# Questions tagged [theory]

For questions about statistical theory. Always include a more specific tag as well.

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### Incorrectly rejecting a null hypothesis

What is the theoretical chance that we (incorrectly) reject the null hypothesis in the following case? ...
533 views

### Expectation of a random variable and the indicator random variable proof

I need to show that $E[T 1_A] = E[T|A]P(A)$. What I've got so far is $E[T1_A] = \int_{\Omega}T.1_A dP = \int_AT dP$. Now I know that $\int_AdP = P(A)$ however I'm lost as to how do I get the ...
5k views

### How to Prove that an Event Occurs Infinitely Often (Almost Surely)?

Exercise: There is a fair 6-sided die and a biased coin that has probability p > 0 of coming up heads on each toss. The die gets rolled infinitely often, and whenever you roll a 6, you then toss ...
684 views

### Theoretical results behind Artificial Neural Networks

I have just covered Artificial Neural Networks on Coursera's Machine Learning course and I would like to know more theory behind them. I find the motivation that they mimic biology somewhat ...
4k views

### Non-normal distributions with zero skewness and zero excess kurtosis?

Mostly theoretical question. Are there any examples of non-normal distributions that has first four moment equal to those of normal? Could they exist in theory?
4k views

### Performing k-means clustering on a set of lines

I have a set of lines (y = numbers between 1 and 100, x= discrete) that I am trying to cluster to group similarly-shaped profiles. I have found that the profiles seem to cluster the cleanest when ...
400 views

### Generalization error of PCA and kernel PCA

I've been recently reading Shawe-Taylor et al. 2005, On the Eigenspectrum of the Gram Matrix and the Generalization Error of Kernel PCA, where the authors analyze the squared residual of kernel ...
817 views

### What is the VC Dimension of a Naive Bayes Classifier?

How do you calculate the VC dimension of a Naive Bayes classifier with say K features?
111 views

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### Are complete sufficient statistics unique?

I'm under the impression that up to a one-to-one function complete sufficient statistics are unique. How can I show this?
435 views

### Estimating $R^2$ when some coefficients are forced (i.e., restricted coefficients)

I am running a regression in R, and wanted to find the right way to calculate the $R^2$. I have an identity that I am empirically testing with data that is y = x1 - x2 + x3 (unfortunately dont have an ...
3k views

### What is the definition of generalization error and its justification?

I was trying to understand rigorously what the goal of machine learning is. One could frame that one of the central goals of machine learning is to obtain the best possible function ever. But what ...
142 views

### Diferrencing vs Moving Average

Moving Average and differencing a series can both be used to remove seasonality. Does the difference of these two lie in the model they are used? Moving Average used in classical decomposition and ...
731 views

### Forecast error for AR and MA process

AR(p) process is denoted by: $X_t=\mu+\alpha_1(X_{t-1}-\mu)+\alpha_2(X_{t-2}-\mu)+...\alpha_p(X_{t-p}-\mu)+Z_t$ I don't understand forecast error. Let $\epsilon_{t+l}$ be the forecast error at $l$ ...
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### SARIMA model equation

Can someone please tell me in the book here how is this SARIMA equation obtained? I know that AR(1)=$Y_t=\alpha_1Y_{t-1}+e_t$ Non Seasonal AR(1)=> $Y_t(1-\alpha_1B)=e_t$. My question is what ...
664 views

### What makes constant function an estimator?

This is a theoretical one. This question is inspired by recent question and discussion on bootstrap, where a constant estimator, i.e. a constant function $$f(x) = \lambda$$ was used as an example of ...
25 views

### Recaculating control limits when an out of control situation is displayed

When plotting control charts if a point plots outside the control limits then the reason for that point to plot outside is checked and if an assignable cause is found to be the cause then after ...
28 views

### Changing a target value that minimize the product being outside the specifications in quality control

I was doing problems in the article here. In the question on page 16 I don't understand part b) how to find a target value that minimize the product being outside the specifications. Can someone ...
98 views

### Distribution of Minimum of RVs

I'm having trouble seeing why for RVs $X_{1}, \ldots, X_{n}$ it is true that: $$Pr(min(X_{1}, \ldots, X_{n}) > x ) = Pr(X_{1} > x, \ldots, X_{n} > x)$$ In other words: Why is the event that ...
3k views

### Choosing among proper scoring rules

Most resources on proper scoring rules mention a number of different scoring rules like log-loss, Brier score or spherical scoring. However, they often don't give much guidance on the differences ...
89 views

### Calculating $\bar c$ in C chart

In studying C chart I came accross this problem in Statistical Process control by Douglas .C.Montgomary. In this exercise it says: A control chart is used to control the fraction of non confirming....
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### Mixed model weight matrix

Consider Henderson’s classic mixed model equations: What is the nature of the weight matrix, S-1? I believe that this is composed of the covariance matrix, but at the same time it is diagonal. How ...
115 views

### Is it valid to apply large scale statistics to a single case?

I have a small black box that displays a 0 or a 1 every time I press its button. I can tell you, truthfully, that over the past 20,000,000,000 times it has been pressed, 19,999,999,990 times it has ...
248 views

### When Monte Carlo simulation can't be used to simulate a statistical system?

My question is simple. Which are the general conditions for which a Monte Carlo simulation can be used to represent a statistical system? Or conversely, which are the statistical system that cannot be ...
277 views

### Approximating a binomial distribution with a mixture normal

This is purely a theoretical question (I legitimately can't think of a real application), but if you wanted to approximate a binomial distributed variable with a two-component mixture normal, is there ...
597 views

### Non significant Pearson correlations included in hierarchical regression?

I would like to perform hierarchical regression in which all variables are based on previous research/theory. But when I perform Pearson correlations, I found that some variables did not correlate to ...
784 views

### Does the presence of an outlier increase the probability that another outlier will also be present on the same observation?

**Edit: (10/26/13) More clear (hopefully) mini-rewrites added at the bottom** I'm asking this from a theoretical/general standpoint - not one that applies to a specific use case. I was thinking about ...
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### Is there a statistical application that requires strong consistency?

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is ...
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### Theoretical expected value and variance

Let $X$ be a random variable having expected value $\mu$ and variance $\sigma^2$. Find the Expected Value and Variance of $Y = \frac{X−\mu}{\sigma}$. I would like to show some progress I've made so ...
260 views

### How do we understand the relationship between independent probabilities and real-world independence?

From what I have come to understand, the events A and B are considered independent for purposes of probability theory when $$p(A \cap B) = p(A) \cdot p(B)$$ Now, supposing I flip two coins. I ...
107 views

### Frequency of data: is quarterly data better than half-year data? Why?

Hi have a dataset which I constructed using quarterly observations (from bank accounts). I could have also used half-year or yearly data, but I chose quarterly because I thought that higher frequency ...
Suppose that $X$ comes from an exponential family $$p_\theta(x) = h(x)\exp(\theta x - A(\theta)),$$ and that, conditional on $X$, $Y$ also comes from an exponential family of the form  p_\eta(y\...