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

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How to interpret $\chi^2$ values obtained with $\chi^2$-test?

I have the following observed and expected values and I am trying to determine the goodness of fit with $\chi^2$ test. On calculating the $\chi^2$ I obtain the value 0.9999742, but how is it possible ...
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1answer
42 views

Deriving the Ridge Regression $\boldsymbol{\beta}\mid \mathbf{y}$ distribution

Apparently the estimate $\hat{\boldsymbol{\beta}}$ for ridge regression comes up as the mean or mode of the posterior distribution given by $f_{\boldsymbol{\beta}\mid \mathbf{y}}$. This is the ...
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1answer
18 views

Methodology: breaking multi-regression apart

I have to perform a multiple regression where my independent variable is store visits, while the dependents include hour of day, day of week, and others. I need to do this in Excel. Excel limits ...
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18 views

Two-alternative forced choice [on hold]

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...
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33 views

Rigorous theory behind overfitting

I am taking an intro to ML class, and in my limited experience, training ML algorithms (validation, overfitting etc.) feels a bit like black magic. For instance, you aren't supposed to touch the test ...
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1answer
17 views

On what basis we are evaluating variance is high or low? [on hold]

I am having a below case where N = 300 , Mean = 67 and SD = 30. With the above data we can say the variance is high because of SD = 30. My question is how we are defining it as high on what basis we ...
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1answer
34 views

What is meant by using a probability distribution to model the outputs for a regression problem?

Often a theoretical text will say something like, 'a probability distribution may be used to model the outputs' or, 'assume a probability distribution such as normal or Lognormal for the outputs'. ...
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21 views

Geometric proof of partical correlation with pythagoras theorem [on hold]

Please help me with my statistics lesson. I need to prove the partical correlation formula with the definition and the pythagoras theorem. The definition (partical correlation): ...
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20 views

R Commander: how to test correlation across all variables? [closed]

If I want to test correlation between two variables I would run the following command in R: ...
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26 views

What does make SVM a “soft computing” method?

Soft computing is defined in [1] by the capability of "operating with uncertain, imprecise and incomplete information in a manner that reflects human thinking". So, based on my limited understanding, ...
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9 views

References for Combine Seasonal Period in Seasonal Time Series

I'm in a middle writing my thesis. I got confused for find the references which said period of seasonal could be chosen from the small period which already containing another periods. Here is my ...
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14 views

Calculate state change probability

I am having telco order management data and need to calculate the probability of each order going through different stages. data is like this: Order No; product_type; time spent in step1; time ...
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37 views

Expectation of a conditional density

I'm trying to figure out why the following equation holds: $$f_{Y}(y) = E(f_{Y|X}(y|X))$$ I have sort of "worked out" the RHS to be: \begin{align} f_{Y}(y) &= E(f_{Y|X}(y|X)) \\[5pt] ...
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1answer
23 views

Algorithm to determine a point in time series data, after which probability of increase in value is very low

I am working with dataset which contains number of movie tickets sold per day. This is basically a count of total number of tickets sold, for a particular movie, for each day after its release date. I ...
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0answers
6 views

How to calculate sample size for testing H0: Y is related to X by a sigmoidal function?

I'm assuming I would need to specify a given functional form and conditional standard deviation for Y at each X. That would give me some parameters to be estimated and I would need to set a threshold ...
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1answer
41 views

Interchanging limit and derivative for CDFs

Let $F_{\theta}(x)$ denote a cumulative distribution function indexed by the parameter vector $\theta$. Given this definition is the following equation correct (and if so under which conditions)? ...
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2answers
129 views

For least squares estimation, what is the difference between using the estimator $\hat{\beta} = X^{T}Y$ vs $\hat{\beta} = (X^{T}X)^{-1}X^{T}Y$

For least squares estimation, the estimator $\hat{\beta} = X^{T}Y$ is an unbiased estimator while $\hat{\beta} = (X^{T}X)^{-1}X^{T}Y$ is also an unbiased estimator given that $X$ is well-defined. Is ...
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1answer
39 views

Is it possible to determine how many effects I can estimate in a least squares problem just by looking at the correlation matrix?

I currently have a model matrix $X$ with $6$ columns, which is being used for a factorial design problem, with each column associated with an effect. The ultimate goal is to be able to estimate as ...
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3answers
68 views

Express correlation matrix of $X$ in terms of $X^{T}X$ (in the OLS context)

In least squares estimation where $Y = \beta X$, how can we find the correlation matrix of $X$ in terms of $X^{T}X$? It seems that $X^{T}X$ is very close in structure to the correlation matrix, but ...
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44 views

Convergence in distribution and degenerate random variable

Let $\{Y_n\}$ be a sequence of random variables with an associated sequence of CDFs $\{F_n\}$ given by : $$F_n(y) = \begin{cases} 0 & \textsf{for}&y <0 \\ (\frac{y}{\theta})^n & ...
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0answers
11 views

Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with ...
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1answer
51 views

Percentage interpretation of negative values when you can't use log transformation

I have a data set of 5 indicators of the stock market. 2 of the indicators have negative values: e.g. they range from say -50 to 100. After running a regression I would like to be able to compare the ...
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1answer
49 views

data normalization after dimension reduction for classification

The classifier is KNN or RBF-SVM. After doing dimension reduction (e.g., PCA, LDA or KPCA, KLDA), does it need to do normalization before classification? In LIBSVM ...
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0answers
15 views

What mean to use in odometer calculation based on model?

Im trying to figure out what would be considered a valid calculation. Imagine I have a dataset that looks like that: ...
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1answer
32 views

On One-to-One Functions of Complete Statistics

Why is a one-to-one function of complete statistic also complete? How might you go about proving this?
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1answer
23 views

On the proof of admissibility of constant estimators under squared loss

The question concerns the discussion in Wasserman, All of Statistics, Section 13.6. He defines: An estimator $\hat{\theta}$ is inadmissible if there exists another rule $\hat{\theta}'$ such that ...
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1answer
54 views

Limiting distribution of $\frac{\sqrt{n}\left(\bar{X_n}-\mu\right)}{\sqrt{\bar{X_n}}}$ from mean of Gamma$\left(\mu,1\right)$?

Given $\bar{X_n}$ is mean of random sample with size $n$ from Gamma distribution with parameter $\alpha=\mu$ and $\beta=1$. I wanna find the limiting distribution of ...
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17 views

What statistical test should I use to compare the binary outcome of two groups (with pre- and post- test)?

I have two groups of people: experiment group and control group. People in each group accepted pre-test and post-test. The outcome of the tests is binary: "good" or "bad". In other words, each people ...
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26 views

Show that weighted least squares estimator for a specific model is not consistent

Here is the background for this problem: $\qquad\qquad\qquad$ $Y_{1},...,Y_{n}$ iid $N(\mu,c^2\mu^2)$, $\,\,$ $c^2$ known. $\,$ The problem is as following: Consider the above model. Define ...
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1answer
10 views

Is there a framework for reinforcement learning with states and actions in the same domain?

In reinforcement learning, there are states, actions, initial states, terminal states, a progress function and a reward function. Is there a theoretical framework or setting where states and actions ...
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25 views

Does the UMVUE have to be a minimal sufficient statistic?

I'm studying point estimation and I have found this question that seems pretty tricky to me. If $T$ is a minimal sufficient statistic for $\theta$ with $E(T) = \tau(\theta)$, can you say that $T$ ...
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7 views

Significant change over time between categories

I am trying to establish whether change over time is significant at an archaeological site. Basically, there are seven different time periods (lets call it A-G) and in each time period there are six ...
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1answer
54 views

How should I calculate the variance of a circular random variable?

Consider the following function being the PDF of a circular random variable (orientation angle from the zenith) ...
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31 views

How to compare PCA with KPCA for dimension reduction?

Both linear principal component analysis (PCA) and kernel principal component analysis (KPCA) are unsupervised dimension reduction methods. I have a dataset with $4000$ training samples and $40000$ ...
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113 views

Pooled Variance for Dependent $t$-test statistic?

I'm looking into finding a way to calculate Cohen's $d$ for correlated samples. Assuming pooled variances, we end up getting $$\text{SE}\left( \Delta \text{ of means}\right) = ...
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1answer
33 views

Random unit vectors in $\mathbb{C}^n$ and $\mathbb{R}^{2n}$

Suppose $\mathbf{u}\in\mathbb{C}^n$ is a complex random vector with circular symmetry, uniformly distributed on the unit complex $n$-sphere, so we have $\|\mathbf{u}\|=1$. In other words, $\mathbf{u}$ ...
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1answer
125 views

Marginal normality and joint normality

Let $X$ and $Y$ be two independent standard normally distributed random variables $N(0,1)$ .If we define a new random variable $Z$ such that : $$Z = \begin{cases}X & \text{if} &XY > ...
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1answer
62 views

Compute $E\left[ \Phi \left(X \right) \Phi \left(Y \right) \right]$ for a bivariate normal distribution

Assume that $X$ and $Y$ follow the bivariate normal distribution with correlation coeffcient $\rho > 0$, zero means and scale parameters equal to one. I am looking for an elegant way to compute ...
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1answer
17 views

Derivation of formula for sample size of finite population

I found here the formula for computing the sample size $n$ of a finite population $N$ $$ n = \frac{n_\infty}{1 + \frac{n_\infty - 1}{N}} $$ where the sample size for an infinite population $n_\infty$ ...
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0answers
28 views

calculating PMI for co-occurrences of words

I am in the process of building a question answering system. I am interested in calculating the PMI for words $x$ and $y$ occurring within 5 words of each other in a document. I have the formula and ...
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0answers
8 views

Count of the biggest bin in histogram, C#, sharp [migrated]

I want to make histogram of my data so, I use histogram class at c# using MathNet.Numerics.Statistics. ...
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0answers
23 views

What are the correct ways to check if a variable is significant?

I've conducted a series of computational experiments varying a set of discrete parameters. I see on the figures that some of the parameters don't affect the result in any consistent way and look like ...
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2answers
49 views

$\mathbb{P}(A_1)≤\mathbb{P}(A_1)$ in Boole's inequality ($n=1$) proof?

Why does this proof use $≤$ in the $n=1$ (induction base case) case for Boole's inequality, when in fact it's an equality? That is, why claim, $\mathbb{P}(A_1)≤\mathbb{P}(A_1)$, when it should be a ...
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1answer
19 views

Question on Integrating Over Joint Probability

Let us say that we have a joint probability density denoted as $$P(x_1,x_2,...,x_n)$$ If we are trying to find the probability that every $x_i$ is greater than $0$, is it correct to say that the ...
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1answer
37 views

How to do this transformation $Y= g(x) \sim \mathcal U(0,2)$?

If $X$ is a continuous random variable with probability density function $f_X(x)=2(1-x)$ for $0 < x < 1$, find the transformation $Y=g(X)$ such that the random variable $Y\sim \mathcal U(0,2)$.
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1answer
43 views

Hypothesis testing for the (Pearson) correlation coefficient

I don't understand why we have to assume ρ=0 in to get the probability density function? If I say null hypothesis p is something like 0.3, I can still use the probability density function, can't I?
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1answer
29 views

Boosted Trees: Objective Function clarification

Reading through this overview of boosted trees, I'm having trouble understanding how the second line was derived. $$ Obj(t)=\sum_1^n{loss(y_{i} - \hat{y}_i^{(t)})} + \sum_1^t{\Omega(f_i)} \\ = ...
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43 views

How can I determine whether a change is statistically significant ONLY based on confidence intervals and points of estimate?

The table above shows the proportions (%) of a given country's population that have experienced theft from vehicles each year along with respective confidence limits. How can I tell if a change ...
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0answers
18 views

Obtaining distribution function for positive linear combination of independent random variables with Poisson distribution [duplicate]

Let $X_1,X_2,...,X_n,Z$ be i.i.d poisson random variables,respectively with mean $\lambda_i$ and $\lambda$ where $"n"$ is definite, $c_i\geq 0$. Assume all the parameters $\lambda_i$'s and $\lambda$ ...
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1answer
35 views

Probability of winning and losing match in a group of five teams?

We are conducting a tournament in our office. For that purpose we have made groups and each group is comprised of five teams. Each team will play four matches with each other. Now we have to pick top ...