Questions tagged [mathematical-statistics]

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

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Correlations between two sequences of irrational numbers

If $x$ is an irrational number and $b$ an integer, let's define $g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$. Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. ...
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1answer
11 views

Linear Quadratic Regulator

from http://people.eecs.berkeley.edu/~somil/Papers/lqrlecture.pdf Why must the matrices be positive semidefinite? What is the input authority cost? What is the purpose of multiplying the transpose ...
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U~U(0,1) and Xt = sin(2*pi*u*t), t = 1,2,3,… Does Xt (t is a suffix of X) is strictly stationary, why or why not? [on hold]

The time series X_t = sin (2*piUt), where t=1,2,... and U follows uniform (0,1). Does X_t is strictly stationary?
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11 views

Reconstruct data of the past of a time serie

I'm trying to construct a time series model in which the water flow (caudal) is a function of temperature and precipitation. I want to forecast the results of the water flow. My problem is that I don'...
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1answer
25 views

How to calculate tf-idf for a single term

I am following the tf-idf method described in this paper: Measuring, Predicting and Visualizing Short-Term Change in Word Representation and Usage in VKontakte ...
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1answer
147 views

Normalization constant for uniform distribution over categorical distributions

Suppose we have a uniform distribution over all categorical distributions p for m categories, where the pdf has the form $$ f(x) = \left\{\begin{aligned} &c, && 0 \le p_i \le 1, i = 1, ......
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28 views

Intuitive explanation of scaled inverse chi-distribution

I am having a hard time understanding the scaled inverse chi-distribution. I looked for Wiki and other resources which are pretty math heavy without an intuitive ...
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106 views
+50

Non-negativity of interaction information for special trivariate case

Consider a discrete trivariate distribution $P(X_1, X_2, Y)$, which satisfies $$ p(x_1, x_2, y) = \min( p(x_1,y), p(x_2,y) ), $$ for all $x_1$ and $x_2$ for which $p(x_1, x_2) > 0$ and for all ...
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6 views

How do I apply the difference between two simplexes to a third simplex and have the result still sum to 1?

My question concerns simplexes, but I'll give an applied example instead. Imagine we are polling a national election. Three political parties compete. At the last election, the overall result was as ...
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1answer
149 views

Do random variables follow the same algebraic rules as ordinary numbers?

In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim ...
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1answer
38 views

what is statistics uncertainty on a quizz score?

Remark : I have created the question from scratch : it is not my teachers who asked me that. I'm wondering myself about a general problem of uncertainty. Let's consider a 'school' quizz with D ...
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61 views

What does it mean if the sobol main and total effects indices are the same?

What does it mean when the total and main effects ANOVA indices are the same? Does it mean there is zero interaction of the different inputs? Is there some other way to quantify or understand that? I ...
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80 views

Sufficient statistics - how can the conditional pdf not depend on $\theta$ when $\theta$ is in the equation?

I understand the heuristic definition: say you know a statistic, $T$, of some sample that you want to use to estimate the corresponding population parameter - but you don't know the data points of the ...
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1answer
87 views

How to measure the consistency of improvement on different conditions?

I want to measure whether the speed improvement of method 1 over method 2 is consistent on different conditions. Below are two examples of the speedup values of method 1 over method 2 on 5 conditions. ...
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2answers
157 views

Regression model when the dependent and independent variables show exponential distribution

As the Title suggests i am trying to figure out what would be the regression model to use when both the dependent and independent variables show an exponential distribution. Do I have to perform a ...
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1answer
9 views

Calculation of critical values

Are the critical values for the two-sample t-test with unequal variances calculated the same as for the two-sample z-test?
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2answers
326 views

Treating continuous variables

I attended a conference on ML and Data Science and I have a general question that was not answered in the conference. If we have a continuous variable, let's say age. What is the best way to handle ...
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29 views

Problem on sufficient statistics

Let the distribution of $X_1,X_2,...X_n$ depend on two parameters $a, b$ such that there exists a single sufficient statistic, for either parameter when the other is fixed/known. Show that there is ...
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176 views

Which likelihood function is used in linear regression?

When trying to derive the maximum likelihood estimation for a linear regression, We start by a likelihood function. Does it matter if we use either of these 2 forms? $P(y|x,w)$ $P(y,x|w)$ All pages ...
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A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in [closed]

A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in how long they remain in the body before they are excreted or metabolized. The pharmacologist ...
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1answer
1k views

Basu's Theorem Proof

I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below: Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
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1answer
38 views

A basic question about a randomized test involving the error type I

I have a basic question in the context of testing statistical hypotheses, more specifically, about randomized tests. Suposse that I have two actions (altenatives) about a certain unknown parameter $\...
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2answers
45 views

Asymptotic normality of random vector

If each component of a random vector is asymptotically normal by the central limit theorem (CLT), can anything be said about the asymptotic distribution of the random vector? The components are not ...
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14 views

What is the difference between lasso and WOE encoding in logistic regression?

I know lasso is one of the best method to select important variables and make variables sparse. But WOE encoding does the same thing, making variable smooth. I would like to know what is the ...
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2answers
1k views

Confused about the realizability assumption and equations of upper bound

I'm reading the the first chapter of Understanding machine learning from theory to algorithms and they said that: Let be the set of "bad" hypotheses, that is (e is the accuracy ...
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1answer
34 views

Minimizing an Integral of function by derivation

Currently I am reading ESL. I am struggling to unterstand the step 2.15 to 2.16 in Chapter 2.4. The authors mention that the stationary function can be found by differentiation. The commentary by ...
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22 views

Pooled variance versus pooled standard deviations squared?

Typically, one estimates the pooled variance using something like $$ \sigma^2 = \Sigma_i^n \ w_i \sigma_i^2 $$ where $w$ corresponds to some kind of weight. Note how we average variances here. ...
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1answer
139 views

Accuracy of empirical cumulative distribution function

I have a random variable with an unknown distribution and I want to find its cumulative distribution function. I sample the distribution $N$ times, with $$X_1, \dots, X_N$$ being iid random ...
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7 views

Make exclusive class intervals from given class intervals [closed]

Expenditure(Rs): 0-19 20-39 40-59 No of families : 4 8 12 Note :Be sure you don't make class interval from -0.5 as Expenditure is on Rs which cant be negative.
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2k views

Solution to German Tank Problem

Is there a formal mathematical proof that the solution to the German Tank Problem is a function of only the parameters k (number of observed samples) and m (maximum value among observed samples)? In ...
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12 views

how entropy works in continuous distribution using the K nearest neighbour?

So calculating entropy in continuos distribution was developed by Kozachenko and Leonenko (1987). I have seen one implementation here. however I have problem understanding the approach. basically, ...
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2answers
275 views

Product of Gamma by Beta rv

If $X$ has a beta distribution $ \beta(\alpha,b)$, $Y$ has a gamma distribution $\Gamma (K,\theta)$ and $X$ is independent of $Y$. What is the distribution of the product $P=XY$ . Thanks!
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1answer
27 views

Determine the highest values in a set [closed]

Let's say I have a set of 5 values: [0.1, 0.3, 1.2, 1.6, 2.3] What statistical technique could I use to return the highest values, say ...
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3answers
204 views

which distribution should be used in this question?

A basketball player succeeds in making a basket three tries out of four. How many times must he try for a basket in order to have greater than 0.99 probability of making at least one basket? In this ...
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10 views

How to define a good threshold based on a numbered dataset

I have a set of ordered number list as follows. 25789, 20500, 18888, 15000, 13200, ...., 118, 8, 1, 0 In my dataset most of the values are in the range of (1000-...
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1answer
184 views

If the VIF is 2 then what is the value of correlation coefficient $R^2$

If variance inflation factor is 2 what is the value of correlation coefficient $R^2$? $$VIF = \frac{1}{1-R^2}$$ Given $VIF =2$, then is this calculation correct? $$\begin{align} 2 &= \frac{1}{1-...
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1answer
18 views

Why does the intercept of Principal Component Regression differ from the orginal regression?

I am comparing two regressions: $y= \beta_0+ \beta_1 x_1 + \beta_2 x_2 + \epsilon $ and $ y = \gamma_0 + \gamma_1 PC_1 + \gamma_2 PC_2 +\eta $. The regressors in the second regression, $PC_1, ...
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1answer
769 views

Asymptotic Distribution of the Wald Test Statistic

I am trying to understand the asymptotic distribution of the Wald test statistic, specifically under the alternative hypothesis which I've found little reference to. For clarity, the binary ...
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13 views

Random variates generation for disrecte distribution by inverse transform method [duplicate]

I have solved this question but still confused. I don't know whether this method is correct or not and also the answer which I get is right or wrong, as I was told to generate random variates for ...
2
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1answer
35 views

Is this series divergent?

I have $$G_N = \sum_{i=1}^{N} \left\{ \frac{1-\pi_i}{\pi_i} + \frac{1-\pi_i}{T\pi^2_i}\right\} (y_i-\theta)^2=\sum_{i=1}^{N} V_i$$ where $2\le T\le 10$, $0\le \pi_i\le1$ and suppose $y_i\sim N(\theta,...
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33 views

How many classmates does a freshman have?

The freshmen at East China Normal University has just received their student ID. Let the last three digits of a student ID be ABC, then A is the class he is in, whereas BC is his number in the class. ...
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25 views

How to perform regression with different error variances? [duplicate]

I have two series of measurements values, first series is X and second is Y. I need to model Y as a function of X, where I know the method that was used to measure X is two times better then the ...
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1answer
24 views

What is a good measure of the similarity of 6 different time series?

Essentially, I have 6 different data time series that were each generated first using an industry standard methodology (call it method m.A) and then again using my technique (call it method m.B). ...
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2answers
302 views

Can the differential entropy be negative infinity?

Define the (differential) entropy for density $f$ as $$ H(f) :=-\int_{0}^{1} f(x) \log_{2}(f(x)) dx \, .$$ I am trying to find a Lebesgue measurable $f$ defined on $[0,1]$ such that $f\geq 0, \...
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1answer
45 views

Proof for how the drift estimator, for a random walk with drift, is unbiased?

Random walk with drift formula is: (Yt = α + Yt-1 + εt ) How do I go about checking that the drift estimator α-hat is unbiased.. which is proving that E(α-hat) = α? Is this something I would need ...
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3answers
225 views

Showing characteristic function is infinitely differentiable

Suppose that X is a real-valued random variable. Suppose that there exists a constant M > 0 such that the support of X lies entirely in the interval $[-M,M]$. Let $\phi$ denote the characteristic ...
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1answer
38 views

Mean residual life in terms of incomplete moments

In a paper by Braga, Cordeiro, and Ortega 2017, they representing the Mean Residual Life in Terms of the incomplete moments $$\operatorname{MRL}(t)= \frac{1-m_1(t)}{1-F(t)}-t$$ where $m_1(t)=\int_0^...
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1answer
17 views

Statistical test to check whether an item meets the specification

I have a data on emission of an automobile which a company manufactures. The emission test was conducted on roads near towns and villages and the data is as follows. According to a new rule passed ...
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1answer
155 views

Tr of matrix in Kullback-Leibler divergence PRML2.13

I've learned PRML recently but get stuck in a math problem. this exercise is to calculate $KL$ divergience of $2$ multi Gaussian $$ N(\mu,\Sigma)\ \ \ and\ \ \ N(m,L)$$ but I do not know how to ...