# Questions tagged [mathematical-statistics]

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

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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,...
1 vote
12 views

### 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 ...
13 views

### 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 ...
13 views

### 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 ...
102 views

### 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 ...
1 vote
78 views

### 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 &...
1 vote
43 views

### 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 $p$ ...
58 views

### 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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### 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 vote
44 views

### 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 ...
1 vote
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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 ...
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 \... 1 vote 0 answers 28 views ### 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 ... 3 votes 2 answers 276 views ### 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. 2 votes 1 answer 78 views ### 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 ... 9 votes 4 answers 1k views ### 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 69 views ### 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 ... 1 vote 0 answers 16 views ### 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 ... 2 votes 1 answer 73 views ### 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? 2 votes 2 answers 135 views ### 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 ... 1 vote 0 answers 39 views ### 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 votes 1 answer 116 views ### 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)=\... 1 vote 0 answers 25 views ### 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 ... 2 votes 0 answers 66 views ### 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 ... 2 votes 2 answers 117 views ### 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(\... 0 votes 0 answers 4 views ### 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. ... 2 votes 1 answer 31 views ### 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} $$where \... 6 votes 2 answers 286 views ### Are these two definitions of the coefficient of determination R^2 equal? I want to do multiple linear regression as explained on this Wikipedia site: I am given the following data:$$ yx=(~(y_1,x_{11},\ldots,x_{1p}),\ldots, (y_n,x_{n1},\ldots,x_{np})~) $$of n-many ... 0 votes 0 answers 36 views ### Conditions for existence of KL divergence and unique minimum Consider two probability density function g(y) and f(y: \theta), \theta \in \Theta. The KL divergence of f and g is defined by$$ D_{KL}(g|f) := \int \log \frac{g(y)}{f(y: \theta)} \, dy = \...
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}^...