistic for continuous random variables given in Introduction to Mathematical Statistics by Hogg and Craig (7th edition). Let $X_1,X_2,...,...
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31 views

The frog problem with negative steps

In this question The Frog Problem (puzzle in YouTube video) a frog has to jump forward from leaf to leaf with equal probability. It is computed that the expected number for the number of steps he has ...
2
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1answer
31 views

Regression with summary statistics

The following setting should already be familiar: Let $X$ be some space, $\mathbb R^d$ for simplicity, and let $Y\subseteq \mathbb R $. An unknown distribution $\mu $ is defined over $X\times Y$ and ...
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33 views

Intuitive explanation of scaled inverse chi-distribution

I am having a hard time understanding the scaled inverse chi-distribution. I looked for Wiki and other resources which are pretty math heavy without an intuitive ...
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27 views

Statistics books with applications in quantitative finance

I am a Pure Maths PhD. As I would like to venture myself into quantitative finance jobs, I need to pick up statistics, programming and finance. I have been reading time series analysis by Hamilton ...
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17 views

Can cross validation be applied to threshold based outlier detection model?

I have a threshold based outlier detection model. I apply PCA then calculate the distance from the centre of the features, and use the MSE to differentiate if the datapoint is a outlier. However, I ...
2
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33 views

Switching $H_0$ and $H_1$ by replacing $p$ with $1 - p$

I was reading the source code of tseries::adf.test, and it writes ...
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38 views

Gaussian process and its limitations

I once saw the following statement on Gaussian process, ...
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39 views

An elementary proof of the equivalence of measure theoretic and density expected values

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $X\colon \Omega\to\mathbb{R}$ be real-valued and measurable. Suppose there exists $f\colon \mathbb{R}\to [0,\infty]$ such that $P(X\in A)=\...
2
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58 views

Bias Variance Decomposition 2.7 in Elements of Statistical Inference

I try to derive 2.7 from the book. I expose my demonstration $E_\tau[(y_0-\hat{y}_0)^2]=E_\tau[y_0^2]-2E_{\tau}[y_{0}\hat{y_{0}}]+E_{\tau}[\hat{y_{0}}^{2}]$ $= y_{...
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25 views

High dimensional linear regression inference

I am reading through high-dimensional literature currently but I got confused. Especially about statistical inference with LASSO, anyone can clarify me the main difference among Van De Geer(2014)ON ...
2
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1answer
98 views

Effect of class imbalance on logistic regression (mathematical basis)

A number of posts, and papers, state that logistic regression (LR) is robust in the face of class imbalance. Unless the imbalance is extreme (e.g., events=0.01 or less), with adequate sample sizes ...
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44 views

Help Identifying the Name of a Theorem

Recall the following theorem: Let $X$ be a random variable with probability density function $f_x(x)$ and let $g$ be monotone over the support of $f_x$. The random variable $Y=g(X)$ has density $$...
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43 views

Showing that $(l'\Sigma l)\chi_p^2 = (l'S l)\frac{(n-1)p}{n-p} F_{p,n-p}$

Let $X_1,\cdots X_n$ be i.i.d. $N_p(\mu,\Sigma)$. I have that when $\mu$ and $\Sigma$ are unknown that the Scheffe type method gives $(1-\alpha)100$% confidence intervals for all linear combinations $...
2
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1answer
56 views

Calculating portfolio volatility from portfolio returns vs. from covariance matrix

I'm having trouble understanding the difference in calculating portfolio volatility via the portfolio returns vs. via the covariance matrix. To be more specific: I understand that on the individual ...