0
votes
1answer
33 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
0
votes
0answers
5 views

Matrix Completion feature vectors

In the noisy matrix completion problem, with enough number of revealed entries say $|E| = nr^2 log(n) $, can we have a bound on the error in the singular vectors of a sub matrix. For, example say ...
0
votes
1answer
10 views

Weighted variance of a small sample

I am trying to calculate the variance of a small sample. I have the data: ...
1
vote
1answer
14 views

Confidence interval for difference in means

I need to obtain a 95% confidence interval for the indifference in the mean score overall I have the following data which states subject (A-L) and then Test and Retest A B C D E F G H I J K ...
0
votes
1answer
18 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
1
vote
1answer
32 views

Doubts on Bayes hypothesis test

I meet one problem on hypothesis testing in statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$. $H_i$, $i=1,2$, is the hypothesis that $T$ is from the statistical ...
3
votes
0answers
59 views

Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions ...
0
votes
0answers
21 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
0
votes
0answers
33 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
0
votes
1answer
33 views

Regarding jointly multivariate normal X1,X2…X5

So I have a question from statistical inference that I need some help with: $X_1,X_2,...,X_5$ are jointly multivariate normal with means = $\mu_i$, variances = $\sigma^2_i$, correlation = $\rho$ ...
1
vote
2answers
77 views

Determining constant in a CDF

I have a question that I literally have no idea how to begin, I was hoping someone could help me: It says $X_1,X_2,\ldots,X_n$ is a sample from a distribution It says that the Cumulative ...
0
votes
1answer
43 views

Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
0
votes
2answers
72 views

Likelihood Function for the Uniform Density $(\theta, \theta+1)$

Let the random variable X have a uniform density given by $f(x;\theta)$~$R(\theta,\theta+1)$ What is the maximum likelihood function according to the samples $X_1,\ldots,X_n$? The question is much ...
6
votes
1answer
98 views

What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
1
vote
1answer
188 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
0
votes
0answers
67 views

Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
1
vote
1answer
106 views

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,...,$X_m$ be i.i.d. F, $Y_1$,...,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z'_1$ + $\Delta$, where $\psi$ is an unknown ...
0
votes
1answer
42 views

Explain how $ p(a,b,c,d) = \frac{\phi(a,b,c)\phi(a,b,c)}{Z} $ leads to $ Zp(a,b,c) = \phi(a,b,c) \sum_d \phi(b,c,d) $

this might be a quiet basic question: Let $ \phi(\chi^i) $ be a potential. Then we have $ p(a,b,c,d) = \frac{\phi(a,b,c)\phi(b,c,d)}{Z} $ By summing we have: $ Zp(a,b,c) = \phi(a,b,c) \sum_d ...
0
votes
1answer
43 views

Fisher-Tippet Theorem: how to compute the limit

Here is the question: The PDF of $X_{1},X_{2}...X_{n}$ are $f_X(x)=1-e^{-x},x>0$. They are independent. Let $M_{n}=max(X_{1},X_{2}...X_{n})$ and ...
1
vote
0answers
54 views

Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
1
vote
0answers
66 views

Properties of almost sure convergence

If, $\Sigma$ is the population covariance matrix and $S$ is the sample covariance matrix, $p$ is the number of variables, $\frac{p}{n} \rightarrow c$ as $n \rightarrow 0$, $\frac{1}{p}|S|_{F}^{2} ...
3
votes
0answers
62 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
1
vote
0answers
53 views

Interval overlap maximisation problem

Consider a line of equally spaced sensors and a disturbance which travels unidirectionally along the line at a fixed speed such that the disturbance takes time $\tau$ to travel between adjacent ...
0
votes
1answer
103 views

Estimating the total attendance

Suppose you do not know how many people are attending a convention, but you do know that as each person entered he was given an identification tag with a number on it. The tags are numbered serially ...
-1
votes
1answer
169 views

Jensen inequality

Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random ...
1
vote
1answer
365 views

Likelihood Function for the Uniform Density.

Let the random variable $X$ have a uniform density given by $$ f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]} $$ where $-\infty\leq\theta\leq\infty $ the likelihood function for a sample of ...
2
votes
1answer
78 views

Likelihood Functon.

$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ . What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ ...
0
votes
1answer
375 views

Central Limit Theorem (multivariate)

It's known that a sum $\mathbf S_n$ of iid random vectors $\mathbf {X_1,X_2, X_3,...X_n} $ which are $\mathbb{R^d} $ normaly distributed with covariance matrix $\Sigma$ and mean $\mathbf 0 $, will ...
2
votes
1answer
103 views

Fun with Powerball history. What probability theories apply?

I wanted to play around with probability theories and see if I find any statistical inferences from the Powerball lottery history going back to 11/1/1997. What theories could I apply besides simple ...
2
votes
1answer
233 views

A representation theorem for a minimally sufficient statistic by Bahadur

The Statement of the Problem I'd appreciate help in proving the following, unproven theorem from a classic article by Bahadur ([BAH], Theorem 6.3) (the expressions in square brackets are my ...
1
vote
0answers
24 views

What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]

Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads? How do I apply Chernoff Bounds to this? I really need help understanding Chernoff Bounds.
1
vote
1answer
88 views

Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
1
vote
0answers
64 views

Survival Analysis

I have some survival times which are exponentially distributed for two groups G1 (treatment) and G2 (control). The data are censored with a censoring distribution given by h(c), so I only observe: 1) ...
1
vote
1answer
223 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
1
vote
1answer
59 views

Likelihood Cramér-Rao Bound.

How can I show the following necessary and sufficient condition? An unbiased estimator $ \hat{\theta} $ of $ \theta $ achieves the Cramér-Rao Lower Bound if and only if $$ \frac{\partial ...
6
votes
2answers
311 views

Taylor series approximation statistics

how can I show the following: Let $X_1, X_2,\ldots, X_n$ be i.i.d Poisson with mean $\lambda$. Let $Y = |\{i: X_i =0\}|$. Then $\lambda$ is estimated by $$\eta = - \log(Y/n)$$ Use Taylor series to ...
0
votes
1answer
40 views

independence of uniform random variables1

let $X_j \sim U(0,1)$ if $$Y_j=\frac{X_j}{X_1+X_2+\cdots+X_n}$$ I want to show that: $Y_j $are independent $\operatorname{Var}(Y_1)=\dfrac{c}{n^2} +o\left(\dfrac{1}{n^2}\right)$ then calculate ...
1
vote
1answer
98 views

Joint Convergence and Donsker's Theorem

I have a question about joint convergence results derived from an FCLT (i.e., a Functional Central Limit Theorem). To motivate my question, consider the following setup: Let $y_t$ be a random walk ...