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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91 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
6
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37 views

Find a function such that follows to normal in distribution

Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each ...
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30 views

single variable is significant but overall test is not

I do a multiple regression with 3 independent variables $X_1$, $X_2$ and $X_3$. The correlation between $Y$ and $X_1$, $Y$ and $X_2$, and $Y$ and $X_3$, are each large and statistically significant. ...
4
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92 views

Divergence based robust inference

The term 'divergence' means a function $D$ which takes two probability distributions $g,f$ as input and puts out a non-negative real number $D(g,f)$. I have learnt that the inference based on ...
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27 views

Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
4
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77 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
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39 views

Bayesian information criterion from measure theoretic point of view?

Bayesian information criterion (BIC) is well known and it is derived from the maximizing the posterior density function which is equivalent to solving the marginal likelihood integral. My question is: ...
3
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41 views

Pareto distribution confidence interval

$X$ is distributed by Pareto with $$f_X (x) = \frac{\alpha k^{\alpha}}{x^ {\alpha +1}},\alpha,k>0,x>k.$$ Derive a 95% confidence interval for $k $. My friend said I gotta do this $$Pr ...
3
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73 views

t-distribution and Degrees of freedom

Why t- distribution have n-1 degrees of freedom? I know that it is used when population variance is not known but what determines n-1
3
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46 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
3
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91 views

Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions ...
3
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269 views

The distribution of the ith order statistic for discrete random variables

Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$. We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim ...
3
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157 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
3
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115 views

Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
3
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67 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
2
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17 views

Comparing two textbooks for machine learning

I am a Ph.D student in Electrical Engineering. I am going to study the field of machine learning and I found some textbooks to study this field. 1) Probabilistic Graphical Models: Principles and ...
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30 views

How does one find the Fisher Information of a MLE?

The Statement of the Problem: Suppose that $Y_1, Y_2, \ldots, Y_n$ constitute a random sample of size $n$ from an exponential distribution with mean $\theta$. Find a $100(1-\alpha)\%$ confidence ...
2
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20 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
2
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123 views

Sample median of Cauchy distribution is consistent. How?

When we use chebyshev's inequality to show whether an estimator is consistent or not, we require the mean square error of the estimator and I do not know sample median's probability distribution. So ...
2
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52 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
2
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45 views

Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
2
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23 views

Why not use always a binomial exact test to compare two proportions instead of chi square?

I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that ...
2
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37 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
2
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29 views

Finding a general form of the density function when we have a four dimentional random variable.

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
2
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131 views

Stratified Sampling: Total and mean estimate of population

Question: In a sample survey designed to estimate the total number of cattle, the universe of 2072 farms was stratified into 5 strata on the basis of the total acreage of farms. In the hth stratum (h ...
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18 views

Show that $Y = 2\sqrt{X_1 X_2}$ has a $\Gamma(2p, 1)$ dist.

$X_1$ and $X_2$ are independent with $\Gamma(p, 1)$ and $\Gamma(p + 1, 1/2)$. Show that $Y = 2\sqrt{X_1 X_2}$ has a $\Gamma(2p, 1)$ dist.
2
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220 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
2
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23 views

Sampling from a graph

Suppose you have a graph $G=(V,E)$ that is unobservable globally and you wish to take a sample from the vertices of that graph to infer something about its global properties from local properties. ...
2
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26 views

Sum of random variables - is there an efficient way to do inference?

Suppose I have a series of variables $X_1, X_2, \ldots, X_n$. I have $S = \sum_i^n X_i$. Now suppose that I have a constraint on the distribution of S, for example from some data. Looking at any of ...
2
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46 views

Normalizing multiple different features from unknown distributions

I'm doing some "exploratory" data analysis over a large set of classes/proteins, with a few hundred different features (I.E. Continuous variables) extracted from the data. The features are calculated ...
2
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73 views

Estimating a sub-population characteristic based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ...
2
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270 views

hypothesis testing practice question

This is a practice question for a final exam. Two types of cordless weed-trimmer batteries are tested. One group of 5 batteries averaged $1.4$ hours, while the other group consisting of 8 batteries, ...
2
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0answers
72 views

degrees of freedom for a chi squared goodness-of-fit test

For a statistics project, I gave out a 20 question multiple choice quiz with each question containing five answers. I would like to run some hypothesis tests on the data by using a Chi squared ...
2
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103 views

Probability distribution for a digit of a number

If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
2
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70 views

How do I must do it?

For a sample of size n from a random variable with density function $$f_{\theta}(x)=\frac{2x}{\theta^2}, x>0$$ find the confidence interval for $\theta$ of average length consistently lower level ...
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28 views

Inference the time a process would take depending on the number of threads

I have several datasets consisting on: Number of threads n Start process time t1 Stop process time ...
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353 views

Is there a statistical hypothesis test that uses the mode?

Is there a statistical hypothesis test that considers the mode rather than the mean or median?
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43 views

According to Chebyshev's rule, how many observations should lie within one and a half standard deviations of the mean?

Using the formula : $p = 1 - k^{-2}$ I calculated that $p = 1 - 1.5^{-2} = 0.56$ , which equals to $56\%$. Because I have $24$ data points I go ahead and solve the number of points is $56\%$ of ...
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38 views

Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-degenerate random variable.

Let $X_1,X_2,...X_n$ be iid with the same chi-square distribution with one degree of freedom. Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a ...
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70 views

Theoretical distribution of a random variable

Martin has $n$ words, and he wants to make a computer program that chooses for him $k$ words (and shows them to him), where $k \le n$, for as many times as he clicks a button until all of the words ...
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54 views

Does a Markov process have memory zero?

I have the following question: The Markov process with two states and a transition matrix $$P =\begin{pmatrix} 0.3 & 0.7 \\ 0.3 & 0.7 \end{pmatrix}$$ has memory zero. Is it true? My ...
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26 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy.

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
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21 views

testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...
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36 views

Rank of a random matix

This arises in Time-Series modelling. Suppose $Y_i \sim N_p(0,\Sigma_i)$ and they are not necessarily independent (but assuming $\Sigma_i$ to be p.d.). Then for any ${a}\neq 0 \:\:\:\:$ $Y_i'a\neq0$ ...
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20 views

Conditioning multivariate Gaussian on a function of coordinates

I have a pretty general question and I would really appreciate if you give me any hints or point me towards some relevant literature. Suppose $X$ is an $n$-dimensional Gaussian vector. What is the ...
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65 views

Estimating Parameters using Method of Moments and Maximum Likelihood and Finding expected values/variance

Let's say we have a dataset $(x_i, Y_i)$ on each randomly n chosen non cities in a country where $x_i$ i=1,...,n is the known population size in city i with cancer. Say $Y_i$ has a Poisson ...
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15 views

Penalty function of multi-peak fit?

The question I have is about the answer from here by @Silvia: http://mathematica.stackexchange.com/questions/26336/how-to-perform-a-multi-peak-fitting I can only understand some of the code but the ...
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0answers
13 views

Denominator in Maximum Posterior Estimation - How to Interpret?

Suppose we're given a sequence $x_1,\ldots,x_n$ of realizations of i.i.d. $\mathcal{N}(\mu,\sigma^2)$ random variables and we want to apply maximum posterior estimation to estimate the parameters ...
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22 views

Covariance estimation and Graphical Modelling

I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions. Suppose, $X_i \overset{iid}{\sim} N_p(\mu,\Sigma)$, ...
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48 views

Likelihood ratio test for a normal distribution with unknown mean

Suppose $X_1,X_2,…,X_n$ is a random sample from a normal population with mean $μ$ and variance 16. Let sample size=16. Find the likelihood ratio test for $H_0:μ=10 $ against the simple alternative ...