Check of moments for pseudo random number generators using CLT

I’m trying to get to grips with my lecture notes but cannot understand the idea behind it. I’d really appreciate some help: “Generate a sequence of size $n$ from stated distribution. Consider an independent sample $X_1,X_2,X_3,..,X_n$ from pdf $f_x(x)$, with mean $\mu$ and variance $\sigma^2$. Then the CLT says that… (def. continued – I don’t … Read more

Not able to understand KL decomposition

The bias-variance decomposition usually applies to regression data. We would like to obtain similar decomposition for classification, when the prediction is given as a probability distribution over C classes. Let P=[P1,…,PC] be the ground truth class distribution associated to a particular input pattern. Assume the random estimator of class probabilities ˉP=[ˉP1,…,ˉPC] for the same input … Read more

Help: What is the theory underpinning unit-treatment additivity for ANOVA

I have read extensively about the assumptions for ANOVA, and I am experiencing confusion with the concept of unit-treatment additivity. I would be deeply appreciative if anyone could please explain the theory in laymans terms. Many thanks in advance if this is possible. Here is a short excerpt from my text book The equal variance … Read more

Is the covariance between the product of two variables and one of the variables zero?

For two centered (zero expectation) random variables X and Z I am interested in the covariance of the product XZ and either X or Z. Cov(X,XZ)=E(X(XZ−E(XZ)))=E(X2Z) I think the last quantity can be non-zero. However, for any choice of X and Z I have tried, in simulations, I find the quantity to be approx. zero, … Read more

Pattern Recognition Time Series via FFT

I came across this interesting article where the author used FFT to discover some patterns in a time series. I am new to this kind of analysis and have maybe some basic questions about it. How do you compute the Frequency when getting the FFT? I used the fft function in MATLAB with some data … Read more

Finding UMPT for uniform distribution with varying support

$\textbf{Problem}$ Let $X_1,\dots,X_n$ be a random sample from $f(x;\theta) = 1 / \theta$, where $0 < x < \theta$. We want to test $H_0: \theta \leq \theta_0$ versus $H_1: \theta > \theta_0$. Obtain the uniformly most powerful test with size $\alpha$. You must describe how to calculate $\alpha$ to get a full credit. Here I … Read more

Adjust decay rate dynamically

Say I have a stream of values $\langle s_1, s_2,\ldots\rangle$ coming in and a function $$E_{s_1:s_n}(t) = E_{s_1:s_{n-1}}(t-1) + \alpha\cdot (s_t-E_{s_1:s_{n-1}}(t-1))$$ that compute their exponential moving average as the values flow in. I would like the alpha, i.e the decay rate, to adjust dynamically as a function of the last $h$ values we have seen. … Read more

Large deviations results for cosine of two samples from Normal?

I’m looking for large-deviations style results for cosine of two independent samples drawn from N(0,Σ) . IE, q=⟨X,Y⟩‖ More specifically, are there any interesting bounds on the probability of this value being large in terms of properties of \mathcal \Sigma ? Intuitively it seems this value should be small when \Sigma has small condition number. … Read more

Prove $X_t =$ $\sum_{j=-\infty}^\infty \psi_j * Y_{t-j} = \psi (B) * Y_t$ is stationary?

I was going through my time series text and I found a curious identity that the book just says is true without actually going in depth to prove it. I am unable to figure out how this works, so I would appreciate pointers. The theorem states: “Let {$Y_t$} be a stationary process w/ mean 0 … Read more

Connection between canonical correlation and distribution of roots of characteristic equation

I’m trying to make sense of the following sentence from introduction “Multiple discoveries: Distribution of roots of determinantal equations” “The distribution of the squares of the canonical correlations when the population canonical correlations are zero is the same as the distribution of the characteristic roots of one sample covariance matrix in the metric of … Read more