The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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probability of rank of a number

Suppose I have 10 numbers. I want to perform two experiments. First experiment: I pick one of the numbers and compute the probability of being rank from 1 to 10 of that number, i.e, what is the ...
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21 views

comparing two means using independent samples

A survey will ask baseball fans if they think the Kansas City Royals will return to the World Series this year. We will estimate the difference in the proportion of males and females, ($p_{\texttt{...
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estimation for analytic stochastic processes

this is for experts in probability and stats. There is a theorem, I have seen once: Given a stationary analytic random process, one can show that from the values of a sample path in a finite interval ...
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1answer
11 views

Asymptotic Error bound

Asymptotic error bound is the limit on the error when the size of sample goes to infinity. Am I right about this? If not can somebody explain what Asymptotic error bound is? And the situations in ...
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19 views

Prediction Algorithms in Statistics

Here it is what I'm looking for: Having a random variable, with an unknown distribution of its values, is there any smart algorithm that can predict the next value of the variable based on n samples ...
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1answer
45 views

Bayesian Example

Ex. suppose that $x=2$ denotes the number of successes in $n=5$ independent trials with probability $θ$ of success, that is $x$ has a binomial distribution with the parameters $n=5$ and $ θ$. ...
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60 views

can i get rich if i learn statistics ?? [closed]

can i get rich or at least earn money :) if i learn statistics ? i have a degree on physics but can not find a job so my question is if using statistics (mathematics) can one get profit from the ...
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8 views

higher order asymptotic expansion for likelihood ratio

I have been studying Hayakawa(1975) and (1977) and was wondering if anyone has already computed higher order terms for his expansions following his framework. I'd be very happy if someone could ...
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Estimating Distributed Cache Hits Based On Local Cache Hit Data

This needs a bit of an introduction. We have 6 servers with a local cache, in other words, we have 6 different systems of cache that do not speak to each other. A request that comes through our ...
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26 views

Setting up the liklihood distribution for Bayesian Estimation

Here is the exact problem: Suppose that the random variables $Y_1,\ldots,Y_n$ satisfy $$ Y_i=\beta x_i+\varepsilon_i, \quad i=1,\ldots,n. $$ where $x_1,\ldots,x_n$ are fixed constants, and $\...
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37 views

Find a 95% upper confidence bound for lognormal dispersion

Let $X_1,\ldots,X_n$ be a sample from a population $X$. Assume that $X$ has a $lognormal$ distribution $(2,\sigma^{2})$, with $\sigma^{2}$ an unknow parameter. Find, for the variance of the population,...
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When deciding the Hypothesis, do I, in some way, define the rejection region?

This has always troubled me a bit. When I choose my hypothesis, do I define in some way the rejection region [RR], or, do I do that by choosing the test statistic I want to use? By fixing the ...
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1answer
20 views

sufficient statistics to estimate the unknown parameters

I am a beginner in statistical inference and am learning sufficient statistics. As far as I know the distributions conditional on the sufficient statistics doesn't depend on the unknown parameters. I ...
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1answer
20 views

Properties of minimizing statistical distance

I have some data, thus I have its empirical distribution. I want to use a theoretical distribution to fit my data. For example, I observe my data is likely to be distributed as Pareto, so I use Pareto ...
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22 views

find the maximum where gamma is attained

I have this problem and I am not figuring the beginning
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15 views

the posterior under n observations computed as the posterior given a single observation [closed]

I have this problem: Show that, if $p(x|\theta)$ is an exponential family model and $q(\theta)$ its natural conjugate prior, the posterior $Π(\theta|x_{1:n})$ under $n$ observations can be computed ...
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2answers
31 views

Looking for a good, rigorous book on Statistical Inference.

I'd like to brush up on my statistics, but most books are either overly "colourful" or just plain shallow, and certainly far from a Bourbaki-esque style of exposition. Is there any "Graduate Texts in ...
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1answer
56 views

Clarification of a Probability

Suppose I have two continuous, non-negative random variables, $X$ and $Y$ and I have that $$ P(X) = P(X|Y)\cdot P(Y). $$ Can I go on and say that $$ P(X\gt z) = P(X\gt z | Y\gt z)\cdot P(Y\gt z) = ...
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Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
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Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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18 views

How to best compare two different time series with different frequencies

Lets say I have two time series $X_t$ and $Y_{t,q}$. As an examples, lets say $X_t$ is a series that measures year over year changes in the level of output of a good (say number of widgets). So $X_t = ...
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2answers
45 views

Sum over Binomial mass function

In Casella and Berger Book (Statistical Inference), exercise 2.40 is $$\sum_{k=0}^x {n\choose k}p^k(1-p)^{n-k}=(n-x){n\choose x}\int_0^{1-p}t^{n-x-1}(1-t)^xdt.$$ If I replace $x$ by $n$ then LHS ...
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12 views

Invariance to measurements, and invariance to a group of transformations

Why would one demand invariance to measurements(IM) when, previously they've already assumed invariance to a group of transformations(IG). Isn't the IM a special case of IG? Any help would be ...
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12 views

What ranking system should I use for my website? (Wilson vs. ELO/Glicko)

So I have users rate stuff on my site, and so I want to put the "highest-rated" stuff at the top and "lowest-rated" at the bottom. As now, am I only using positive and negative ratings, but this could ...
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Precision-Recall Graph: F1 Score v.s. Break-Even Point

To evaluate two classifiers from the aspects of Precision-Recall, two measures are often used: F1 score and Break Even Point (BEP for short. I failed to find any document about it from wiki, and it is ...
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Inferring the addends of the sum of two random variables

I have three independent Poisson variables: B, C and D, whose parameters $\lambda_B$, $\lambda_C$ and $\lambda_D$ are unknown. I sample once the variable: $$ A_1 \equiv 0.9\cdot B + 0.1\cdot C $$ and ...
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37 views

How to combine correlated estimates to test variable is > 0?

Let X1 and X2 be two unbiased but correlated Gaussian estimators of a true value x. 1. What is the proper way to combine two observations of X1 and X2 to test whether x > 0? 2. How does the answer ...
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18 views

Improving Probability of Event='1' in Logistic Regression (SAS)

I will try to give as much background as I can and if more is needed I will gladly give more. I'm working on trying to find an equation that will tell what probability you have a defaulting on a loan ...
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34 views

Finding the confidence interval of a normally distributed sample

Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of 27 vehicles was 91.29 km/h, with the sample standard deviation being 4.94 km/h. We will ...
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11 views

Stuck on a step in the derivation of the variance of a sample variance

I am trying to understand the formula for the variance of a sample variance $$var(S^2_n) = \frac{1}{n} \left [ \mu_4 - \frac{n-3}{n-1}\cdot \sigma^4 \right ] $$ We start from: $$ var(S^2_n) = E(S^...
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14 views

Bayesian equation: need for priors

As far as I understand, in the problem of Bayesian inference we have a random variable $y$ describing data, which is distributed according to some parameter $x$ via the known conditional distribution $...
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18 views

Cramer roa lower bound for complex numbers

I calculated the Cramer-Rao bounds on variance of these parameter: $VAR_\gamma>\frac{(1-|\gamma(\omega)|^2)^2}{2N}=\sigma_{|\gamma(\omega)|}=\frac{1-|\gamma(\omega)|^2}{\sqrt {2N}}$ I would like ...
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20 views

Find the critical value in Tukey's HSD

I'm trying to find the formula for finding the critical values for Tukey's HSD but I can't find any documentation on how to calculate the critical value based on the number of groups the type I error ...
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73 views

On characterization of MRE estimators

I have some trouble understanding the second equality in the proof of theorem 6;
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1answer
73 views

On randomized estimators [closed]

I been reading the following text on randomized estimators, I cant manage to understand how the randomisation is incoparated into the randomized estimator. How does the random mechanism fit in, ...
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9 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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38 views

Using the Central Limit Theorem to calculate a mean from Poisson distributed random variables

Firstly, I am studying the basic concepts of statistics and so any explanations, advice and suggestions are more than appreciated. Onto the problem- I am given the central limit theorem and understand ...
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25 views

questions about 2 sample t-tests

So I'm just a bit confused about 2 sample t-tests and just want to write out what I think I know and see if that's correct, so if anyone could tell me whether or not what I'm writting is true that ...
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Is this an exponential family of distributions? from casella and berger 6.20

I am trying to do 6.20 in Casella and Berger part d. The solutions manual says that the order statistics are minimal sufficient and not complete. I understand their logic, but why doesn't this work? ...
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26 views

How to inference the conditional probability about LDA?

I'm studying the paper of Blei, "Latent Dirichlet Allocation" ( http://www.jmlr.org/papers/volume3/blei03a/blei03a.pdf ). In his paper(page 1003), given equation is $p(\theta, z|w, \alpha, \beta)= \...
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26 views

Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
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1answer
7 views

Missing approximation to get the Maximum A Posteriori (MAP) estimator of event times with a sparse prior

Assume that a signal $ y $ is a noisy perturbation of time-shifted copies of a given waveform $ f(t) $ defined on K time bins $ \{ 0, \cdots, K-1 \} $: \begin{equation} \forall t \in \{1, \cdots, T\}...
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27 views

Guessing Mathematical Probabilities by Tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as many tests as I want. What is the minimum number of tests required to be at least at 95% sure I have the ...
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19 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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1answer
13 views

Maximum a Posteriori (MAP) Estimator of Time Shifts with Poisson Process Prior

Assume that a signal $ y $ is a noisy superposition of time-shifted copies of a given waveform $ f(t) $ on a finite time interval $ [0, T] $: \begin{equation} y(t) = \sum_{i=1}^{n} f(t - \tau_j) + \...
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Finding independence of two variables

I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared ...
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18 views

Finding Asymptotic Confidence Interval with a condition

I am trying to solve the following problem: Let $X_1, X_2$, and $X_3$ be random variables from the following joint pmf: $$f_{X_1,X_2,X_3}(x_1, x_2, x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} ...
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18 views

finding the sufficient and ancillary statistics

I am trying to find the sufficient and Ancillary statistics for the following problem: suppose $(X_1, Y_1). . . ,(X_n, Y_n)$ be iid random vectors from the pdf: $f_{X,Y} (x, y) = {\frac{1}{2π \...
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48 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = {\frac{1}{n}}{\...
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77 views

Expectation or Integration of the normal cdf

Can any one help me how to solve this pronbelm? 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), $Z \text{ is Normal(0,1)}$, and $k$ ...