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

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36 views

what should be the null hypothesis?

conduct a test hypothesis to show that it takes longer for people with average fitness to successfully complete physical therapy as compared to those with above average physical fitness. Test at ...
5
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2answers
58 views

Why is the standard deviation defined using differences^2 instead of differences^4? [duplicate]

I've read some post about why to take the std. deviation and its benefits, here's one of them: Why square the difference instead of taking the absolute value in standard deviation?, and another by ...
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0answers
37 views

“All of Statistics” questions [on hold]

I am not able to solve a couple of questions asked in the exercise 10.11 of book - All of Statistics. Question 1 Question 2 How do I start ? Any hints would be helpful ? I do not want complete ...
2
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1answer
14 views

Adjusting Monte Carlo estimates to generate correct even moments - improving on antitthetic draws

If I want to generate a matrix 10,000 (row) samples of 3 uniform (uncorrelated) variables it is trivial to use antithetic draws to ensure the odd moments such as the mean equal their "true" value. ...
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0answers
21 views

t-student distribution [duplicate]

I've got this problem: Here, if $Z,W$ are independent random variables, and $Z$ has normal standart distribution and $W$ has $\chi^2$ with $n$ degrees of freedom, $T=\frac{Z}{\sqrt{\frac{W}{n}}}$. I ...
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2answers
34 views

Predictive Accuracy formula in Excel or R [duplicate]

I have posted this question, not sure how to move that question to this stats.stackexchange.com. ...
5
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2answers
242 views

Example for a prior, that unlike Jeffreys, leads to a posterior that is not invariant

I am reposting an "answer" to a question that I had given some two weeks ago here: Why is the Jeffreys prior useful? It really was a question (and I did not have the right to post comments at the ...
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0answers
10 views

How can I prove the variance of residuals for an unbalanced one-way layout?

How can I prove the variance of residuals for an unbalanced one-way layout? var($e_{ij}$)=$σ^2$($1- \frac{1}{n_i})$ here i = 1,...., a j = 1,....$n_i$ where $e_{ij}$'s are the residuals. where ...
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0answers
19 views

Level of a time series and adding daily dates to plot

just wondering if you can help me with explaining this plot Just wondering what does level tell me? is it the trend of the data with seasonality taken out, which is the slope right? Can't seem to ...
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0answers
38 views

Calculating $E\big((X_1X_2)/(\sum_{i=1}^n X_i^2)\big)$ [on hold]

Let $X_1,X_2,...,X_n$ be a random sample of $N(0,\sigma^2)$. How can I calculate $$ E\bigg(\frac{X_1X_2}{\sum_{i=1}^n X_i^2}\bigg)\ ? $$
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28 views

Linear Regression using statistical method or gradient descent? [on hold]

Parameters for a regression line can be calculated using gradient descent , normal matrices operation and also it can also be calculated using statistical means where slope and intercept are ...
0
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1answer
51 views

finding upper bound for $P(T=0)$

Let $X_1,X_2,...,X_n$ is a random sample of Poisson distribution with parameter one. if $T=\bar{X}(n-\bar{X})$ how can find upper bound for $P(T=0)$
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0answers
17 views

How to show invariance for this test statistic

$$ \frac{|s^*M^{-1}z|^2}{(s^*M^{-1}s)\big[1+\frac{1}{K}(z^*M^{-1}z)\big]} \:\:\:----->1 $$ $$K \text{is real scalar},\: \quad s,z \in \mathbb{C}^{N \times 1} \quad \: and \quad M \in ...
0
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1answer
28 views

bias and sampling

This was an interview question I encountered. can some one answer this When you sample, what bias are you inflicting? How do you control for biases? What are some of the first things that come to ...
0
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1answer
39 views

assumptions made for a normal distribution- a interview question

I was asked in my interview: What assumptions can you make? Why and when? (i.e When is it safe to assume "normal"). Can some one answer this .
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1answer
36 views

What does z* represent in statistics?

Could somebody please give me an explanation as to what Z-star means for statistics, where it's used, and how to calculate it? My book is very confusing and I don't really know what it means. Even a ...
0
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0answers
20 views

how to calculate Kolmogorov-Smirnov test for one sample in R [migrated]

I have a matrix in which I would like to calculate Kolmogorov-Smirnov test on each column separately. Let's say my matrix is like below ...
3
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2answers
96 views

What does an error in ANOVA indicate?

When I came across ANOVA, the instructor talked about df(Error), ss(Error), etc. What do these error terms indicate? Do error terms differ for two-way ANOVA with dependent and independent variables? ...
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0answers
33 views

Huber's M estimator for contaminated Gaussian noise

Huber discussed in this seminal paper "Robust Estimation of a Location Parameter" link that if we have some observations $x_i$ as follows: $$y_i = \theta + \nu_i, ~~i=1,\cdots,N, \tag{1}$$ where ...
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0answers
19 views

Two-way ANOVA: negative interaction sum of squares?

I am doing a two-way ANOVA. The two variables are both categorical, one with 3 and the other with 2 categories. The count for each category is not equal. The data (only category means) is here: ...
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0answers
19 views

decomposing distribution function of sum

I am thinking about decomposing distribution function of sum of two random variables into sum of distribution function. Let assume $Y_1\sim F_1$ and $Y_2 \sim F_2$ and $Y_2\in \mathbb{R}$ and ...
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0answers
22 views

What is the reason behind lift measure?

While reviewing some slides I see the following formula for calculating lift (which is defined as "measure of dependent/correlated events"): $$lift(A,B) = \frac{s(A\cup B)}{s(A)\cdot s(B)}$$ which ...
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16 views

Multivariate Gaussian, rearranging means

Looking through the the matrix cookbook, a collection of matrix identities, I came across this one called "rearranging means" in the multivariate Normal distribution (Sec. 8.1.5 or Eq. #356, also ...
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0answers
15 views

Fisher information matrix of an arbitrary statistic

I have a proof (that is a bit long) that shows that for a single parameter $\Theta$, $J_\mathbf{X} (\Theta) \geq J_\mathbf{T} (\Theta)$ where $J_\mathbf{X} (\theta)$ is the fisher information of the ...
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4answers
296 views
+50

What are the appropriate Statistical tests when dealing with two or more categorical outcome variables in a model

I'm familiar with GLM techniques when dealing with continuous outcomes and categorical or continuous predictors. I've used Logistic regression with a single dichotomous categorical outcome variable ...
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0answers
6 views

Confidence Interval for mean difference with non-mutually exclusive 1 to many matches in R

This is somewhat both a programming question, and a stats question. So sorry for the overlap (although it seems that there is a large overlap between the realm of stats and programming of ...
1
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1answer
29 views

Hazard function for proportional odds model

The Cox proportional hazards model for survival data with covariate ${\bf z}$ is defined through the hazard function $h(t,{\bf z})$ by $$ h(t,{\bf z}) = h_0(t)~\cdot\theta~~,~~~\theta = \theta(\beta, ...
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0answers
9 views

Conjugate Distributions

I read one way to derive saddle point approximation uses the so called conjugate distributions densities $h_t(x)$ for a given density function $f(x)$. It is defined as follows. $$ h_t(x) := ...
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1answer
17 views

Accounting For Increase In Volume

I'm not very good at statistics but today at work we were in a meeting where manager stood at the top of the room and told us our quality performance in manufacturing our goods had dropped but our ...
0
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1answer
21 views

Variance calculation RELU function (deep learning)

weight initialization is important for modern deep learning. To understand [1,2], I would like to understand the following: $$ E[x^2] = 0.5 Var[y], $$ where $x= max(0,y)$, $E[.]$ is the expectation, ...
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0answers
13 views

Alternative answer to choose k people out of n, then choose 1? [migrated]

The question is fairly simple, we first choose k people out of n, that is $C(k,n)$ as the combination function, then we choose 1 person out of k, we have k choices. The total number of choice is given ...
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2answers
17 views

Division of vectors [closed]

In R: aff<-c(4,8,12) bff<-c(2,4,6) aff/bff [1] 2 2 2 But vectors' division is undefined. Where I am wrong?
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0answers
9 views

Implementing the Bivariate survival function (Dabrowska estimator etc)

I am trying to implement the Dabrowska estimator to apply to some data that has been collected and I am wondering if anyone can help. The original paper is Kaplan-Meier Estimate on the Plane by Dorota ...
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0answers
14 views

evaluate the similarity between two time series

I have two time series, $\mathcal{T}_1$ and $\mathcal{T}_2$, each time series is of two dimensional. One time series is collected from two sensors (SA, SB), and the other is collected from other ...
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0answers
25 views

How to interpret D-Efficiency in Experimental Design?

I created an orthogonal design in R using AlgDesign package. I got 14 profiles and some output: D =0.2519353; A = 5.462121; Ge = 0.748; Dea = 0.714. How to interpret D , A , Ge and Dea? Also, In ...
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0answers
21 views

What does 's' stand for in this definition of Fractional Brownian Motion?

It's taken from Mandelbrot & Van Ness' (1968) definition of Fractional Brownian Motion. I believe it is a definition of the difference between values of the process at t1 and t2, but I don't ...
0
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1answer
34 views

Difference in value between Biostatistics or Bioinformatics Masters? [closed]

I'm a mathematics and economics double major and I've been very interested in attending Georgetown's MS programs in Bioinformatics and biostatistics. However, I'm not so sure which one I should ...
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0answers
17 views

Probability of taking a random sample of 24 measurements and getting a mean of at least 103.6 of true population

A random sample of size n = 24 measurements is drawn from a normal population. The sample has a mean of 103.6 and a standard deviation of 12.5. If the true population is 100, find the probability of ...
0
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0answers
21 views

predict sales using naive bayes and handle sparse data problem

Problem I am trying to use naive bayes for ranking products in a search application. I would like to predict the sales of a given product given the search keyword and the category. the current formula ...
12
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1answer
358 views

Same Mean, Different Variance

Suppose you have eight runners run a race; the distribution of their individual run times is Normal and each has mean $11$ seconds, say. The standard deviation of runner one is the smallest, two the ...
1
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0answers
18 views

calculating the correlation coefficient

Let for random variables of identically distribution $X_1,X_2,...,X_n$ have $$(X_i,X_j)\stackrel{d}{=}(X_1,X_2):\forall i\neq j , \sum_{i=1}^n X_i=0$$ how can caculate the correlation coefficient of ...
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0answers
6 views

Sample complexity for agnostic PAC learning for real valued functions

How many samples are needed for ERM to have $\epsilon$ excess risk relative to the best hypthosis $h^*$? Assume a bounded (and Lipschitz, if needed) loss function. The only survey I have been able to ...
0
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0answers
29 views

Finding expected value

I am not sure of how to find the value asked in below question. Any help would be appreciated. Suppose that the joint distribution of X and Y is the uniform distribution on the circle x2 + y2 ...
1
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2answers
187 views

Is Statistics Mathematics? [duplicate]

I am from the mathematics community on StackExchange. I should say that I know absolutely nothing in statistics. I have never taken a statistics course or read any book on it. I am familiar with basic ...
0
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0answers
28 views

Multivariate Cramér-Rao inequality: intuition for positive semidefiniteness

Here's what Wikipedia says about the Multivariate Cramér-Rao inequality: If $\boldsymbol{T}(X)$ is an unbiased estimator of $\boldsymbol{\theta}$, then the Cramér–Rao bound reduces to ...
3
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1answer
54 views

Are the order statistics minimal sufficient for a location-scale family?

Suppose we have a location-scale family with pdf $$\frac{1}{\sigma}f(\frac{x - \mu}{\sigma}).$$ Let $X_1$, $X_2$, $X_3$,..., $X_n$ be a random sample from the location-scale family. Is the statistics ...
3
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1answer
43 views

Intuitive interpretation of Bayes risk $R(\delta, \lambda) = \int_{\Omega}R(\theta, \delta) \lambda(\theta) d\theta$

Consider the risk function R of an estimator (statistic) $\delta(X)$ trying to estimate parameter $\theta$: $$R(\theta, \delta) = E_{X \sim P_{\theta}}[Loss(\theta,\delta(X)]$$ Which can be ...
1
vote
1answer
41 views

probability distribution estimation from correlated samples [closed]

I am looking to solve the following estimation problem. Consider a blackbox where (given below) given an scalar input $X \in \mathcal{X}$, its $N$ observations, $Y_1, \cdots, Y_n \in \mathcal X$ are ...
0
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0answers
44 views

Any machine Learning models to predict dates?

I have a general question regarding machine learning models. The idea is to predict what DATE the customer is likely to make transactions or purchases. Variables present in the data set are item, ...
4
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5answers
82 views

Which expansions and identities are useful to applied statisticians? [closed]

Simple mathematical relationships like $V(X) = E(X^2) - E(X)^2$, aside from being theoretical results, are useful because they allow analysts to do back-of-the-envelope calculations, restate results ...