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

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28 views

Cosine similarity between a clean signal and its noisy version

Given a $D$-dimensional datum that is an iid sample from a spherical Gaussian distribution, and the noise-corrupted version of that datum generated by adding spherical Gaussian noise, is there a ...
3
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0answers
24 views

A stochastically increasing exponential family for which $\lim_{\theta\rightarrow\inf\Theta}\mbox{P}_\theta(X\leq x)\neq 1$

Question A little something that I've been wondering about for a while: Let $P_\theta$ be a stochastically increasing (one-parameter) exponential family on the sample space $\mathcal{X}$ with ...
1
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0answers
17 views

Meaning of marginal prior distribution

I have a variable $\mu$ with distribution, given $\phi$ of $ \mu|\phi \sim N(\phi, \sigma^2)$, and the distribution of $\phi$ is $\phi \sim N(\nu, \tau^2)$. (Here, $\sigma^2,\nu$ and $\tau^2$ are ...
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0answers
7 views

Fligner Wolfe test and 2-sample Wilcoxon test

What is the significant difference between the Fligner Wolfe test and 2-sample Wilcoxon test? How can I compare the treatment groups with the control groups of different studies after pooling?
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0answers
10 views

sequential testing and hierarchical tesing

With respect to the hypothesis testing, I once heard sth such as sequential testing, hierarchical testing, and multi-level testing. But I could not find a good reference discussing these topics. Any ...
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1answer
33 views

stick breaking model of Dirichlet process

I have a question regarding sticking-breaking model of Dirichlet process, which is defined as follows: There are further statements that I am not clear that how to derive equation 1 from that ...
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2answers
45 views

Function of random variable

If I have a random variable $X$ which has mean $\mu$ and variance $\sigma^2$, what is the approximate expression of $log(X)$ and $\sqrt{X}$? Do I assume normal approximation or use Taylor expansion?
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1answer
143 views
+50

Clarification in information geometry

This question is concerned with the paper Differential Geometry of Curved Exponential Families-Curvatures and Information Loss by Amari. The text goes as follows. Let $S^n=\{p_{\theta}\}$ be an ...
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1answer
48 views

What are the consequences of knowing (or not knowing) the range of possible answers from a test

Lets say we are interested in some unknown variable x. We poll a x at different intervals and get a set of values that x has been. However we know that if we poll x enough times we will eventually get ...
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0answers
18 views

Question about the frequency definition of probability

In the book, it is written that the relative frequency or the frequency ratio gradually tends to become more or less constant as N becomes larger and larger. It is also mentioned that this is an ...
1
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0answers
49 views

calculating distribution of $U=-\ln Y$

suppose $X_1,\ldots,X_n$ be random sample of continuous distribution with distribution function $F$ if $$Y=\prod_{i=1}^k F(X_i) \prod_{i=k+1}^n [1-F(X_i)]$$ how can I calculate distribution of $U=-\ln ...
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1answer
29 views

Expectation expansion query

I am trying to do a proof. Define the best Bayesian estimator by $\theta^B=E(\theta|x)$. Prove that for another estimator $\gamma$ of $\theta$, we have $MSE(\theta^B)\leq$$MSE(\gamma)$. Proof: ...
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1answer
26 views

finding joint Moment generating function of $U,V$

suppose $X_1, X_2$ are independent random variable of $N(0,1)$. suppose $$U=X_1+X_2, V=X_1^2+X_2^2$$ how can I calculate joint Moment generating function of $U,V$
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0answers
15 views

Possibilities of accelerating EM algortihm

I'm trying to use the EM to estimate some parameters. I've programmed and it delivers. The problem however is that for each run of my programme, it can take either 5 seconds, 1min, 3min or more to ...
11
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2answers
476 views

Why maximum likelihood and not expected likelihood?

Why is it so common to obtain maximum likelihood estimates of parameters, but you virtually never hear about expected likelihood parameter estimates (i.e., based on the expected value rather than the ...
3
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1answer
99 views

finding probability density function

Suppose $X_1,\ldots,X_n$ are random samples of a $U(0,\theta)$ distribution. If $Y_n$ are the largest order statistics of the sample, and $V=\displaystyle\frac{n\bar{X}}{Y_n}$, how can I calculate ...
1
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1answer
43 views

How can I find a level $\alpha$ most powerful test?

Let $X_1,X_2,\dots ,X_n$ be a random sample from a distribution with pdf $f(x,\theta)$. Find a level $\alpha$ most powerful test of $H:\theta=\theta_0$ against $K:\theta = \theta_1$ when ...
2
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0answers
31 views

A test for a binomial r.v. vs. sum of two binomial r.v.'s?

Suppose I draw a sample from one of the two possible random variables $X$ or $Y$ where $X\sim\text{Binomial}(p,N)$ and $Y=A+B$ with $A\sim\text{Binomial}(p,M)$ and $B\sim\text{Binomial}(q,N-M)$. ...
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0answers
61 views

Joint distribution of a random variable and the sample maximum

This is one necessary part of a slightly larger problem, but this part has me stumped. We have that $X_1, X_2, ..., X_n\stackrel{iid}{\sim} U(0,\theta)$. What is the joint density of the first ...
2
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1answer
46 views

Imbalanced training dataset and Random Forest regression model

I have a large dataset (>300,000 observations) that represent the distance (RMSD) between proteins. I'm building a regression model (Random Forest) that is supposed to predict the distance between any ...
0
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1answer
31 views

Expectation operator and logarithmic function

$$\frac{1}{C_t}=E_t\left[\beta \frac{1}{C_{t+1}} \right]R_{t+1}$$ How to log linearise the function? $C_{t+1}$ is the stochastic term; $\beta$ is known.
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2answers
47 views

Can I think of the level of a hypothesis test as being the probability the null hypothesis is true? [duplicate]

I am trying to understand the level of a test better and was wondering if the level of a hypothesis is essentially equal to the probability that the null hypothesis is true. I have been trying to ...
3
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2answers
41 views

If the level of a test is decreased, would the power of the test be expected to increase?

If we have some test, and we decrease its level, would the power be expected to increase? I have given some thought to this question before but haven't been able to convince myself of the correct ...
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0answers
31 views

Precision and recall Break-Even performance

I am using WEKA to do classification. I wish to compare with an existing paper that uses the Precision and Recall Break-Even performance as their results. Can someone please help in order for me to ...
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1answer
44 views

What is it meant by the “rejection region” and “power” of a likelihood ratio test?

Suppose that $X_1,...,X_n$ are i.i.d. data from a $N(\mu, 100)$ distribution. I am trying to find the rejection region for the likelihood ratio test for level $\alpha= 0.10$ of the test: $H_0: \mu = ...
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1answer
25 views

What does “$R^2$ estimates the combined dispersion against the single dispersion of the observed and predicted series” mean?

In a paper I came across the description of $R^2$ as "it estimates the combined dispersion against the single dispersion of the observed and predicted series". I am not able to understand this ...
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0answers
25 views

Understanding about Edgeworth expansion

I don't quite understand "by the properties of the Fourier transform, $it^r\psi(t)$ is the Fourier transform of $(-1)^rD^r\Psi(x)$" from Wikipedia about Gram–Charlier A series and Edgeworth ...
2
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2answers
113 views

Can a multivariate distribution with a singular covariance matrix have a density function?

Suppose a multivariate distribution over $\mathbb R^n$ has a singular covariance matrix. Can we conclude that it does not have a density function? For example, it is the case for the multivariate ...
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1answer
70 views

For what class of distribution is this function concave?

consider the following sorted sample: $x_1<x_2<\ldots<x_n$ and the kernel function: $h(j)=\log(x_j-x_1), j>1$. Now, it turns out that when I draw data from many continuous ...
4
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0answers
57 views

How to find conjugate prior for a given distribution?

Given a distribution $\{p(X|\theta),\theta\in\Theta\}$: how do you show the existence/non-existence of its conjugate prior? what are some general ways/principles to construct/find its conjugate ...
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0answers
19 views

Statistics probability with python [migrated]

Let X1,X2,X3 be independent random variables, each of which has the exponential distribution with parameter λ=2. Use random simulation to estimate the probability that X1+X2+X3≥1. I was told to use ...
2
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1answer
68 views

estimate a distribution parameters only by data mean and std. dev

I need to estimate a truncated gamma distribution parameters (shape , scale). But, I only know the data mean and std. dev. I do not know the data set. Given the mean and std. dev. of a data set from ...
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0answers
13 views

Why does the p-value of a composite null hypothesis have a supremum attached it? [duplicate]

I noticed that there is a definition of the p-value in my textbook. It is defined as the p-value of a composite null hypothesis and it says the following: I have no idea why it is written with a ...
0
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1answer
53 views

Why is the p-value written with a supremum?

I noticed that there is a definition of the pvalue in my textbook is defined says the following: I have no idea why it is written with a supremum. I've spent hours pondering this, does anyone have ...
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0answers
26 views

Error of data fitting by gamma.fit() in Python

I need to find gamma fit for data in Python 3.2. param = gamma.fit(samp) // samp is a list of floating point numbers I got error: ...
2
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1answer
75 views

How to find the distribution of data with data fitting

I need to do data fitting to find the distribution of given data. I need to find the PDF function of the distribution. I can use data fitting functions in MATLAB and Python. It looks like a ...
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0answers
35 views

How many samples are needed to estimate a p-dimensional covariance matrix?

In general, how many points are needed to estimate a p-dimensional covariance matrix? Does it depend on how the data are spread out across the different dimensions? Does it depend on the true ...
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0answers
23 views

Geometric mean of uniform variables

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
2
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0answers
10 views

Geometric Mean of Uniform random variables convergence [duplicate]

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
3
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1answer
37 views

Variance of absolute value of a rv

Suppose that $X \backsim iid (\mu, \sigma^2)$. We are interested in $E (|X|)$ and ${\rm Var}(|X|)$. Can you suggest a way to proceed? I thought of rewriting $|X|$ as : $|X| = Xd - X(1-d)$, ...
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0answers
9 views

Fidelity of measurement using conditional probabilities

Let me begin by saying that I'm not entirely sure if this is the correct forum, or if Mathematics would be more suitable. The problem I'm about to describe is statistical in nature, so I suppose it ...
4
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1answer
113 views

Variance of Z for Z = X + Y, when X and Y correlated

So I'm trying to show that ${\rm Var}(Z) \le 2({\rm Var}(X)+{\rm Var}(Y))$ for $Z = X + Y$. This seems to be pretty easy to show given that $X$ and $Y$ are uncorrelated. But I'm running into trouble ...
0
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1answer
56 views

Expectation / minimizing variance of weighted sum of means

(I'm assuming the second $\bar x$ should be a $\bar y$), but I'm mostly confused how to solve this problem because it seems like since $\bar x$ and $\bar y$ are values, not random variables, $W$ is ...
2
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1answer
111 views

Confidence Interval for a Random Sample Selected from Gamma Distribution

Working on a homework question and having some trouble... Any help would be greatly appreciated. Based on a sample 1.23, 0.36, 2.13, 0.91, 0.16, 0.12 from the GAM$(2,\theta)$ distribution, find an ...
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0answers
16 views

Comparing two distributions using Generalized method of moment estimation

If I consider a data X, I want to fit a distribution Y which has a density f(x) but It's not possible to express the moments of Y as function of the parameters. My problem is : 1- to estimate the ...
3
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1answer
41 views

Why the trees generated via bagging are identically distributed?

I have problem in intuitive understanding of following arguement: "The trees generated via bagging are identically distributed, thus the expectation of the average of a set of trees is the same as ...
2
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2answers
179 views

How to create ROC curve to assess the performance of regression models?

I knew that, ROC curve are use to assess the performance of classifiers. But is it possible to generate ROC curve for the regression model? If yes, How?
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2answers
94 views

Suppose $X_1,…,X_n$ are iid from $N(\mu,\sigma^2)$, how can I find $P(T(X)>200)$?

Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ with $\mu$ unknown and $\sigma = 2$ known. Letting $T(X) = \sum{(X_i-\mu)^2}$ from i = 1 to 40, how can I use the central limit theorem to ...
2
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1answer
47 views

questions regarding the order statistics of a normal distribution

Assume $x_1, x_2, ..., x_n$ are realizations from a normal distribution with mean $0$ and standard deviation $1$ and assume they are ordered from small to large. Let $y_i=x_i-x_{i-1}$ be the ...
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0answers
30 views

Restriction on Cholesky decomposition [closed]

In my model I'm using a Cholesky decomposition to produce a certain covariance structure of a vector of normal variables: $\Omega=LL'$ Where L is the lower triangular Cholesky factor and $\Omega$ is ...