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

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How to make sense of the EM algorithm expressed in terms of Kullback-Leibler divergence?

In the textbook by Wasserman, All of Statistics, the Algorithm is expressed as: 1) Pick a starting value $\theta^0$. 2) (E-Step). Calculuate: $$ J(\theta|\theta^j) = E_{\theta^j} \left(log ...
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14 views

For a Fisher Information matrix $I(\theta)$ of multiple variables, is it true that $I(\theta) = nI_1(\theta)$?

For a Fisher Information matrix $I(\theta)$ of multiple variables, is it true that $I(\theta) = nI_1(\theta)$? That is, if $\theta = (\theta_1, \ldots, \theta_k)$, will it be the case that the fisher ...
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22 views

I need help with my statistics final [on hold]

i need to know what type of test to use for the following questions 1) Does Mark’s yards relate to the temperature during the start of the game? 2) Does Mark’s year in college impact the amount ...
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8 views

How can we find the Fisher Information Matrix of a linear regression model?

Suppose we have a sequence of random variables $y_1, \ldots, y_n$ such that $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$ where $\epsilon_i$ is assumed to be i.i.d. $N(0,\sigma^2)$, with $\sigma^2$ ...
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7 views

Why is the Fisher Information sometimes written as a conditional expectation?

I was looking at the Fisher Information entry on Wikipedia just now, and saw that the Fisher Information Matrix was written as: $$ \mathcal{I}(\theta) = - \operatorname{E} \left[\left. ...
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19 views

good practice for cluster analysis

I want to find out what are the best practice in conducting a reliable cluster analysis: 1. outliers: Is it necessary or not to remove the outliers in the variables to be used for cluster analysis? ...
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10 views

EM algorithm increase after E step?

It might be a silly question, but here it goes. The short version of my question is whether the marginal likelihood calculated after every E steps should be increasing or not. More details: Using ...
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13 views

How to compare two test according to power?

I know the expectation calculates the power of the test, but how can i find a test that is as good as the given one based on the power?
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12 views

Mahalanobis distance for highly multivariate random variable

I have to compute the Mahalanobis distance for a $10^6$ dimensional multivariate random variable. What is the best (and fastest) way to do this? I am currently taking cholesky decomposition of the ...
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1answer
30 views

In a mixed effects model, how do you determine when the slope and intercept should be independent?

This is a question regarding the theory underlying mixed effects models, specifically a general rule of thumb that can be used to determine the structure of random-effect portion. Here's what I ...
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1answer
60 views

If X~Exp(λ), what is the expected value of Y=X²?

I am trying to compute this using the integral definition of expected value but I don't think I am doing it right as I am getting a very hard integral that I can not solve. When computing ...
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18 views

Converting math classifier-combination equation to code [on hold]

In this paper, section 2.1, they provide an approach to gain new reasoned probabilities from a set of classifier results. I can understand the concepts but am having difficultly completing the ...
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30 views

STATANSWER PLEASE AND THANK YOU [closed]

The accompanying table shows results of challenged referee calls in a major tennis tournament. Use a 0.05 significance level to test the claim that the gender of the tennis player is independent of ...
3
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28 views

Is M-estimation valid only for regression models?

Is M-estimation valid only for regression models or does it's working hold good for robust estimation of parameters in other statistical models? I understand that M-estimators are asymptotically ...
4
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48 views

Is unbiased maximum likelihood estimator always the best unbiased estimator?

I know for regular problems, if we have a best regular unbiased estimator, it must be the maximum likelihood estimator (MLE). But generally, if we have an unbiased MLE, would it also be the best ...
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4 views

finding and comparing temporal and location-specific pass patterns in a soccer game

I have data on several soccer matches, where ball passes have been recorded in terms of XY-location where the pass starts, XY location where the pass ends, the team making the pass, the player making ...
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19 views

What's the definition of multivariate mode?

In the case of grouped data where a frequency curve have been constructed to fit the data, the mode will be the value (or values) of $x$ corresponding to the maximum point (or points) on the curve. ...
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26 views

Finding the distribution and covariance matrix of a distribution

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 ...
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10 views

Variance estimation for Levy process

Let $(X_t)$ be a Levy process. It then holds that $$ E(X_{t+\Delta} - X_t) = \Delta \nu, \\ V(X_{t+\Delta} - X_t) = \Delta \mu, $$ under sufficient regularity conditions in terms of moments. For ...
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22 views

Confounding variable vs lurking variable

I realize that there there are countless examples of this on the web however I am still have some difficulty with it. I was just hoping to get some feedback on my interpretation. In a lurking ...
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22 views

quantile of standardized t distribution

How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is ...
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1answer
19 views

Can feature with high positive correlation have opposite weights sign? [duplicate]

I have two features for a binary classification problem which are highly positively correlated. (0.79) But when I build a logistic regression classification then I see their weights are opposite in ...
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30 views

show asymtotic normality

Let $x=(x_1,x_2,...,x_n)$ be a sample from a multivariate normal distribution, with mean vector $\mathbf{\mu}$ (n by 1 column vector, all elements equal to $\mu$) and covariance matrix $\Sigma$ ...
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2answers
28 views

Distribution of sum of squared normals, scalar form

I am working on a problem and hit a wall. I don't need the whole problem answered. just this part. $X_1,...X_n \sim N(\theta, \theta^2)$, what is the distribution of ${\sum_1^nX_i^2}/n$? It's ...
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2answers
94 views

Probability of a claim being rejected or accepted using binomial distribution

I am self-learning statistics. I was going through Probability and Statistics For Engineering and the Sciences by Jay L. Devore and stumbled upon the following example: An electronics manufacturer ...
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14 views

Comparing algorithms: runtime vs accuracy

I am comparing several algorithms with respect to their runtime and accuracy on several datasets. I follow [1] and compute the average rank using the accuracy and apply the Friedman and Nemenyi tests. ...
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7 views

How do I determining the exact percentile among a sample given the 10, 25, 50, and 90 percentiles

Background: I'm trying to provided employees or decision makers with comparative wages analysis based on national or regional wage estimates. I'm developing a tool to provide employees and clients an ...
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33 views

How to properly show the efficiency of a process?

I'm no statistician but my background is in computer science. At work, we are trying to improve the efficiency of a system where 5 people (A-E) each produce one part of a report and send it to 2 key ...
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20 views

Likelihood ratio test, given distribution

I am trying to find the implementable form for a likelihood ratio test and ran into a problem. Could anyone look at it an give me a hint? I'm stuck. Here's the problem... $X1,...Xn$ are distributed ...
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2answers
193 views

Convergence in distribution of the following sequence of random variables

$X_n\sim Beta\left(\frac{\alpha}{n},\frac{\beta}{n}\right)$ with $\alpha>0$ and $\beta>0$. Does $X_n$ converges to a distribution? How do I approach to show that this converges to a ...
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0answers
10 views

Topics needed to learn to for predicting future of a data set?

I am planning to work this summer by writing a code which will eventually become an app, that has the ability to predict what the prices will be atleast 6 month ahead of time. I have a data set of the ...
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29 views

Confidence Interval using likelihood-ratio statistic

The lifetime $T$ of a light bulb follows a distribution with probability density function $$f(t;\theta) = 0.5\theta^3t^2e^{-{\theta}t}$$ If $n = 20$, $\sum_{i=1}^{20} t_i = 996$ Determine the 95% ...
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1answer
26 views

Weight or impact of variable in correlation

Consider I've have two curves (catgeory1,category2) that are strongly negatively correlated( -0.88). for 1, I've few sub-category (o,p,q,r,s,t,u,v,w,x,y,z). I want to know the weight of each one in ...
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26 views

Machine Learning Algorithms For Poor Quality Statistics

I'm looking for references on any work done in the area of machine learning for automated detection of poor quality papers that use statistics (for example p-value hacking, misunderstanding confidence ...
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12 views

regarding generating synthetic data simulating the real data

We are trying to develop some predictive models. The current scenario is that we have to rely on synthetic data at first since the real data set will not be available quite soon. It is understandable ...
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14 views

Weighted Box Plot

I have a summarized data in r that looks like the following: ...
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1answer
25 views

Cronbach's alpha vs. Average Item-Sum correlation?

I have heard that Cronbach's alpha is used to measure reliability of a test (to check if a set of items/question are measuring the same construct). How does the Cronbach's alpha compare with the ...
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16 views

Statistical analysis of the results calculated from a stochastic global optimization

I recently use the Basinhopping algorithm [1], a Monte Carlo Markov Chain variant, to find the global minimum of numerical functions (which are not necessarily smooth). Though the global optimization ...
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9 views

The probability distribution of rth order statistic after transformation

If I have a sample from a Rayleigh distribution, then I transform this sample to a sample from Gamma distribution by using the fact that the summation of the square of a Rayleigh variable is a Gamma ...
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33 views

Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that $$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$ I am interested in a metric that ...
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1answer
27 views

Ratio of two gamma distributions

Assume that $X \sim Ga(\alpha_1, \beta_1)$ and $Y \sim Ga(\alpha_2, \beta_2)$. Define $Z= X/Y$. What 's the distribution of $Z$?
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1answer
52 views

Identification from minimum value of truncated distribution

Suppose that a given population is endowed with a pair of characteristics $T$ and $K$. Let's think of these characteristics as random variables $$(T,K) \sim \operatorname{BiNormal}((\mu_T, \mu_K), ...
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21 views

Integrating to get the posterior distribution for the precision

Our model is the following: $x_i|\mu,\tau \sim N(\mu,\tau)$, where tau is the precision. $\mu|\tau \sim N(\mu_0,n_0\tau)$, $\tau\sim Ga(\alpha,\beta)$. In the picture below, how is the ...
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1answer
25 views

Expectation of $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$?

What is the expectation: $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$ when $X_{i,j} \sim N(\mu, \sigma^2)$ and $X \in \mathbb{R}^{n \times k}$ where $n>k$ and $A$ is a given p.s.d matrix (not ...
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12 views

How to introduce a weighted percentage?

We wanted to know the knowledge of people about how to eat fruits. For this we provided a party and invited 1000 people. There were 3 kind of fruits on the tables. Banana, orange and melon. If ...
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20 views

Should I estimate a covariance mattix or a precision matrix (inverse covariance matrix)?

More specifically, what are some pros and cons of estimating a precision matrix over a covariance matrix?
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35 views

Are MVUEs and MLEs always functions of a minimal sufficient statistic?

Is it the case that both minimum variance unbiased estimators (MVUEs) and maximum likelihood estimators (MLEs) are always functions of a minimal sufficient statistic? If so, how do we know? If not, ...
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1answer
30 views

Simple question on exchangeability

We say that $(x_1,x_2,\dots)$is an infinitely exchageable sequence of random variables iff for any permutation $\pi$, $p(x_1,\dots,x_n)=p(x_{\pi(1)},\dots,x_{\pi(n)})$. Let $(x_1,x_2,\dots)$ be an ...
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41 views

Novice question - Interpreting Linear Regression Analysis

I'm looking for a very simple explanation for a project I'm working on. I'm comparing unemployment data and median income based on level of education (Bachelors, Masters, College, Trade Certificate, ...
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117 views

How to graph throughput

UPDATE 2: maybe a simple way to state it: for web application performance data consisting on: start timestamp | end timestamp | response time I'd like to compute a non-aggregated data set consisting ...