Articles of statistics

Fitting of Closed Curve in the Polar Coordinate.

I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I’m thinking of deducing a similar formula using Maximum Likelyhood, but the problem is I don’t know what kind of hypothesis to choose. […]

Birthday Problem: Big Numbers and Distribution of the Number of Samples involved in Collisions

A lot of questions about the birthday problem can be found here, but none seems to address my problem: Background I am thinking of a hash-type data structure design which accepts a certain number of collisions to occur. Collisions shall be detected and handled in a second data structure with substantially lower collision probabilities. The […]

VC dimension for Rotatable Rectangles

It can be shown that VC dimension of rotatable rectangles is 7. The problem is I cannot understand how to approach the solution. So far I used bruteforce to solve this kind of problem, I was drawing points in different shapes and check whenever the hypothesis shatters the points. In this case the heptagon is […]

probability density function of $Y=X_1\dotsb X_n$, where $X_n \thicksim U$

What is the probability density function of the following product of uniformly distributed random variables: $Y=X_1\dotsb X_n$, where $X_n \thicksim U[1,2]$ (Uniform distribution); the $X_n$ are independent. OBS: It is not a duplicate. I found the answer only for the $U[0,1]$ and here, there is no analytical equation for the pdf.

Sufficiency of $X_{(n)}$ for random sample of scale uniform variables.

Consider a random sample $X_{1}, \dots, X_{n}$ where $X \sim \mathrm{unif}[0, \theta]$ for $\theta \in (0, \infty)$. Usually we prove that $T = X_{(n)}$ is a sufficient statistic for $\theta$ by appeal to Neyman’s Factorization Thoerem. I’d be interested in demonstrating the sufficiency of $T$ by showing that the conditional distribution $X\mid T$ does not […]

Generalized birthday problem (or continuous capture recapture?)

Let’s say I don’t know how many days there are in a year (N) and want to figure this out by asking people for their birthday. I ask D random people for their birthday and find there are E duplicate birthdays. How can I estimate N? For example, say I draw: 24, 49, 52, 28, […]

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: >>> import numpy as np >>> from scipy import stats >>> a=np.random.random_integers(1,9,4) >>> a array([3, 7, 4, 3]) >>> b=np.random.random_integers(1,9,5) >>> […]

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.

My question is: do you know any examples when $X$ and $Y$ are both normally distributed, but the two dimensional vector $(X,Y)$ is not? I found some example in book, but I don’t understand it. The example is: Let us assume that $f_1$ and $f_2$ are both the densities of $2D$ normal distribution with $0$ […]

Iteratively Updating a Normal Distribution

Is there a way to update a normal distribution when given new data points without knowing the original data points? What is the minimum information that would need to be known? For example, if I know the mean, standard deviation, and the number of original data points, but not the values of those points themselves, […]

Showing a distribution is not complete for a parameter $\theta$

I am trying to show that a normal distribution with parameters $\mu = 0$ and variance $\theta$ is not complete. I am looking for a function $u(x)$ that is not equal to 0 such that $\mathbb E(u(x)) = 0$. I have done some research on this problem and I have found that $\bar{X}$ and $S$ […]