Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
7,699
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Hierarchical forecasting - demand classification required for prediction?
I have product sales data for which I would like to predict what will be the sales for each product at the product level, product store level, product store and region level etc.
To solve this problem,...
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0
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Hierarchical forecasting packages - No trend removal or stationary check?
I have sales data for which I would like to predict what will be the sales for each product at the product level, product store level, product store and region level etc.
To solve this problem, I came ...
- 3,008
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0
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13
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How can I compare model performance across datasets of varying sizes?
I have a person wearing 2 sensors. I create two models, one using Sensor-1 and other using Sensor-2 data
I have multiple people repeating the same experiment with varying numbers. How do I a ...
- 31
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0
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13
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CO2 impact and offset by tree planting calculations: inference and data [migrated]
Suppose CO2 emissions are such that factory growth is on the increase, and trees, are either in slow increase or decrease.
Suppose, this occurs at a global level, and CO2 flows easily between nations ...
6
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1
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Why are complete statistics named "complete"?
I get why sufficient statistics are named "sufficient", but what about "complete" statistics?
I have this definition from F.J. Samaniego, Stochastic Modelling and Statistical ...
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1
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Statistics Inference Question: "Prob(An equation) = 1" compared with "The equation holds" [duplicate]
When I study the textbook Statistical Inference by Casella and Berger, I have often seen expressions in the form of P(an equation) = 1. However, some other textbooks or lecture notes will instead say &...
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1
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0
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43
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Why is the variance smaller for the same coefficient in a reduced regression model vs. full regression model?
Let's say we have two estimators for $\beta$.
$\beta$ denotes all a full set of coefficients, one for each covariate in a dataframe.
$\beta$ can be split into $\beta_p$ and $\beta_r$, where $p$ ...
0
votes
1
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58
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Unbiased estimator for parameter of random variables following a uniform distribution [duplicate]
Suppose $X_i$ are i.i.d. and have density $f_\theta(x) = \frac{1}{\theta}$ if $x \in (\theta, 2\theta)$ for positive $\theta$.
$(\min_iX_i, \max_iX_i)$ is a sufficient statistic for $\theta$?
To ...
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0
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3
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Loan Data: Bucket recoveries 1-D array
Some context: When someone defaults on their loan, we keep track of the recoveries as a percentage of the exposure (loan amount), we have a limited time T (legally) to collect recoveries, those ...
1
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1
answer
44
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Adaptive sample size for determining the significance of difference between two non-parametric distributions
I have two non-parametric distributions, A and B, and I would like to determine if they are significantly different (say, by the p-value of a U-test). Normally, I would sample $n=1000$ from ...
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Convergence in probability and boundness in probability with respect to sample mean and sample variance
This is a question about the convergence in probability and boundness in probability.
Suppose $X_i \overset{\textrm{i.i.d.}}{\sim} (\mu, \sigma^2 )$ for $i=1,2, \cdots, n$.
Denote $\overline{X}$ and $\...
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1
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Please can someone explain the notation of this multivariate Taylor expansion?
Kamanzi-wa-Binyavanga, 2009, wrote the following paper, Calculating Cumulants of a Taylor
Expansion of a Multivariate Function:
What I am confused about, is how precise the notation. I understand ...
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2
votes
2
answers
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How to understand that there are n - 1 degrees of freedom in calculating sample variance? [duplicate]
According to Wiki:
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.[1] Estimates of statistical parameters can be ...
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0
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1
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Survival Analysis: Treatment not assigned at the beginning of the observation period
I am new to the field of survival analysis and looking for recommendations on how to deal with the following scenario:
I have data on patients suffering from a certain disease. Some patients receive ...
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1
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Consider $X_1, \cdots, X_n$ which are iid and have common pdf $f(x | \theta) = e^{-x+\theta}$ $(x > \theta)$
Consider $X_1, \cdots, X_n$ which are iid and have common pdf $f(x | \theta) = e^{-x+\theta}$ $(x > \theta)$. Why is $$\frac{f(x|\theta)}{f(y|\theta)} = \exp \left\{{\sum_{i=1}^{n} (y_i-x_i)}\right\...
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calculate joint cdf from joint pdf, joint pdf of (x,y) is 2*e^(-x)*e^(-y) with domain 0<x<y<inf [closed]
I have already calculated the first part which is 2, but i have no idea how to calculate the joint cdf from joint pdf, my professor gives solution like this , but i am still confused about it.
in the ...
0
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1
answer
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how can I evaluate the unbalanced data set
The output here shows the titles and descriptions of the comments written for the evaluation of the top 100 books in amazon with nltk vader, and the total reviewer rating for those analyzes, but there ...
1
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1
answer
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What is the dual function of this non-overlapping Group Lasso's penalty?
I'm trying to find the dual function of this function (non-overlapping Group Lasso's penalty function):
$$ \mathfrak{h}: \mathbb{R}^p \to [0,\infty], \ a \mapsto \sum_{j=1}^{k} \left\| a_{\mathscr{A}...
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0
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Is There a Standard Metric for Evaluating Treatment Impact Considering Action Cost in Uplift Models?
I'm currently exploring Uplift modeling, specifically the use of the Conditional Average Treatment Effect (CATE) metric:
$$ \tau(t', t, x) := \mathbb{E}[Y | X=x, T=t'] - \mathbb{E}[Y | X=x, T=t] $$
...
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What is a good journal for submitting my article on a conjecture in theoretical statistics, re: ancillary complement for correlation?
I'm working on a draft of a statistics article, and I'd like to plan for the journal where I'll ultimately submit. My problem is, the article topic is somewhat abstract—it's a conjecture in ...
4
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1
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154
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Regression coefficient on a triangle using geometry
I am encountering a question as follows:
Let $X, Y$ be two independent uniform random variable on $(0,1)$. We consider the regression model $Y = \beta_1 X + \beta_0$, given the restriction that $X + Y ...
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2
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490
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Frobenius norm of a product of Gaussian matrices
Suppose $$C_n=X_1 X_2\cdots X_n,$$ where $X_i$ is $d\times d$ matrix with IID entries normally distributed with mean 0 and variance $\frac{1}{d}$.
The following appears to be true for large $d$, why?
$...
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Given conjugate prior and posterior distributions, what is the PRIOR predictive distribution? [closed]
I am doing an assignment on my statistics class. We had 1 lecture about bayesian parameter estimation, where we were taught about the following formula (and it's discrete form, if $h(\theta)$ was ...
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2
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1
answer
232
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Is the cumulative distribution function of a r.v. X strictly increasing (X -) almost everywhere?
Let $X$ be a random variable and $F_X(x) = P(X \le x)$ its cumulative distribution function (cdf). $P_X$ is the probability measure induced by $X$, which is defined by $P_X((a,b)) = P(X^{-1}((a,b))$ ...
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0
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Repeated Sampling and Confidence Interval Theory
I thought I'd ask a fairly fundamental question regarding confidence intervals at the risk of potentially furious backlash from the stats.stackexchange community.
However, I've never quite yet found a ...
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What if regressor self correlated
I am wondering if the self-correlationess in X of the linear regression will impact the estimation or the t-statistics of the coefficient? For example, we have a linear regression model Y(t) = a1 + a2*...
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Posterior Predictive Distibution [closed]
How do we actually calculate (what are the operations that need to be done) the posterior predictive given a vector of observations; can we do away with the assumption of independence?
Let's say we ...
3
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1
answer
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Prove that the Deviance and the Generalised Pearson Statistic are asymptotically equivalent
I am reading the paper Exponential Dispersion Models from Jørgesen and at page $137$ I have encountered a claim that I don't know how to prove.
The author claims that the Generalised Pearson Statistic,...
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transformation of uniform random variables
Let $U_1, U_2,...,U_n$ be a sequence of independent random variables with Uniform distribution over the interval $(0, 1)$ and let $Y = -\frac{1}{\lambda} log(U_1)$ . what is the distribution of Y? i ...
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1
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Completeness of Gamma family
Let $X_1,...,X_n$ has a Gamma$(\alpha,\alpha)$ distribution. Find the minimal sufficient statistics. Is this a complete family?
My attempt: I found the Minimal sufficient statistics is $T(x)=(\...
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Are there margins such that, while the "correlation" parameters of a Gaussian copula are positive, the correlations between the margins are negative?
Let there be a multivariate distribution $F$ with margins $F_1,\dots,F_n$ and a Gaussian copula with "correlation" matrix $\Sigma$. Let the off-diagonal elements of $\sigma$ be positive.
Let ...
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Finding P-value and power of the Most Powerful Test
You observe a sample $X_1, \quad, X_{20}$ with the density
$$
f(x, \vartheta)=2\left(x / \vartheta^2\right) I_{[0 \leq x<\vartheta]}
$$
with an unknown parameter $\vartheta>0$, yielding
$$
\min \...
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The variance of data set A is twice that of data set B; does this imply that the dispersion of data set A is twice that of data set B? [duplicate]
If the variance of data set A is 4, and the variance of data set B is 2, can we say that the data in set B is twice as dispersed as the data in set A? If the standard deviation of data set A is twice ...
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2
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Why do we need a consistency assumption in causal inference?
Why do we need a consistency assumption in causal inference? I think the consistency assumption is quite obvious and it is more like a definition for the observed outcome.
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2
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1
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Proof of Rejection Sampling: Flawed reasoning about continuous random variables
I recently studied Rejection Sampling as part of one of my University courses. When justifying why Rejection Sampling makes sense (to "prove", so to speak, that samples drawn using Rejection ...
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4
answers
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Does learning thorough statistical theory require learning analysis?
Does learning thorough statistical theory requires learning analysis before that?
I looked at the textbook for statistical theory. So far I don't know if analysis is required, but I think I have heard ...
1
vote
1
answer
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calculation of Kendall-Theil-Sen intercept
There are two approaches for estimating the Kendall-Theil-Sen intercept b0.
Approach 1
β0 = Median{yi - β1*xi : 1 ≤ i ≤ n}
Approach 2
β0 = Y_median - β1*X_median
where X_median is the median of the x ...
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0
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Covariance calculation in a DiD estimation
I am estimating a difference-in-difference model estimating the effect of a parental leave reform on female wages. It is not possible to take the logarithm of the varaibles as a lot of wages are 0.
I ...
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In linear models, why are we focused on BLUE rather than UMVUE?
It seems more natural to talk about UMVUE (following the basic statistical estimation theory). But when we turn to lm, we only care about BLUE, why? Are there any insurmountable difficulties here?
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Cramer-Rao bound for biased estimators
So the Cramer-Rao bound gives us a lower bound on the variance of an estimator, now if the estimator is unbiased then we have a bound on the mean square error. While I can see the utility of the bound ...
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Probability of Random variable less than a quantity containing random variable [closed]
What is the probability of the following: $P\left(Z_j>\frac{\epsilon}{a_j*R}\left(\sum^{M}_{m=j+1} ~ a_m*R*X_m+1\right)\right)$
where $Z_j$ and $X_m$ are independent and identically distributed ...
2
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1
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Conjecture: $\frac{1}{2}\frac{\mathbb{E} |X-X'|^p}{\mathbb{E} |X|^p}\leq1.$
Let $X$ be a real-valued random variable that is nonzero (that is, not identically zero). For any positive real number $p$ such that $\mathbb{E} |X|^p<\infty,$ define
\begin{eqnarray}
I_p(X)=\...
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0
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Show that the conditional variance $\{h_t\}$ of a GARCH process is an ARMA($m,p-1$) process
This is question 10.6(c) from Time Series and Forecasting (2nd Edition) by Brockwell and Davis. The parts (a) and (b) of this question can be found here in another SE question.
Question: Suppose that $...
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Asymptotic normality of the maximum likelihood estimator with dependent data
In the setup, assume $\left(\mathbb{R}, \mathscr{B}\left(\mathbb{R}\right), P\right)$ is the underlying probability space and suppose that $\left\{\mathcal{F_n}\right\}_{n\in \mathbb{N}}$ is a ...
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2
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Why is the asymptotic bias of the maximum likelihood estimate $b(\theta) = \frac{b_1(\theta)}{n}+\frac{b_2(\theta)}{n^2}+...$?
Firth (1993) states in his introduction that for a $p$-dimensional parameter $\theta$ the asymptotic bias of the maximum likelihood estimate $\hat{\theta}$ may be written as:
$b(\theta) = \frac{b_1(\...
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0
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4
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What type of rotational invariance information can I get from interest points (coordinates) of corners from an image?
Assume that you are using the FAST algoritm for corner/feature detection. You pick an image and run the FAST algorithm.
Question:
All these green dots are actully coordinates in x and y direction. ...
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votes
1
answer
31
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Is there a proof of the equivalence between Cronbach's alpha and reliability calculation from a CFA?
Given a uni-dimensional CFA, reliability can be calculated as:
$$
\omega = {\left(\sum_{i=1}^p \lambda_i\right)^2\over \left(\sum_{i=1}^p \lambda_i\right)^2 + \sum_{i=1}^p\sigma^2_i}
$$
where $\...
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votes
2
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286
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Are these two definitions of the coefficient of determination $R^2$ equal?
I want to do multiple linear regression as explained on this Wikipedia site: I am given the following data:
$$
yx=(~(y_1,x_{11},\ldots,x_{1p}),\ldots, (y_n,x_{n1},\ldots,x_{np})~)
$$
of $n$-many ...
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0
votes
0
answers
36
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Conditions for existence of KL divergence and unique minimum
Consider two probability density function g(y) and f(y: $\theta$), $\theta \in \Theta$.
The KL divergence of f and g is defined by
$$
D_{KL}(g|f) := \int \log \frac{g(y)}{f(y: \theta)} \, dy = \...
0
votes
1
answer
73
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Sufficient conditions for second-moment ergodicity
Let $(Y_t)_{t \in Z}$ be a covariance-stationary stochastic process. According to Hamilton (page 46-47), we say that the process is
Ergodic for the mean if
$$\overline{Y}\equiv \frac{1}{T}\sum_{t=1}^...
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