Generalised inverses – Solve normal equations

I am still not fully able to handle the concept of generalised inverses when applied to OLS. Is there a way to show that β =  (X′X)−X′y+(Ik−(X′X)−X′X)z solves the normal equations:  X′Xβ = X′y so that it holds for any z? Note:  X is an (n × k)-regressor Answer AttributionSource : Link , Question Author : … 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

Probability of getting a seat in the train car

A train has got five train cars, each one with N seats. There are 150 passengers who randomly choose one of the cars. What is the probability that everyone will get a seat? I think that what is asking me is “what is the probability that each wagon is chosen by no more than N … Read more

Sufficient statistics in multiparameter exponential family

I’m trying to work through a theorem in the Lehmann statistical inference book and I’m confused about a proof. They are proving that a set of tests are UMP unbiased level-alpha tests for a series of hypotheses in a multiparameter exponential family. Write the following exponential family as: f_{X}(x\mid {\boldsymbol {\theta }})=h(x)g({\boldsymbol {\theta }})\exp {\Big … 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

Can we do a chi-squared test of independence on nominal(output) and ratio(input) data?

I have a data set with Age and Survival Status column. I have categorized the ages into Age Groups namely: 30-47 years old, 48-65 years old and 66-83 years old and have aggregated the Survival Status for the ages which fall in the same group. Survival Status can have the value as True or False … Read more

Exercise on cointegrated processes

I have this time series problem that I cannot solve: Considering two I(1) independent stochastic processes zt and xt and a stationary stochastic process wt, I have the following DGP: yt=xt+zt+wt. Suppose that I misspecify the DGP and regress yt only on xt and zt. Will I be able to find cointegration between xt and … Read more

Conditional Gaussian Distribution

I am trying to study this paper on Linear Gaussian Models. I’m a little stuck on the following result for finding the conditional of a Gaussian: In the paper it states this is done ‘simply by linear matrix projection’, could anyone possibly provide me with a starting point or some hints so I can try … Read more

Define model and interpret multinomial regression

Question What should my (multinomial logistic regression) model look like for analysis of the data described, and how can I interpret it and report it in an acceptable way for journals that typically expect a P value. Background I’m a grad student. I have taken the only stats course available in my department, so I … 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