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

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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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25 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
57 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 ...
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27 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?
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1answer
15 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
54 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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14 views

Help please with some questions for school [on hold]

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
75 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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20 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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0answers
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
83 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
74 views
+100

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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131 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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214 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 ...
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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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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 ...
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0answers
22 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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0answers
20 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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0answers
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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0answers
61 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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0answers
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
37 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 ...
4
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1answer
60 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 ...
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1answer
35 views

Are $\hat{\beta}_{\text{ls}}$ and $S^2$ independent if errors are not normally distributed?

When estimating a linear model $$ Y_i = X_i\beta + \varepsilon_i \quad \quad 1\leq i\leq n$$ We have $\hat{\beta}$ the least squares estimation of the slope and the estimation of the variance, $S^2 = ...
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1answer
34 views

Closed form solution for t-stats and p-values in multiple regression

I am trying to build a spreadsheet that will perform multiple linear regressions on a number of data series using the closed-form solution. It was fairly straightforward to write the solution for the ...
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0answers
18 views

Why do we take the absolute value of weight of evidence when computing the information value?

I was looking at the explanation for Information Value calculation in STATISTICA and I find it a bit confusing: ...
3
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1answer
43 views

More than one unbiased estimator for a single unknown parameter?

Is it possible to have more than one unbiased estimator for a single unknown parameter?If "Yes" then how and if "No" the why?
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1answer
100 views

Distribution of the quotient of two gamma random variables with different rate parameters?

I have a question about how to derive the distribution of the quotient of two random gamma variables drawn from two different Gamma distributions with the same shape, but different rates. For example, ...
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0answers
28 views

Standardization (z-score) across the “Samples” or across the “variables”?

I found in literature that one of the most common way of standardization data is to compute z-scores (mean subtraction and division by standard deviation). Can anybody tell me if it is ok to compute ...
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20 views

Deducing an approximation to the Gini coefficient

Assuming we do not know the Lorenz curve function, If $(X_k, Y_k)$ are the known points on the Lorenz curve, with the $X_k$ indexed in increasing order $(X_{k – 1} < X_k)$, so that: $X_k$ is the ...
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0answers
13 views

Class of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
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1answer
40 views

Finding $Cov(2X+7, X^2 +3X - 12)$

So I have this pdf, $f(x)=3x^2$ for $x\in (0,1)$ and I need to find $Cov(2X+7, X^2+3X-12)$. My main concern about how I answer this is, what is the joint pdf for these two distributions? I guess ...