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

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10 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 ...
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0answers
9 views

What types of statistical analysis technique available to compare two different time series

I am currently looking for suggestion to compare or study the two different period time series like sales in 2000 and 2001
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0answers
36 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
20 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
49 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} ...
3
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1answer
73 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
6 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
90 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 ...
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1answer
18 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
18 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
85 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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30 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
109 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 ...
2
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1answer
39 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
33 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 $Y_i\sim Y$. $\mathbf{X},\mathbf{Y}$ ...
8
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2answers
225 views

What is the maximum likelihood estimate of the covariance of bivariate normal data when mean and variance are known?

Suppose we have a random sample from a bivariate normal distribution which has zeroes as means and ones as variances, so the only unknown parameter is the covariance. What is the MLE of the ...
2
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2answers
51 views

Which property of count data make mean-variance dependency?

I have read about the fact that, there is dependency of variance on mean of count data.In most of cases they do variance stabilization transfomration as preprocessing step of data modeling. I wonder, ...
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2answers
28 views

Tool for generating correlated data sets

Does anyone know of a tool that I can use to generate a set of data with known correlations (and to put the icing on the cake - output this in json,csv,txt or some common format)? I am working on ...
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0answers
15 views

Using odds ratio to evaluate a confusion matrix generated from a text classification

Can the concept of odds ratio be applied in the confusion matrix to measure the degree of association between responsive and non responsive documents labelled by the system and user? Should the terms ...
0
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0answers
8 views

Express Multinomial as vector sum of bernoulli trials?

So we know we can think of the binomial as a sum of iid bernoulli. Can we similarly express the multinomial as a vector sum of dependant bernoulli's and get the asymptotic distribution that way? I ...
2
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1answer
37 views

Show that a statistic is ancillary

Let $X_{i} \sim U(0, \theta) $ and $X=(X_1,\dots,X_n)$. Show that $$ \frac{X_{(1)}}{X_{(n)}}$$ Is ancillary for theta I coulxnt find a way of doing it that looks convenient. Any idea? P.s: ...
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0answers
59 views

Restriction matrix for a VAR

In New Introduction to Multiple Time Series Analysis by Luetkepohl (2005), section 5.2.1, it says that one can specify linear restraints for a VAR, $Y = \beta X + U$, in the form $$ ...
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3answers
44 views

What is the pre-requisite knowledge required to study econometrics? [closed]

I'm going into 3rd year and one of the modules I am currently planning to take is econometrics. However, since my degree is almost solely based on mathemtical modules thus far, I have limited ...
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24 views

How calculate average probabilities in MLP or SVM?

I have a system that find best model (best inputs and parameters of MLP/SVM) model in a financial problem for every inserted database and create a specific model for a specific data sample. I'm using ...
0
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1answer
26 views

Does Normal distribution theory originate in the psychometric literature, especially reliability theory?

The basic statistical literature does not talk about the exact background of the normal distribution. Is the basis of this assumption in psychometry or it has an origin in pure statistics i.e. ...
3
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1answer
32 views

From joint cdf to joint pdf

We can get the joint pdf by differentiating the joint cdf, $\Pr(X\le x, Y\le y)$ with respect to x and y. However, sometimes it's easier to find $\Pr(X\ge x, Y\ge y)$. Notice that taking the ...
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1answer
22 views

Consistency of unbiased estimator of error term variance in Multiple regression

Let $Y=X\beta+\epsilon$. We know that $\frac{e'e}{n-k}$ is an unbiased estimator of $Var(\epsilon)$, where $e$ is the vector of residuals, and $\epsilon$ is multivariate normal distributed in this ...
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1answer
31 views

Asymptotic distribution of uniform order statistics

It can be shown that for an iid sample from a Uniform(0, 1) distribution, \begin{equation} n(1-U_{(n)}) \rightarrow exp(1) \\ n(U_{(1)}) \rightarrow exp(1) \end{equation} To see this just try finding ...
2
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1answer
27 views

Predictive Model from Counts Data

I have some data that is the number of times a person visited a doctors office over a course of $5$ years. I want to create a model that would be able to predict the most likely number of counts that ...
2
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0answers
30 views

Hessian of Laplace distribution

The density of the Laplace distribution is given by: $$f(x;\mu,\sigma)=\frac{1}{2\sigma}\exp\left(-\frac{\vert x- \mu\vert}{\sigma}\right).$$ It is easy to see that this function is not ...
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1answer
46 views

Finding distribution function [duplicate]

Let $X_1,X_2,...$ denote iid random variables such that $X_1$ is continuously distributed with density $p(x)$ and distribution function $F(x)$ where $F(0) = 0, F(x) > 0$ for all $x > 0$, and ...
3
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3answers
166 views

Why it is good to take log on Finance data? Does it have nice properties? [duplicate]

Just like what I am asking in the title. I see nearly all the financial datas take logs before the data analysing step, Why? Dose it have nice properties?
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0answers
14 views

What is the procedure to compare two different period time series

I am currently working on the task that I would like to compare two different period time series like Sales in 2012 vs Sales in 2013. Kindly suggest me any statistical procedure.
4
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1answer
58 views

Sample Mean of AR(1) model

Consider the AR(1) model with iid innovations with finite mean and variance. Also, let $X_0 = 0$. \begin{align} X_t = \phi X_{t-1} + \epsilon_t \end{align} The goal is to derive the asymptotic ...
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0answers
16 views

Confusing related to kernel method of Regression

In the Support Vector Machine (SVM), non-linear data are mapped to a higher-dimensional feature space to be separated linearly, a kernel is used to compute the inner product in the lower dimensional ...
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0answers
13 views

Can variance in error term be estimated without fitting a line?

In this exercise, I was asked to prove that the simple linear regression - ( $Y_i=\beta_0+\beta_1X_i+\epsilon_i$ with all the usual basic conditions) - of $\{Y_{ji}\}$ for each $X_i$ is the same as ...
4
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4answers
171 views
+100

Texts on Various Topics in Statistics (GLMs, MCMC, Decision Trees, etc.)

I am currently looking for texts (or preferably a specific text) which have a good balance between theory and application and are as comprehensive as possible and are at an introductory level, ...
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1answer
31 views

Marginal probability density of x (obs) obtained by integrating x(missing)

I am reading a text book on missing data, and a sentence below is slightly challenging for me to understand. The marginal probability density of $ \left ( x_{obs}\ \right)$ is obtained by ...
3
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3answers
102 views

Taylor's expansion on log likelihood

As far as I know, Taylors expansion works for fixed functions. I was wondering why it is justified to use it on the log likelihood. Even if we consider it as a function of only $\theta$, doesn't it ...
1
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0answers
23 views

Low pass filter to maintain edge information

I am looking for a kernel as low pass filter that satisfy as:I must find a kernel that statisfies as follows: In the my reference paper, the author suggest gaussian kernel that is The gaussian ...
0
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1answer
30 views

Area below probabilities

Let $p$ be probabilities and $D$ is the real How can I proof that the areas $$\int p \; d F_{p}(p|D=1) = \int (1-p) \; d F_{1-p}(1-p|D=1)$$ are equal. Where $F_{p}$ is the empirical distribution ...
3
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1answer
127 views

What do vertical bars mean in statistical distributions?

What do the vertical bars mean in the first and third formulae? $$v_i|z_i=k,\mu_k\sim\mathcal{N}(\mu_k, \sigma^2)$$ $$P(z_i=k)=\pi_k$$ $$\pi|\alpha\sim \text{Dir}(\alpha/K1_K)$$ $$\mu_k\sim ...
1
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1answer
26 views

Doubt on interpretation of Neyma-Pearson lemma

In my book: $\mathbf{X}=(X_1,\ldots,X_n)$ $f(\mathbf{x})$ is the joint density, where $f$ is either $f_0 \text{ or } f_1$. Suppose we want to test $H_0: f=f_0$ or $H_1: f=f_1$. The test, whose test ...
1
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1answer
32 views

Geodesic distance and mean

My data-set consists of points in globe. Suppose a User visits locations $l_1,l_2,\dots l_n$ (each location in $(lat, long)$ in the city with probability $p_1,p_2,\dots,p_n$ and I want to calculate ...
1
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1answer
265 views

Which is kernel similar gaussian kernel?

I must find a kernel that statisfies as follows: In the my reference paper, the author suggest gaussian kernel that is The purpose of that kernel is that it will take a weight for each points ...
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2answers
64 views

Limiting joint distribution of estimators; Functional Statistics; Influence curves;

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
2
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1answer
45 views

$E(X_i \cdot I(X_j>\theta))$, where theta is the median?

Let $X_1, ..., X_n$ be iid with a distribution F. Let $\theta$ be the median of F. What is the value of $E(X_i \cdot I(X_j>\theta))$? If $i\neq j$, then $E(X_i \cdot I(X_j>\theta))= 1/2 \cdot ...
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0answers
34 views

How to convert daily variances to a monthly volatility and then annualize it?

I work out the conditional variance using a GARCH model based on daily returns as follows: ...
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1answer
16 views

Get Characteristic Function from MGF? [duplicate]

I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. ...