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

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21 views

can we generate a random words from english latters that follow the bigram of 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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0answers
12 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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1answer
26 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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1answer
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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17 views

What will be the pdf of 2D gaussian? [on hold]

If $(X_i,Y_i)$ are iid sampled from 2D gaussian with $N([0;0],[1, θ; θ, 1])$ $\theta$ $\epsilon$ $\Omega(-1,1)$; Find 2d minimal statistic?
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26 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 [on hold]

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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1answer
40 views

I got a dataset, there are about 400 variables,most of them are quite similar [on hold]

I got a dataset, there are about 400 variables, most of the variables are quite similar. Actually, I would like to say their number are pretty close, but I think they won't too correlated, since the ...
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2answers
113 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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0answers
11 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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0answers
49 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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27 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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1answer
41 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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0answers
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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2answers
72 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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0answers
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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0answers
16 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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0answers
11 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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0answers
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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0answers
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
20 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
34 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
86 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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0answers
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. ...
2
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1answer
49 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 ...
2
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1answer
142 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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2answers
60 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 ...
3
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1answer
39 views

Use law of total variance to find unconditional variance of overdispersed Poisson?

First, I need to prove that the distribution of a RV X, where X|lambda ~ Pois(lambda), and lambda ~ gamma(a, B), is a negative binomial. I know that it is, but why negative binomial instead of another ...
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0answers
31 views

How to derive this problem with soft-thresholding method?

The problem is defined as $$ \min_{x} \Bigg\{ a{\|x\|}^2+\frac{b}{2}{\|x-c\|}^2 \Bigg\} $$ where $x\in R^{n \times 1}, c \in R^{n \times 1}$ and $a,b$ are scalars. Equations 2.5 to 2.8 of this paper ...
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31 views

General theoretical properties of empirical Bayes estimates

I was wondering if someone could provide reference (if such exists) for the theoretical properties of empirical Bayes(EB) point estimates, in the sense of what can we say about their risk under ...
6
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1answer
123 views

Derivative of order statistics

--For further background into the question, one can refer to equations 2.3 and 2.6 (page 1275 of [0])-- Define: $$g_G(x)=\mbox{med}_Y|x-Y|$$ where $X$ and $Y$ are independent stochastic variables ...
4
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0answers
47 views

Inverse covariance matrix, off-diagonal entries

Let $\Sigma$ be a covariance matrix. According to the material in this link, If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative ...
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1answer
40 views

ANOVA Theorem Explanation [duplicate]

I am trying to figure out why the following holds: Given $y_{i}=E[y_{i}|X_{i}]+\epsilon_{i}$ that $E[\epsilon^{2}_{i}] =E[E[\epsilon^{2}_{i}|X_{i}]] = E[V[y_{i}|X_{i}]]$ Specifically I am trying ...
2
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1answer
50 views

Consistency of OLS in presence of deterministic trend

For consistency of OLS estimator for linear model $$ y_i = \beta^T x_i + \epsilon_i, \; i = 1,\cdots, n, $$ the model assumptions are usually (the ones I am familiar with) The sequence of random ...
13
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4answers
234 views

What is the intuition behind the independence of $X_2-X_1$ and $X_1+X_2$, $X_i \sim N(0,1)$?

I was hoping someone could propose an argument explaining why the random variables $Y_1=X_2-X_1$ and $Y_2=X_1+X_2$, $X_i$ having the standard normal distribution, are statistically independent. The ...
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1answer
33 views

Hodrick-Prescott derivation in lay terms

I am currently working with the Hodrick-Prescott filter. I would like to understand the equation in lay terms.
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7answers
1k views

MLE in layman terms

Could anyone explain to me in detail about maximum likelihood estimation (MLE) in layman's terms? I would like to know the underlying concept before going into mathematical derivation or equation.
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1answer
54 views

Explanation of density rewriting?

Can somebody please explain the math behind this statement to me? I am not sure how they represent the left hand side by that integral and finally how it is proportional to that. \begin{align} ...
4
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1answer
83 views

Closed form expression for count of “runs” in binary sequence sharing same length, number of 1's, and location of final 1

I am struggling with the following combinatorial problem related to research I am doing. Take a binary sequence $(y_1, y_2, \ldots, y_n)$ of length $n$ with $x$ $1$'s, where the final $1$ is in ...
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0answers
11 views

Sample size for Cluster sample

I'veen asked to claculate a necessary sample size and I tought the best way to reduce costs and time is using cluster samplig protocol but I have few information and I don't have much experience with ...
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3answers
112 views

testing logistic regression coefficients using $t$ and residual deviance degrees of freedom

Summary: Is there any statistical theory to support the use of the $t$-distribution (with degrees of freedom based on the residual deviance) for tests of logistic regression coefficients, rather than ...
2
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1answer
19 views

Is Outlier detection in two separate databases is equal to one combined database?

Suppose that we have two databases : Database_1 and Database_2 . Database_1 has 300 samples ...
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0answers
40 views

How to get correlation between two categorical variable and a categorical variable and continuous variable?

I am building a regression model and I need to calculate the below to check for correlations Correlation between 2 Multi level categorical variables or 2 binary variables Correlation between a Multi ...
3
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2answers
106 views

Normalization to non-degenerate distribution

I am reading de Haan's Extreme Value Theory (2006). In the discussion of distribution of sample maximum, he said "in order to obtain a non-degenerate limit distribution, a normalization is necessary". ...
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41 views

Can a very bad Coefficient of determination ($R^{2}$) not be indicative of model performance?

Thanks in advance for the advice. I am trying to build a generalized linear model that has many predictors. The $R^{2}$ value of the model is quite low (.21), but when I use the model to predict ...
6
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1answer
115 views

Comparison between MAD and SD

I am reading Huber's Robust Statistics (2nd). On page 2 and 3 he gave an example. The basic facts are summarized here. Let $(X_n)$ be a sequence of random variables and define two measures of spread ...
3
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1answer
45 views

Ancillary statistic: $X_i \sim N(\theta, \theta^2)$

Let $X_1, X_2, ... , X_n$ i.i.d random variables with probability density function $N(\theta, \theta^2)$. Show that $$T(X) = \frac{X_{(1)}-X_{(n)}}{X_{(2)}-X_{(n)}}$$ is ancillary to $ \theta$. My ...
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1answer
34 views

Reconciling two definitions of 'uncorrelatedness'

In this paper, the authors defined uncorrelatedness in the following way: Let $\mathbf{X}=(X_1,...,X_n)$, and $\mathbf{Y}=(Y_1,...,Y_n)$, where $X_i\sim X$ and $Y_i\sim Y$. $\mathbf{X},\mathbf{Y}$ ...