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

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how can i calculate $E(Y)$ [on hold]

suppose $X$ is random variable non negative with Distribution function $F_{X}(x)$ and $Y$ is random variable with Distribution function $G_{Y}(t)=1-E(e^{-tX}) ,0\leq t$ . how can i calculate $E(Y)$
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16 views

what kind of statistical test should I carry? [on hold]

Hie, I have a set of data as below: What kind of statistical test should i carry out. Concurent users are virtual users. byrequest, busyness, traffics etc are algorithm of each web proxy such as ...
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1answer
53 views

Why does Bengio's deep learning theory book claim $\hat{y} = x w_1 … w_i … w_l$ is a non-linear function of $w_i$?

In chapter 8 section 8.7.1 it tries to explain batch normalization. In the second paragraph of that section it tells us to consider the simple example: $$ \hat{y} = x w_1 ... w_i ... w_l $$ and ...
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1answer
24 views

Reference Books on Asymptotic theory of Statistics and Probability

Can anyone suggest me some good reference books on Asymptotic Theory of Statistics and Probability for students pursuing a post-graduate degree in Statistics ? It would be very much helpful if the ...
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16 views

Why the shape of mahalanobis values of a set of data is elliptical? [on hold]

When I searched about mahalanobis distance, i came to know that the shape of mahalanobis values of a set of data is elliptical. Then i searched why the shape is elliptical? But i didn't get any ...
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1answer
13 views

Comparing Crude Death rate of two population

As we all know that the formula for crude death rate is $$CDR = \frac{D}{P} \times 1000$$ where $D$ is the totla number of deaths(during a calender year) among residents of a community and $P$ is the ...
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53 views

What's the difference between asymptotic unbiasedness and consistency?

Does each imply the other? If not, does one imply the other? Why/why not? This issue came up in response to a comment on an answer I posted here. Although google searching the relevant terms didn't ...
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17 views

Appropriate testing for outliers

I want to perform an outlier-test for my data. The distribution of the data looks like a hyperbola (there are many values near 0, all values are >0). Sample size of data is 5000. I want to find an ...
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1answer
33 views

Machine Learning - How to use a classifier to find the most likely model

I have been learning about the use of machine learning algorithms and their application to particle physics. Now, I have some doubts concerning what to do with the results. Let me explain: imagine ...
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14 views

statistical tolerance

My request is: I have data (sums of money) and someone calculated tolerance limits for the data. I want to validate these tolerance limits with statistical methods/arguments. Does anyone know ...
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39 views

Role of Central Limit Theorem in creating confidence intervals and hypothesis testing

How is central limit theorem useful in creating confidence intervals as well as its role in hypothesis testing around the population parameter.
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1answer
60 views

How does the support of a continuous random variable change under transformations?

Let $X_1$, $X_2$ have a joint pdf and support set $S$. Suppose random variables $Y_1$, $Y_2$ are given by $Y_1=u_1(X_1,X_2)$, $Y_2=u_2(X_1,X_2)$ where the functions define one to one transformation ...
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7 views

Equal Error Rate (EER) and Receiver Operating Characteristic (ROC) curve

I have a one-vs-all classifier set. This set consists of, let's say, 3 classifiers (LibSVM SVMs) each trained on data for a class and all other class data. The current setup for a sample is that the ...
5
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1answer
42 views

Estimating a variable from its cosine corrupted by additive Gaussian noise

I observe $y_i=\cos(\theta)+z_i$, $i=1,\ldots,n$, where each $z_i\sim\mathcal{N}(0,\sigma^2)$ is an i.i.d. zero-mean Gaussian random variable. I am interested in estimating $\theta\in[0,\pi]$ with ...
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1answer
20 views

Confidence Interval for Regression Line (simple linear regression)

So I've constructed a confidence interval for my regression line. However, because I have 2500 data points it is a very, very narrow interval (I can barely see it next to the regression line when I ...
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1answer
11 views

Lower point estimate than point estimates of the mean [closed]

If After calculating a point estimate x of µ and I notice that it is substantially lower than the point estimates of the mean, what conclusions can I make from this and using which argument?
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1answer
96 views

how can i calculate $E\left(\Pi_{i=1}^{n}\displaystyle\frac{X_i}{X_{(n)}}\right)$

suppose $X_1,X_2,...,X_n$ be a random sample of $U(0,\theta)$. how can I calculate $E\left(\Pi_{i=1}^{n}\displaystyle\frac{X_i}{X_{(n)}}\right)$. $X_{(n)}$ = $max_{1\leq i \leq n}X_i$
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20 views

my Wilcoxon is contradicting my Spearmans Rho [duplicate]

In brief, I'm trying to find out if 62 people perform better on test one or test two. I ran a spearman's rho and it came back positive and significant with a moderate strength. In other words, when ...
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0answers
10 views

Which statistical test for data comparison?

I have several sets (e.g. 100) of essential genes + non-essential ones, all were extracted from a certain database of essential genes for a particular disease under different conditions. How can I ...
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0answers
14 views

Finding Power Function for a LRT test

I am trying to solve the following problem: Let $X_1, . . . , X_n$ be a random sample from the $Uniform(θ, θ + 1)$ distribution. Derive the power function for the LRT for $H_0 : θ = θ_0\ vs.\ ...
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20 views

Specific Differences B/T SPSS Grad Packs

I have been using the Trial Version of IBM's SPSS Statistics Desktop v. 24 I would like to purchase the most affordable option that enables a few but very specific features my research requires. ...
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26 views

How to determine if a set of data is Student-t distributed

So I know how to use goodness of fit tests generally. For example if I wanted to check whether my data came from a Poisson distribution I would first estimate the parameter using MLE (so just the ...
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1answer
40 views

100-Dollar Game Responses as Predictor

Outcome = Feature Sentiment Participants will rate on 1-5 point scale how much like a particular feature. Potential Predictors = Developments A - E Participants will be asked to imagine they have ...
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29 views

Questions about explanatory variable and E(Y|X=x)

In the context of simple linear regression, is the explanatory variable a random variable? In simple linear regression, we usually write $E(Y|X=x) = a + bx$. I understand the meaning, but how can we ...
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147 views

Expectation or Integration of the normal cdf [closed]

Can any one help me how to solve this problem? I have a random variable $W$, i.e., $$W=\Phi(X)^k\Phi(-X)^m=P(Z\le X)^kP(Z \ge X)^m,$$ $X$ is Normal($\mu$,1), $Z$ is Normal(0,1), and $k$ and $m$ are ...
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1answer
30 views

How to deal with Features having high cardinality

I am building a simple classification model, the classification model is used for identifying negative or positive feedback on a ticketing system. The tickets have alot of categorical data with high ...
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1answer
56 views

how can I calculate $P(X_1\leq X_2\leq X_3)$ [closed]

suppose $X_1,X_2,X_3$ are random sample of distribution with probability function $P(X=k)=(1-\theta)\theta^k , k=0,1,2,... ,\theta \in (0,1)$ how can calculate $P(X_1\leq X_2\leq X_3)$
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2answers
21 views

Find out a huge number of coordinates is uniformly distributed or not

Assume that we have a huge number of locations like $10^{6}$ locations in two-dimensional space. The coordinates are generated randomly. what I want to do is to make sure that the data distributed ...
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6 views

property of check function

I'm working on a problem about check function and I'm stuck but even after doing some research about it, I cannot find any material explaining about this check function and I'm totally clueless about ...
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1answer
14 views

what method to test percentage against population percentage?

I have population of 6 regions, and also customers from these regions. What is the best way to test if the percentages of customers from these regions are about the same as the population ...
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19 views

Spatial analysis of the network data

I'm a bit new in the spatial statistics (althought I'm not new in statistical analysis, particular in Bayesian inference). When we have point measurements in space - geostatistics/spatial statistics ...
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24 views

How to draw samples from a multivariate Gaussian distribution without having access to a function that does the job? [duplicate]

I am using the programming language Lua which does not have any built-in function for drawing samples from a multivariate Gaussian distribution. So I wonder, how can one implement a function that does ...
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1answer
26 views

Convergence in Distribution for a sequence of standardized chi square random variables

Let $\{X_n\}$ be a sequence of random variables where $X_n \sim \chi ^2_{_n} \forall n $ . The sequence $\{X_n\}$ has an associated sequence of MGFs given by $\{M_{x_n}(t)\}$ ,where $ ...
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7 views

Representer Theorem: Is the RKHS $H$ the feature space or the estimator space?

I don't know how theoretical this page is, but I don't know a better place to ask this. It is more an intuitive question than a formal one. In reproducing kernel hilbert space theory we normally have ...
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7 views

How do I find impact of independent variable change in dependent variable change?

Suppose I am having a function revenue = NumberOfUsers * ConversionRate * AvgOrderValue. If revenue is changed to 24%, How I can know %impact of NumberOfUsers,ConversionRate and AvgOrderValue?
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9 views

Replacing NA columns with Median in R [migrated]

I keep getting errors with the codes, which would be correct?
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0answers
4 views

Should you use converted currency when fitting a Generalized Linear Model [migrated]

So long story short, my data set has a bunch of currency values (ie: Dollars, Euros, Pounds) over the span of several years. The original idea was that I should use a daily conversion rate to ...
3
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0answers
29 views

Statistical test to compare two groups: Does my proposed process make sense?

Here is the situation. I am facing a comparison problem where I am trying to compare an insulated pipe with a non insulated pipe. I need to test if the use of insulation creates a significant ...
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0answers
6 views

How can I get Equal Error Rate?

I have FRR and FAR values for a threshold that ranges from 0 to 1 at 0.1 intervals. This gives me 20 values. I know I can construct a ROC graph but is there an equation i can use to simply get the EER ...
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0answers
11 views

How do you calculate the pareto shape parameter?

I have a set of data that shows a pareto distribution. I am trying to figure out the probability of an event within that data set. I need to know the pareto shape parameter in order to calculate this ...
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0answers
17 views

Difference between statistical moments and inertia moments

do you know the difference between statistical moments and inertia moments and the contribution for to find centroids of surfaces? I read on Varignon's theorem, but I did not understand. For ...
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1answer
55 views

How can i know the function is convex or not?

As an examples i am sending you following two functions $f1(x)$, and $f2(x)$ to find either functions are convex or not? These are well-known functions used in robust estimation.My question is how to ...
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16 views

Definition of Stable convergence in law: why do we need an extension of the probability space?

I am trying to understand the definition of stable convergence in law. I have found the following definition. Definition. Let $Y_n$be a sequence of random variables defined on a probability space ...
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22 views

Conditional Variance of a vector given a matrix

I'm trying to reproduce a proof and I've gotten stuck in the following section. Given that $\phi$ is an $N \times 1$ vector, $X$ is an $N \times M$ matrix, $\beta$ is an $M \times 1$ vector, and ...
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19 views

What is the difference between a “natural” sufficient statistic and a sufficient statistic?

Typically, in dealing with properties of exponential families, which are parametrized as the following: $$ f_X(x| \theta) = h(x)\ \exp\Big(\ \eta(\theta) T(x) - A(\theta)\ \Big) $$ where $h(x)$ and ...
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55 views

Markov model parameter concentration and Fisher Information Matrix

For iid data, the posterior on the parameter $$ p(\theta \mid x_{0:T}) = \prod_{t=0}^T p(x_t \mid \theta) p(\theta) $$ is known to become independent of the prior which is the Bernstein-von Mises ...
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15 views

How to make sense of the EM algorithm expressed in terms of Kullback-Leibler divergence?

In the textbook by Wasserman, All of Statistics, the Algorithm is expressed as: 1) Pick a starting value $\theta^0$. 2) (E-Step). Calculuate: $$ J(\theta|\theta^j) = E_{\theta^j} \left(log ...
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50 views

For a Fisher Information matrix $I(\theta)$ of multiple variables, is it true that $I(\theta) = nI_1(\theta)$?

For a Fisher Information matrix $I(\theta)$ of multiple variables, is it true that $I(\theta) = nI_1(\theta)$? That is, if $\theta = (\theta_1, \ldots, \theta_k)$, will it be the case that the fisher ...
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20 views

How can we find the Fisher Information Matrix of a linear regression model?

Suppose we have a sequence of random variables $y_1, \ldots, y_n$ such that $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$ where $\epsilon_i$ is assumed to be i.i.d. $N(0,\sigma^2)$, with $\sigma^2$ ...
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Why is the Fisher Information sometimes written as a conditional expectation?

I was looking at the Fisher Information entry on Wikipedia just now, and saw that the Fisher Information Matrix was written as: $$ \mathcal{I}(\theta) = - \operatorname{E} \left[\left. ...