Questions tagged [mathematical-statistics]

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

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5 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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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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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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A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in [on hold]

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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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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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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Make exclusive class intervals from given class intervals [on hold]

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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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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Determine the highest values in a set [on hold]

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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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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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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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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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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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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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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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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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 ...
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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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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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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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Better algorithmic solution to find the peaks of multiple number series [closed]

Suppose I have set of number series which can be drawn a multiple line graph like this: According to what I have learned, peaks can be found using the theory of local maxima. And I found that the '...
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1answer
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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
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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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Dealing with NAs in Poisson Regression estimators [duplicate]

I am trying to fit a Poisson regression in some soccer matches. I want to be able to predict matches of the first league for a new season, which means that there will be some new teams that have been ...
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Variance of Guassian Products [duplicate]

Suppose I have to vectors $w$ and $x$, each of size $[512,1]$. Each element of $w$ and $x$ is an i.i.d sample from a Guassian Distibution with mean 0 and variance 1. So $x_i$ and $w_i$ follow $N(0,1)$...
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How can I model this?

I have a question about whether I can model in a way that solves this problem: Suppose a swimming coach has 100 athletes and only cares about the distance they can each swim in 5 minutes. From this, ...
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Statistics books with applications in quantitative finance

I am a Pure Maths PhD. As I would like to venture myself into quantitative finance jobs, I need to pick up statistics, programming and finance. I have been reading time series analysis by Hamilton ...
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1answer
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Time Origin in Survival Analysis

I am new to survival analysis, and I am confused about time origin, or what's called calendar data format and survival data format. I have the following dataset, subjects were continuously entering ...
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1answer
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Variance of mean of correlated variables

Page 228 of THIS BOOK provides the formula for the variance of the mean of more than two correlated random variables: where $m$ is the number of variables, $r$ is the correlation between the ...
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1answer
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Variance of difference of two correlated variables when working with random samples of each

I know that the variance of the difference of two correlated variables, $Y_1$ and $Y_2$ , is what the below formula shows, requiring $r$ the correlation between the two variables. But suppose I have ...
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Bias of Pearson correlation estimator of two Bernoulli variables

Crossposting link: https://math.stackexchange.com/questions/3312349/bias-of-pearson-correlation-estimator-of-two-bernoulli-variables Suppose we have two correlated Bernoulli random variables, $X_j$ ...
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What would be the outliers in this specific situation? (given in description)

Recently over a phone interview I was asked this: "I have data consisting of 102 points between 0 and 100 where the first 50 points, starting at 0, are incremented by 0.5 every time. The next two ...
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Why we really need the concept of “Local” Rademacher complexity?

Recently, I have been studying High-Dimensional Statistics: A Non-Asymptotic Viewpoint written by Martin J. Wainwright. In this book, the author uses a special complexity measure which is called Local ...
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1answer
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How to get a unified result on multiple results of a A/B test

I have a A/B test. this test will generate a P_value every day based on the daily data. For example, I had 5 days' significate and 5 days' non-significate. So if I want a unified result from these 10 ...
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Approximate Value / Policy Iteration (Reinforcement Learning)

I am reading Markov Decision Processes in AI : about Approximate Dynamic Programming. Would you like to explain the rationale for introducing the API algorithm, how it compares to AVI ? How would ...
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1answer
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Jurafsky and Martin (2018) Do not understand formula in naiye bayes classifier

Currently I am reading Language and Speech Processing by , Chapter 4 Naiye Bayes and Sentiment Classification. At page $7,$ when the authors discuss worked example. Data set is as follows: Training ...
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1answer
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Proof Verification: Joint variance of the product of a random matrix with a random vector

BACKGROUND QUESTIONS Is the proof of my claim correct? How might my proof be improved? Claim: (1) The joint-covariance matrix of the product of a real random matrix $X$ of dimension $v\times m$...
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How to defend a probability distribution [closed]

Apologies if this isn't the right place to ask this question. While I was an undergrad, I derived a probability distribution. I tried to get this published along with a professor, but she mentioned to ...
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1answer
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How to compare two, time based, measurements of the same object

I have 2 rounds of measurements from a set of 10 capacitors. The measurements are with respect to frequency and are all measured at the same 200 frequencies between 1 kHz and 5 MHz. The data is not ...
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2answers
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How to get the false positive? “1 - true negative” or “1 - true positive”?

wiki gives this Drug testing Example to illustrate Bayes' theorem ${\displaystyle {\begin{aligned}P({\text{User}}\mid {\text{+}})&={\frac {P({\text{+}}\mid {\text{User}})P({\text{User}})}{P(+)}}\\...
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1answer
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What are the % Odds?

NEED SOMEONE GOOD IN MATH! My father, my brother & myself all have AB- Blood Type. According to The Stanford School of Medicine Blood Center around 0.6% of the population has the ABneg blood type....
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1answer
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Determinants of the sum of two matrices and their eigenvalues

I know some basic property of determinant. I read an article and see this formula: \begin{equation} |(\delta-p-1)D+S|=\bigg(\prod_{i=1}^{p}\lambda_{i}(D)\bigg) \bigg(\prod_{i=1}^{p}(\delta-p-1)\...
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1answer
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What is the conceptual difference between Residual Sum of Squares (RSS) and Residual Standard Error (RSE)?

I am very new to statistics and am currently using the Introduction to Statistical Learning with R book. I am having some issues understanding the difference between RSS and RSE. From my ...
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What does beta distribution “has support” mean?

This post says the beta prior has support over all valid probabilities and only over valid probabilities About Support, wiki says In mathematics, the ...
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1answer
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Probability statistics - probability that 10 or more of the donors are Rh-negative?

In a certain population, 18% of the people have Rh-negative blood. A blood bank serving this population receives 95 blood donors on a particular day. So my teacher posted a solution to a problem ...
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Does Wolfram Mathworld make a mistake describing a discrete probability distribution with a probability density function?

Usually a probability distribution over discrete variables is described using a probability mass function (PMF): When working with continuous random variables, we describe probability distributions ...
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Can anyone give a concrete example to illustrate what is an uniform prior?

I googled "uniform prior" and got a link to Prior probability, which uses the term without an explanation or a definition. anther link is to quora, which does not give an concrete example. Can ...
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what is the name of this computation?$e_{10}{(0.1)} = 0.8$

section 1.4.3 of the book "Machine Learning - A Probabilistic Perspective" gives an example about KNN: the input is two dimensional, we have three classes, and K = 10 which contains a computation $...
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1answer
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Variance of beta distribution (fastest way)

Suppose a Random variable $X \sim \mathrm{Beta}(a,b)$ Find the $\mathrm{Var}( \frac{X}{1-X} ) $ My initial approach is to calculate $\mathrm{E}( \frac{X}{1-X} ) $ and $\mathrm{E}( [\frac{X}{1-X}]^...
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Question on a proof in Bickel's covariance estimation paper

Recently, I am reading Peter J. Bickel's paper Regularized Estimation of Large Covariance Matrices. In that paper, the author tries to prove lemma A.3, which goes as follows: Let $Z_{i}$ be i.i.d. $\...