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

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Spatial autoregressive model implementation in R

I need to implement a SAR model with no covariates. To be more specific, the regression I have to estimate is y=bWy+e where: y is the dependent variable; b is the coefficient parameter to be ...
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0answers
12 views

Cross Validation and Logistic Regression

Cross Validation-validation, Over Fitting over-fitting, Logistic Regression and logistic regression (restrictedI am restricted to using SAS only) I have a data set of 60 000 records (binary ...
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1answer
71 views

Mechanics behind deviation from random distribution

The system we are working on is biological, more specifically the distribution of programmed DNA damage events across a chromosome. This can be thought of as 1D array (the chromosome) across which ...
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10 views

Statistically test difference based on time series

Usually in statistical hypothesis testing, we randomly split some unit into treatment group and control group, and we test if there are difference between treatment group and control group based on a ...
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24 views

Cross-validation, over-fitting, and logistic regression

(I am restricted to using SAS only) I have a data set of 60 000 records (binary response, roughly 90 continuous predictors ... the data set is full of legitimate zeros, so I am analyzing the ...
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1answer
48 views

General formula for finding covariance of monomials of multivariate random variables

Suppose that we have independent random variables $X_1,X_2,X_3$ which are gaussian multivariate distributed with a mean of zero vector and a diagonal covariance matrix. $X=[X_1,X_2,X_3] \tilde{} N(0, ...
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1answer
51 views

Applying the T-test [on hold]

So far I have calculated the sample means $\overline{A} = 0.75$ and $\overline{B} = 2.33$. Using these I computed the sample variances: $S_A^2 \approx \frac{28.805}{9} = 3.2005$ and $S_B^2 = ...
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1answer
25 views

A question on notation in the least squares method

I am reading some nice lecture notes on basic statistics, learning some basic topics on estimation of parameters. Reading the chapter about the method of least squares estimation I meet the following ...
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18 views

How to compute the average of the relative weightings?

I have the following problem. I have a sample of daily snapshots of a portfolio grouped by four asset classes: Forex, Futures, Options and Equities. For each day in my sample I compute the weight of ...
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1answer
49 views

Variance stabilisation

$Y$ has mean $\mu$ and variance function $V(\mu)$. If $V(\mu) = \alpha.\mu^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that $Var(h(Y))$ is approximately constant. I tried to ...
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0answers
11 views

Reducing model weight increases accuracy

I have a question. Say i have 3 class labels 1,2,3 Basically, i have about 1 million examples of 1, 1000 examples of 2 and 3 each. Hence, the weight of my label 1 would be 0.001, 2 would be 1 and ...
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21 views

Bound on the total change using Pearson's r

I am given an increasing series $(x_1,....x_n)$ and I know the pearson correlation between $(x_1,....x_n)$ and some (unknown) increasing series $(y_1,....y_n)$. Can I derive an upper and a lower ...
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22 views

Is there any alternative to the EM algorithm?

I am working on biomedical signal analysis and the most used method for parameters estimation is the EM algorithm. My question is : what are the most powerful alternatives to this algorithm?
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0answers
10 views

Creating a decay function to time series data

Here is the challenge. I am measuring performance of an entity each day going back a year. The performance metric (score) is a number between 0 and 1. So it looks something like this, where day is ...
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0answers
9 views

binomial error calculated with formula

i have uploaded a 2D histogram ,, actually it is filled by dividing two 2Dhistograms with same bins.i want to calculate manually each error which is shown with each value i.e binContent.Please ...
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1answer
48 views

Comparing failure rates between 2 products with different number of deployed equipment

I have the following scenario, which I am trying to better understand. There are 2 different brands or groups of devices performing the same functionality as such: - Product/Brand 1 has 4,323 devices ...
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1answer
26 views

Biased and Efficient estimators

Is unbiasedness a necessary condition for an estimator to be efficient? For example, if $\hat {\theta}= \frac{\sum_i^n X_i}{3}$, I assume $\hat {\theta}$ can't be efficient in a Cramer-Rao lower ...
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0answers
66 views

Error in linear regression

Given two series $(x_1,...x_n)$ and $(y_1,...y_n)$, and assume that we know $x_{n+1}$. Given the fact that the pearson correlation won't change in the next observation of $y_{n+1}$, can we bound the ...
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1answer
61 views

What is the value of $\mu_0$ and $\kappa_0 $in $N(\mu_0,\sigma / \kappa_0)$?

In a Bayesian analysis I want to sample $\sigma \sim \text{inverse-Wishart}(\nu_0,M)$ where $\nu_0$ is the degrees of freedom, equal to dimension+1 and $M$ is a scalar matrix Then I will sample the ...
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0answers
24 views

What does improper learning mean in the context of statistical learning theory and machine learning?

I was reading the following paper and it talked about improper learning. I wasn't 100% what it rigorously meant but they do mention: I am not sure what "representation independent" means, but as ...
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3answers
45 views

A hypothesis test question

Let $X_i$ (for all integer $i$)be Bernoulli random variables (which takes either value -1 or 1, with equal probability). Define a random variable $Y$ to be $Y=\sum_{i=1}^d{X_i}$, where $d$ is a hidden ...
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0answers
30 views

Cointegration and variance of time series

If we know that $X_t , Y_t$ are two cointegrated discrete random processes, what can we say about the relationship between variance of the two increments $var(X_{t+h}-X_t)$ , $var(Y_{t+h}-Y_t)$ for a ...
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1answer
22 views

How to compare 2 ARIMA model predictions using mean squared prediction error

How can I compare the predictions of 2 arima models using mean square prediction error in R, given that I know what the observed values are. Help greatly appreciated.
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1answer
45 views

Is it possible to calculate Q1, Median, Q3, StDev from already aggregated data?

We have data that will get aggregated per hour into the following values Q1 Median Mean Q3 Standard Deviation Max Min Count of Values So the data will look more or like this in the end. ...
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1answer
28 views

two tailed unequal variance t-test

Can someone explain to me what is the null hypothesis of the two tailed unequal variance t-test in the below: My understanding is the following: The statement states: "The duration times for the ...
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2answers
91 views

Linear graph turning exponential at a particular point

For a line graph, it behaves linearly upto a particular point and varies exponentially after it. Please suggest me a statistical approach/test to know this threshold point. When I plot a scatter ...
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1answer
44 views

Can a truly randomized procedure (e.g. random treatment allocation) result in unbalanced distributions?

Thousands of randomized trials are ongoing or in the planning. These trials rely on randomization procedures (e.g. patients being randomized to either placebo or active drugs) to yield two balanced ...
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7 views

Usual way of measuring a set's deviation from another set

I have a set of reference daily prices. I also use an algorithm to estimate daily prices. I want to measure the difference between my estimations and the reference. I am quite unfamiliar with ...
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0answers
42 views

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 ...
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1answer
36 views

How to find maximum likelihood of multiple exponential distributions with different parameter values

Let's say that I have a bunch of independent samples, $X_1, X_2, \dots, X_n$ and that they all follow Exponential($\theta_i$) distributions. (So they all have pdf $f(x_i)=\theta_i\exp(-\theta_iy_i)$.) ...
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0answers
39 views

Transform sum of gamma distribution to chi square distribution

Let's say that I have a random sample $Y_1, Y_2, \dots, Y_n \sim\Gamma(\alpha, \theta)$. I can work out using the moment generating function that the distribution of $\sum Y_i$ is $\Gamma(n\alpha, ...
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1answer
51 views

Foundational sufficient statistics

I've been reading through Casella and Berger's Statistical Infererence and have am having a little trouble understanding something in their explanation of sufficient statistics. Here is the passage ...
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9answers
2k views

Are we exaggerating importance of model assumption and evaluation in an era when analyses are often carried out by laymen

Bottom line, the more I learn about statistics, the less I trust published papers in my field; I simply believe that researchers are not doing their statistics well enough. I'm a layman, so to ...
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1answer
77 views

Understanding Fisher Information for vector parameters

Assume that we have a vector parameter $a = [a_1^T \, a_2^T]^T$. I need help to understand the difference between the conditional Fisher information $FI_{a_1|a_2}$ and the regular Fisher information ...
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1answer
245 views

How does the mean function work for a Gaussian Process?

I was reading the notes on Gaussian Processes by Choung B. Do (stanford course CS229) however was unsure of how the mean function worked and what a random variable was on the Gaussian Process So ...
2
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0answers
271 views

Uniform most powerful Test for one-sided hypothesis

I am looking at the proof of the following theorem from Morris H. DeGroot and Mark J. Schervish's probability and statistics. I am having trouble in understanding part of the proof, which I have ...
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2answers
67 views

Normalizing matrix values python/R

I am trying to fill missing values in 1000 x 1000 matrices. Dataset1 contains such 1000 x 1000 matrice with value ranging 1-100. ...
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1answer
19 views

Unbiased estimator and variance

A random sample of n people are asked whether they are against smoking or not. Suppose x are against smoking. What is the distribution of the random variable X (number of those against smoking). State ...
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2answers
57 views

Samples from a multivariate t distribution

Hi I have the following problem. I draw a sample of a multivariate t-distribution with some fixed covariance matrix, so that the realizations are correlated, and $\nu=4$. Now I repeat this $n$ times, ...
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1answer
66 views

Matrix Multiplication to find Correlation Matrix

In this book on matrix factorizations: http://lnfm1.sai.msu.ru/~rastor/Books/Skillicorn-Understanding_complex_datasets_data_mining_with_matrix_decompositions.pdf The author states the following ...
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1answer
39 views

Efficiency becomes 0 when sample size becomes big?

I am trying to solve Robert Hogg's mathematical anaysis 6th exercise 6.2.11. The problem says. Let $\bar{X}$ be the mean of a random sample of size n from a $N(\theta,\sigma^2)$ distribution, ...
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1answer
32 views

Mathematical Basis behind inflation of Standard errors of Regression estimates due to multicollinearity

We know that due to multi-collinearity, the standard errors of beta estimates get inflated. But what is the mathematical basis to it? I am looking for some mathematical relationship or something to ...
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0answers
16 views

Statistical comparison

Im going to test my dataComparison on steganographic images(lossy and lossless) on a large number of samples. I would like ...
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2answers
60 views

Dealing with spikes in data

A company sells chocolates. Demand is recorded weekly. The future demand is estimated using the sales for every week in the previous 3 years. But the sales pattern is corrupted by promotions that have ...
6
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1answer
243 views

what do we mean by hyperparameters? [duplicate]

Can anyone give me full details about what we mean by hyperparameters, and what in the Dirichlet distribution are called hyperparameters? A practice example for the estimation of those parameters ...
3
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1answer
34 views

Karlin-Rubin with Beta distribution [duplicate]

Suppose $Y$ is one observation from a population with a Beta $(\theta,\ 1)$ pdf. Use the Karlin-Rubin theorem to find a UMP level $\alpha$-test (based on $\mathrm{Y}$) of $H_{0}$ : $\theta\leq 1$ ...
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16 views

Variance of regression coefficient

hopefully simple questions From a regression model y=B1(x^2) B1 = SUM(Y*X^2)/SUM(X^4) Please calculate the variance of this regression coefficient ... I'm not sure how to proceed with the ...
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1answer
8 views

Convergence rate when a heavierside function is involved

Let $\bf \alpha \in {R}^p$ be a parameter, $\bf \alpha_0$ is the true and $\tilde{\bf \alpha}$ is its estimator such that $\|\tilde{\bf \alpha} - {\bf \alpha}_0\|_2 = O_p(N^{-r})$, where $N$ is the ...
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19 views

what distribution might be a better fit?

I am trying to fit the data (the green curve) with a mixture of normal distribution. I obtained the fitted distribution (the purple curve) from norMixEM(), but the result is not very satisfactory. My ...
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2answers
170 views

Different CDF Notations

I understand that $F_X(x)$ is the CDF of the random variable X and $F_Y(y)$ is the CDF of the random variable Y. What do these mean? 1. $F_X(y)$ 2. $F_X(X)$ 3. $F_x(Y)$