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

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Observed Fisher information under a transformation

From "In All Likelihood" by Pawitan, the likelihood of a re-parameterization $\theta\mapsto g(\theta)=\psi$ is defined as $$ L^*(\psi)=\max_{\{\theta:g(\theta)=\psi\}} L(\theta) $$ so that if $g$ is ...
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Is the negative binomial not expressible as in the exponential family if there are 2 unknowns?

I had a homework assignment to express the negative binomial distribution as an exponential family of distributions given that the dispersion parameter was a known constant. This was fairly easy, but ...
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25 views

Snow tire tread problem

A local tire dealer wants to offer a refunds without taking too big of a risk. The dealer is willing to refund no more than 4% of customers for what mileage can he guarantee these tires to last? The ...
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1answer
17 views

Multiplicity Adjustment for Correlation Coefficient Confidence Intervals

Has anyone dealt with the issue of computing confidence intervals for the correlation coefficient (parametric or non-parametric) and having to include a multiplicity adjustment factor? I was ...
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24 views

ranking based on lower confidence interval

I have a database of bridge scores from a local bridge club that effectively contains for this question, three fields: name, ...
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21 views

Approximate Probability Distribution Function

I am trying to approximate a large discreet probability distribution function using a histogram with a small number of entries. I.e., create a piece-wise first-order polynomial approximation for a ...
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6 views

SLR Residual Plots vs predictor or fitted values?

In our regression class, the professor said we can either plot the residuals vs the predictor values or vs the fitted values. I asked if there was a difference in the two plots (i.e. might you be able ...
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18 views

Finite Population Variance for a Changing Population

How does the addition of one unit affect the population variance of a finite population if everything else remains unchanged? What are the conditions such that the new unit leaves the variance ...
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31 views

Upper bound for asymmetry using skewness

Can the following quantity be upper bounded by the (standardized third moment) Skewness of $X$ ($\mu_3/\sigma^3$)? $$\left|\mathbb{P}(X \geq \mathbb{E}X) - \mathbb{P}(X \leq \mathbb{E}X)\right|$$ I ...
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6 views

Question on the estimation of a complex matrix

I would like to find a complex matrix C from (huge) set of the data { y }, where every complex-scalar y can be described by a following model which includes the matrix C (whose size is say M x M), y ...
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142 views

Proving the LATE Theorem of Angrist and Imbens 1994

Assume we have a binary instrument $Z_i$ which can be used to estimate the effect of the endogenous variable $D_i$ on the outcome $Y_i$. Suppose the instrument has a significant first stage, it is ...
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42 views

Given N samples from a population, can you determine or approximate the probability of the N+1 sample being within a certain range?

The question is as the title says: is it possible for me to determine the probability of the next sample ending up in a certain range? For example, lets say I pick N grains of sand from a bucket and ...
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21 views

How to calculate multicollinearity of binary variable with other predictors in regression model?

VIF can be used to calculate multicollinearity of continuous variable in regression models. But VIF will only work for continuous variables because this is calculated by running a linear regression ...
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70 views

Can the power of a test of an inconsistent estimator still go to 1 as N goes to infinity?

I'm reading a comment to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the power of a likelihood ratio or ...
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1answer
26 views

Why constrain mean and standard deviation when proving Gaussian is maximum differential entropy pdf?

I'm reading Bishop's Pattern Recognition and Machine Learning. In chapter 1.6: Information Theory (page 53) when trying to derive the maximum differential entropy pdf from the definition of continuous ...
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1answer
24 views

Regression coefficients for correlated output variables

Let $\mathbf{y_1, y_2}$ be standardized $N \times 1$ column vectors, such that $\mathbb{E}\mathbf{[y_1]}=\mathbb{E}\mathbf{[y_2]}=0$, $\text{Var}\mathbf{[y_1]}=\text{Var}\mathbf{[y_2]}=1$. ...
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18 views

Determine a distribution of a gaussian stochastic with different time

I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of $y=x(t_1)-x(t_2)$ is also a Gaussian stochastic with ...
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14 views

getting posterior probability of the hypothesis of hypothesis

At any given time about 5.5% of women (age 15-45) are pregnant. A home pregnancy test is accurate 99% of the time if the woman taking the test is actually pregnant and 99.5% accurate if the woman is ...
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31 views

how to get theoretical center of distribution and theoretical variance of the distribution

Here for 1000 simulations and 40 samples for each, here is random exponential distributor using replicate function ...
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4 views

How many 3 digit even numbers can be formed using (0, 1, 2, 3, 4) and no repetition? [migrated]

My solution to the problem is as follows: The answer I get is 27. My reasoning is that the last digit must be even, so for that position there are 3 choose 1 possibilities. Then the first digit ...
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19 views

Orthogonal projection of a vector which is already orthogonal to part of the basis

Context This question emerged from trying to solve problem 5.1. of Wooldridge, Econometric Analysis of Cross Section and Panel Data. The problem asks to show the equivalence of the estimators ...
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14 views

get a list of predicted variables based on rank

I have a data of device, location, age etc of some users, along with keywords that they clicked (or not). Based on this I want to build a model that predicts a list of keywords for test set that ...
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18 views

Mathematics of simple performance testing

I have a set of sorted tables T that have known but different dimensions. There are two types of functions in this system: f(T) g(T, n), where n is an integer parameter. ... and two types of costs ...
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47 views

can we generate a random words from English letters that follow the bigram of the English language

The main issue is that several research building their solution of detecting and classifying English language is based on bigram distribution. However, I would like to know if it possible to generate ...
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25 views

Confusion about nonlinear transformation of gamma and inverse-gamma distributions

I have a question about the variance of a transformed random variable, illustrated by a particular example. Let $X_1, ..., X_n$ and $Y_1, ..., Y_n$ be independent random variables drawn from an ...
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31 views

Working with few data examples

I have been asked often in some interview, that how we should proceed when we have less data examples(say 50 or 100). What considerations needs to be made while choosing any algorithm. few points ...
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15 views

How can I measure the effect of more than one continuous variable on one dichotomous variable?

I have one dichotomous variable as "success" which I valued as pass (1) and fail (0). I have 3 continuous variables as "study hour", "attendance" and "classroom participation". I want to investigate ...
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29 views

Question about Lagrangian Multiplier (Gradient) Statistic of constrained GMM

I am trying to derive the Lagrangian multiplier statistic (GMM version) under a restriction. The question is given below The quadratic form is given by $Q_n(\theta,\alpha)=[m(\theta)', ...
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26 views

Unknown formula [closed]

I came across a formula presented in a financial context, and have been trying unsuccessfully to try a work out what it means. $SL = 2 * 100 * \sqrt{V_L + (\alpha + \beta)^k \sigma^2_{n - V_L}}$. I ...
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118 views

Calculate $\mathbb{E}[Z_i]$ where $Z_i = \min(X_i, Y_{i-1})$, $X \sim Beta(\alpha,1)$

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim \mathrm{Beta}(\alpha,1)$. Let $Z_i = \min(X_i,Y_{i-1})$, $\forall i >1$. I would like to ...
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14 views

Can someone explain (in simplest words possible) minimum statistics, ancillary,completeness and MUVE

So, I understand what sufficient statistics is but I am unable to extend those ideas to minimal statistics, completeness, ancillary and eventually connect it with UMVU estimators? Can someone ...
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57 views

Problem solving a simple probability exercise

I'm currently studying a Statistics course and I'm trying to solve a simple probability exercise: A production facility employs: 20 workers on the day shift 15 workers on the swing shift 10 workers ...
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28 views

Multiple testing

I am regressing different dependent variables using same set of predictors for all dependent variables such as ...
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6 views

What is the meaning of “finite sample error control”?

I encountered this phrase while reading a paper which goes like this -- "These methods lack finite sample error control due to instability". Although it might not be important, the paper deals with a ...
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44 views

Does the log likelihood become unimodal when the sample size goes to infinity?

I know that, under the usual regularity conditions, the MLE converges to the true parameter values as the sample size gets large. And the scaled MLE tends to being normally distributed. However, in a ...
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24 views

Probability of input word

I have 2 text files containing certain sentences. I calculated the individual probabilities of every single word in both files. for file1: $$p(\text{A})=\frac{(\text{total occurrence of a word in ...
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74 views

Identifiability of normal distribution

I am working on an exercise problem and am stuck in this problem: Suppose that $X_1,\dots,X_n$ are independent with $X_i\sim\mathrm{N}(\alpha_i + \nu, \sigma^2)$. Let $\theta = (\alpha_1, . . . , ...
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39 views

Need help about a longitudinal study with eye tracking data

I am currently conducting a longitudinal research on learner reading development in second language. Let me explain it briefly. I am using eye tracking procedures and trying to describe the ...
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17 views

Are unbiased estimators based on complete statistics unique?

From Cassela and Berger's Statistical Inference, Theorem 7.5.1 (Lehmann-Scheff) Unbiased estimators based on complete sufficient statistics are unique. I wonder if the condition is not ...
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13 views

Calculating Stocks/Shares Odds

I am interested in the calculation of odds for stocks. For example, sports odds are like so: Turkey vs Ukraine hwin, draw, awin 2.20 3.40 3.20 2.20 for a ...
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36 views

Random forest - proof of convergence

I'm having some trouble understanding Leo Breiman's proof that the generalization error of a random forest converges as the number of trees increases (here's a link to the paper). At Appendix I he ...
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23 views

How many models I need?

I am doing an estimation using a bunch of sparse data. Suppose that I have a 100x100 grid and 20 data is available on this 2D grid. One solution is to use a determiastic method and estimate the other ...
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1answer
21 views

expected value of least squares parameters

I'm having some trouble with this equation from the least squares model. $$E[\parallel \mathcal {\hat {B}} \parallel ^2] = [E\parallel \mathcal {\hat {B}} \parallel] ^2 + trace ( \text {cov} ...
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1answer
37 views

Does the average of the square roots of random variables mean anything?

I recently made a plot for work that used a signed square-root scale on the $y$ axis, for visual clarity. The $y$ observations are impulse response functions (IRF) of vector autoregressions computed ...
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1answer
89 views

Can the measurable mapping in the definition of complete statistics depend on sample size?

The definition of complete statistics is from http://en.wikipedia.org/wiki/Completeness_(statistics)#Definition The statistic $s$ is said to be complete for the distribution of $X$ if for every ...
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11 views

Heterozygosity and the Wright-Fisher model

I was reading the textbook [Probability Models for DNA Sequence Evolution][1] by Durrett. In chapter 1, he discusses the Wright ...
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7 views

Definition of the deterministic annealing method

iI have ran into a shape matching problem and one term which I read about is deterministic annealing. I learnt that it would help to convert discrete problems, e.g. ...
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1answer
51 views

Unbiased estimator for $P(X_1=1)$

If $ X_1, ... ,X_n$ are IID binomial with parameters $ n$ and $p, $ find an unbiased estimator for $$G(p)=P(X_1=1)=np(1-p)^{n-1}\, .$$ I need to find this estimator so I can apply Lehmann-Scheffé ...
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1answer
145 views

Covariance of a random vector after a linear transformation

If $\mathbf {Z}$ is random vector and $A$ is a fixed matrix, could someone explain why $$\mathrm{cov}[A \mathbf {Z}]= A \mathrm{cov}[\mathbf {Z}]A^\top.$$
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67 views

Obtaining an estimator via Rao-Blackwell theorem

Let $X_1,...,X_n$ be iid with pdf $$f(x|\theta) = exp(\theta -x) I(x)_{(\theta, \infty)}$$ It is asked to find an unbiased estimator for$ \theta $ that is function of a sufficient statistical for ...