Mathematical theory of statistics, concerned with formal definitions and general results.

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How do i find the P-Value of the this question using TI-83?

In a clinical trial 23 out of 875 patients taking a prescription drug complained of flulike symptoms. Suppose that it is know that 1.5% of patients taking competing drugs complain of flulike symptoms. ...
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9 views

Econometrics - Relationship between cointegration and ECM

I'm pretty new to econometrics and I've been taking a class at University which uses the book "Econometric theory and methods" by Davidson and MacKinnon. It's a pretty good book but there's one thing ...
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3 views

recurrence quantification analysis

I am relatively new to the use of recurrence quantification analysis (RQA) is a method of nonlinear data analysis which quantifies the number and duration of recurrences of a dynamical system ...
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17 views

An Algorithm for Maximal Correlation

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as ...
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1answer
61 views

Slutsky's Theorem to show convergence to Standard Normal Distribution

We are given $W_n = \frac{\bar{X}-\lambda}{\sqrt{\bar{X}/{n}}}$ and need to show it converges to a standard normal distribution. EDIT: The square root in my original post did not extended over the ...
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1answer
16 views

Finding z-scores from z table relating to confidence intervals

I'm having trouble finding the proper $z$ score so that I can find the $99\%$ confidence interval. $\bar{x} = 6.01231$. with an $s$ of $1.96833$ and $n$ of $26$, and I got $2.575$ for ...
2
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19 views

Eliminating a nuisance parameter in likelihood ratio test

I am having an argument with a co-author about how to eliminate a nuisance parameter in a simple likelihood ratio test and am hoping that the community helps us settle it. Our data $\mathbf{x}$ can ...
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40 views

Berry-Esseen Theorem with Continuity Correction

Given independent but non-identical random variables $X_1, X_2, \ldots ,X_n$ with $E[X_i]=0,$ $E[X_i^2]=\sigma_i^2=1$ and finite absolute third moments $\rho_i=E[|X_i|^3].$ Let $$S_n = {\sum_{i=1}^n ...
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1answer
31 views

How to calculate the absolute central moment of a Binomial distribution?

There is an experiment. The coin is tossed $n$ times with $p = 0.5$. The experiment is repeated $k$ times. I need to calculate the average central moment. For example, let $n = 5$ and $k = 3$. $[0, ...
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29 views

find variables that have significant difference between two groups?

I have two grouped data sets with too many variables. As a pretest, I would like to find variables which show significantly difference between the groups. What kind of simple test I can do?
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1answer
19 views

Distribution of sum of mean squared errors - weighted sum of chi squared distributed variables

Suppose $X,Y$ are independent chi-squared distributed random variables with $m,n$ degrees of freedom, $X \sim \chi^2(m)$ and $Y \sim \chi^2(n)$. What is the distribution of $$ Z = \frac{1}{m} X + ...
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32 views

Wilcoxon signed rank test - Normalize data or not?

I am comparing my algorithm's performance with other algorithms, for a minimization problem. There are ten different benchmark problems, for which I have computed the average solution quality for all ...
3
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1answer
59 views

Gamma Distribution and Life of Component?

I came across an old exam question as follows: If the life of one computer component (in years) has Gamma distribution with mean $6$ and variance $18$, how can we find the probability that this ...
2
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0answers
33 views

Gaussian Mixture and K-Means ?! a big challenge?

This is taken from Tom. Mitche Material as Old-Exam. I think the (2) is true and not (3). Who can verify me?
2
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1answer
16 views

Hypothesis Test on Contest, a problems?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
2
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1answer
59 views

Advantages and disadvantages of using population and samples for statistics

I need to know the advantages and disadvantages of using a sample or a population to collect data. I understand what a sample is and what a population is, but I do not know what the advantages and ...
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0answers
16 views

Help please with some questions for school [closed]

Can anyone help me out with this, Consider the data set(yi, x1i,x2i) i = 1,2.....N where E(xji,ui) = 0 j = 1,2. If the model is yi= B0 + Bx1i + B2x2i + ui and ui is normally distributed show how you ...
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1answer
33 views

Show that power of size $\alpha$-test is larger than power of level $\alpha$-test

I have two power functions of the null hypothesis, $Q(\theta_0)$ and $Q^*(\theta_0)$ where $Q$ represents $\psi(\mathbf{y})$, a size $\alpha$-test, and $Q^*$ represents $\psi^*(\mathbf{y})$, a level ...
4
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1answer
77 views

Joint PDF of a Uniform Distribution

The Question I have a sample X1,...,Xn i.i.d. drawn from a uniform distribution $unif[0,\theta]$, θ ∈ Θ = R+; And I'd just like to compute the joint PDF The Solution I have the following solution ...
4
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1answer
158 views

MLE Fundamentals Question

So I'm a little stuck with what I feel is a basic question. The calculation is easy, but I've obviously missed a key concept of MLE. The Question Consider the family of models for the data X1,...,Xn ...
2
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1answer
77 views

I don't understand Chi squared

I think of the contingency table test. In all textbooks I've seen, the test statistic is calculated as the sum of $(O-E)^2/E$ over all cells. But the degree of freedom is not the number of all cells. ...
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1answer
21 views

Coefficient of variation for a subset

How do I calculate the coefficient of variation, CV (or Relative Standard Deviation, RSD) for a subset? e.g. ...
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1answer
15 views

Two random vars, finite mean and variance, represent Var(Y) with conditional expectations

This was asked as an self-assessment question, that I was quite embarrased by, as I had no idea how to start it... Consider two random variables X and Y that are allowed to be correlated and whose ...
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18 views

mean of clients during a specified time interval

I would like to calculate the mean number of guests in a restaurant for example, during one week. I have two cases: 1- we take measures of number of guests each 5 minutes, 2- we take measures each ...
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2answers
48 views

How to deal with 'cut-off' selection bias/sampling bias? (truncated distribution)

In short When measuring an outcome with a normal distribution, but whos mean is below the detection threshold, can you still make statements about differences between populations? Example Say I ...
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11 views

What's the probability that there exists a hyperplane that can split a dataset which have random feature values ?

Given n data points, each with d features, n/2 are labeled as -1, the other n/2 are labeled as 1. Each feature takes a value from [0,1] randomly (uniform distribution). What's the probability that ...
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31 views

Intuition understanding of bootstrap [duplicate]

I want to estimate the population mean using 100 samples. The reasonable estimate of the population mean is the sample mean. I am wondering, why bootstrap can give more accurate result than the sample ...
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1answer
28 views

Weak dependence of a variable

I asked the same question last week, but it seems it wasn't that clear and it was closed, so I try to expand it to explain better what I need. If I cannot do this, sorry in advance. I have a variable ...
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21 views

The mystic “true” bias (bootstrap-method)

This is a problem of understanding. That's why it doesn't include any formula. I have one big data set (n=83 Observations) and a small subdataset (n=15). With the small subdataset, I estimated the ...
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0answers
25 views

From Discrete to continuous Probablities [duplicate]

How does one define Density function for a continuous Random variable? What is the role of density function? How do achieve Density from mass?
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2answers
86 views

What mathematical background do I need for the Gaussian Process book by Rasmussen and Williams?

I started reading the book today, but right off the bat, he mentions infinite Hilbert spaces in the notation, so I feel that it might be out of my league. I am familiar with linear algebra, ...
4
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1answer
82 views

Relationship between hazard function and survival function in the presence of censoring

In survival analysis there is a relationship between the survival function, $S(t)$ and the hazard function, $h(t)$, in that $$ h(t) = -\frac{d}{dt} \log S(t)~~~~~~~~~(1)$$ from which we can form the ...
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1answer
38 views

Finding Probability of Uniform [0,1] Random Variables

If U1, U2, U3 are IID Uniform[0,1] random variables. How can we find the P(U1 < U2 < U3)? Note: the possible orderings of U1 U2 and U3 are equally likely
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1answer
27 views

Why are $M$-estimators NOT scale equivariant?

Consider the following location model. $$ x_i = \mu + u_i, (i = 1,\dots, n), $$ where $u_i$ are $i.i.d.$ with density function $f_0$. Hence, $x_i$ are $i.i.d.$ with density function $f_0(x-\mu)$. It ...
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2answers
134 views

Find the mle of $\theta$

This is from Robert Hogg's Introduction to Mathematical Statistics 6th Edition Exercise 6.1.13. The question is: Let $X_{1},X_{2},...,X_{n} $ be a random sample from a distribution on $\mathbb{R}$ ...
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2answers
244 views

Admissible Estimator for Linear Regression

Is there an admissible estimator for a linear regression model with many parameters without restricting the parameter space? Admissibility will be with respect to Mean Square Error on the regression ...
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1answer
32 views

Convergence in Mean Square

Let $E[y_t|y_{t-j},y_{t-j-1},\cdots] \xrightarrow[m.s.]{}0$ as $j \rightarrow \infty$.Is it necessarily true that $E[y_t] = 0$? My Attempt: \begin{align*} E[y_t|y_{t-j},y_{t-j-1},\cdots] ...
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1answer
14 views

Parametric Estimation of Incomplete survival data/observation

Are there other methods besides the Maximum Likelihood for estimating parameters in an incomplete data when a parametric distribution is assumed?
3
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1answer
58 views

Can someone help solving the 4th order exponential equation subject to the constraint equations?

Let $$f(x)=\exp\left(\sum_{i=0}^4\theta_ix^i\right).$$ Given the following constraints: $\int_{-\infty}^\infty f(x)dx=1$ $\int_{-\infty}^\infty xf(x)dx=0$ $\int_{-\infty}^\infty x^2f(x)dx=1$ ...
2
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0answers
45 views

Why the principal components correspond to the eigenvalues? [duplicate]

Suppose ${\bf{X}} = ({X_1},{X_2},\ldots,{X_n})$ are the original components (also random variables) and ${{\bf{w}}_j} = ({\omega _1},{\omega _2},\ldots,{\omega _n})$ are loadings for the $j$th ...
0
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0answers
27 views

How can I show that $E[(\hat\theta -\theta)^2]<Var(\bar X)=\dfrac{1}{n}$? [duplicate]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
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0answers
8 views

How to determine values of coefficients for a comparsion using factorial design?

I think my problem is best answered by answering the following example but the more general the answer and explanation the better: Given 2 factors X and Y, x with 2 levels x1 x2 and Y with three ...
0
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0answers
23 views

Probability, Predict Percentage of Team A vs Team B

Let's say we have Team A and Team B. Team A wins 60% of the time while team B wins 70% of the time. Team A wins 40% of the time at home and Team B wins 30% of the time at home. Team A has never ...
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21 views

What is the probability that the monthly meeting will be held today, after 3 times postponing?

Today (Sunday) the manager will hold a monthly meeting. But the meeting is postponed one day to Monday, after that the meeting is postponed to Tuesday again, and then it is postponed to Wednesday ...
4
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1answer
61 views

showing that $\bar{X}$ is inadmissible by comparing with $\max(\bar{X},2)$ under squared error loss function

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I show $\bar{X}$ is inadmissible estimator Compared to $\max(\bar{X},2)$ under Squared error loss function
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25 views

finding the risk function $\max(\bar{X},2)$ under Squared error loss function [duplicate]

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I calculate the risk function $\max(\bar{X},2)$ under Squared error loss function
0
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65 views

Expressing a multivariate probability distribution as a univariate equivalent

Is it possible to combine the odds for two or more groups of balls in a multivariate version of Fisher's non-central hypergeometric distribution to give a univariate equivalent? For example, this ...
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23 views

Prob Question-Winning a Game

A and B have a game. There are 7 games in total, who wins 4 games first will succeed the whole game and then the game ends. Given that A has a probability P to win one game, and A already lost the ...
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2answers
38 views

Resource request : How to prove the output of a process is random variables?

I am reading through articles which present the spectral properties of chaotic systems such that they can be candidates for generating pseudo random binary sequences. One such article, is ...
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1answer
62 views

How to generate uniformly random orthogonal matrices of positive determinant?

I've got probably a silly question about which, I must confess, I'm confused. Imagine repeated generating of uniformly distributed random orthogonal (orthonormal) matrix of some size $p$. Sometimes ...