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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UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$
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1answer
29 views

Is my work correct? (Easy problem, confidence intervals)

The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken ...
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1answer
79 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
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10 views

How to show sufficient statistics are complete

By writing out the likelyhood function, I can show that $(X_{(1)}, X_{(n)})$ is sufficient statistics, but how to show they are complete?
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12 views

Is correct my Procedure about Joint Distribution for independent random variables

$ y_i, i=1,2...n$ are random variables are linearly independent For $y_i \sim Ber(p)$ $(p^{x_1}q^{1-x_1})(p^{x_2}q^{1-x_2})\bullet \bullet \bullet (p^{x_n}q^{1-x_n})$ ...
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24 views

Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
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28 views

Sampling substrings of a beaded necklace to determine the necklace composition

I have a necklace composed of 100 beads, where each bead is one of 13 colors. If I am only able to look at one 4 bead sub-sequence at a time (connected, as they would be on the necklace) , how many ...
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10 views

Repeated Measures ANOVA [on hold]

What is the model equation for one-way repeated measures ANOVA? Is that for the 2-way rANOVA similar to the 2-factor experiment with interaction term present in the model equation as follows: Yijk = ...
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1answer
19 views

Wilcoxon signed-rank test

While reading Wikipedia, and my teacher's notes I found that Wilcoxon signed rank test for $n>10$ is given like below: Under null hypothesis, W follows a specific distribution with no simple ...
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24 views

Divergence based robust inference

I have learnt that the inference based on minimizing the following divergence is robust to outlying observations for some specific range of $\alpha\in\mathbb{R}$. $$D_{\alpha}(g,f) = ...
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11 views

Reference needed: Chi-square goodness of fit test, independence test, …

I want to really understand what is a Chi-square test and how it works: when is it needed, what motivates its use, etc. The same thing is needed for "Independence tests" and analysis of variances. Is ...
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11 views

comparing 2 datasets which have different distributions

I'm currently analysing two datasets. They report the same information, but in different ways. I am looking to draw comparisons between the way items fail in each of the datasets. In the first ...
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4 views

assumptions of t-test of correlated terms [closed]

Good Day ,i just want to ask if what are the assumptions of t-test of correlated means.I've been browsing the internet since yesterday because im just so curious if what does this statisticall tool ...
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1answer
26 views

Estimate the variance in confidence interval for difference between two poisson means

I have a problem that goes like this: "The number of bicycle thefts in January this year is 214 which is 48 less then January the year before. Suppose the number of bicycle thefts are independent and ...
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20 views

Degrees of freedom: is this true?

Let $X_1,...,X_n$ be $n$ r.v.'s from a distribution having a certain p.d.f. To the chi-square goodness of fit test, we introduce: $$\chi^2 = \sum_{i=1}^n \frac{(X_i - np_i)^2}{np_i}$$ My ...
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12 views

Hypothesis Testing with given Pearson correlation.

There are two independent sets of samples , female and male. The problem is to calculate the 95 % C.I. of the mean of total( male+ female, two samples) population. -> Female : n=100, mean= 169.1, sd ...
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1answer
18 views

How to find the degrees of freedom for a chi-square variable

How does one find the degrees of freedom for a Chi-square random variable when trying to fit a distribution to a sample? I read an explanation regarding this in this source. I don't understand how to ...
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2answers
54 views

I am running a series of experiments that I expect to have similar outcomes. What is the best method to measure statistical significance?

Following on from this comment on an answer to my previous question, I'd like to know two things: what the best statistical test I can use to measure significance on the experiments I'm running? ...
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1answer
33 views

Limit of median of uniform distribution

Let $X_1,X_2,\ldots$ be a random sample from the uniform distribution on the interval $(0,1)$. Assuming that $n$ is odd, find the pdf of the sample median (say $M_n$). Does the pdf of the r.v. ...
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1answer
12 views

Is this testing a mean or a proportion?

In $200$ families each with $4$ children, we observed the number of boys they had. Summary: $8$ families had $0$ boys, $42$ families had $1$ boys, $67$ had $2$, $70$ had $3$ and $13$ had $4$. Can we ...
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1answer
37 views

'Markovian Property' vs 'Memoryless Property'

The two properties have the commonality in the sense that they predict the future based on the current state, not on the whole history of how the process wandered into the state. Then, what is the ...
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1answer
18 views

Small question about the mean not being in its confidence interval

To study a certain characteristic about a population of people we take a sample of $100$ individuals. The $80$ percent confidence interval for the mean is $(0.9,1.1)$. Part I: Find the sample mean ...
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1answer
20 views

Confusion regarding the C.I of the mean of some population

I calculated (and verified) the confidence interval for the mean of a population, from this sample: $n=100$, $x_1 = ...=x _6= 36$, $x_7 = ... = x_{17} = 37$, $x_{18}=...=x_{43}= 38$, $x_{44} = ... = ...
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32 views

Why isn't the estimator of the square of a parameter the square of the estimator of the parameter?

Let $X_1,...,X_n$ be a sample from a distribution having as a p.d.f: $f(x) = \frac1{\theta} e^{-x/\theta}, x,\theta > 0$ and $0$ elsewhere. The maximum likelihood estimator of $\theta$ is ...
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28 views

How should I solve this? (Can we accept that the proportion is … at risk …) [closed]

This is one part of a question that I couldn't solve. Let $P$ be the proportion of the emergency cases received at a hospital per day. In a sample of $64$ patients, we find that the proportion of ...
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0answers
36 views

Why is $nS_X ^2/\sigma ^2$ $\chi ^2(n-1)$, while the other is $\chi^2(n)$?

Suppose $X_1,...,X_n$ is a random sample from a distribution having $N(\mu, \sigma^2)$. What is the conceptual difference between: $$ \frac1{n} \sum_{i=1}^n (X_i - \bar{X})^2$$ and $$ \frac1{n} ...
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1answer
24 views

Fisher's LSD test.

When we calculate Fisher's LSD test, why do we use Mean Square of Error, $\sigma_e$, which is the variance of all groups (pooled variance), as a variance of each individual group mean? In the ...
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11 views

Advanced Inference, sufficient statistics

I was trying to work on some problems of Statistical Inference and I found these two exercises that I could not solve. 1) Let X be a sample from $P \in \mathcal{P}$ containing p.d.f's $f_p$ w.r.t. a ...
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1answer
37 views

Which test should I use for hypothesis testing with a small sample size?

I've run a test with one control and one experiment group, and am questioning myself on whether or not I've used the right test (or if significance can even be calculated on the following sample ...
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14 views

Looking for some good introductory level resources for Gibbs Sampling

In context of a course in bayesian modelling Im following, im looking for some good resources (videos, lecture slides, texts) about Gibbs sampling.
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Creating a minimal sufficient statistic with Likelihood function

To find a minimal sufficient statistic you can take the likelihood ratio and find a function $T$ so that the ratio does not depend on the parameter $\theta$ , as page 18 here ...
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7 views

How to interpret the Quantified properties of estimator?

https://en.wikipedia.org/wiki/Estimator The link provides a very good explanation of the estimator. I am beginner to statistics and inference , so i have some confusion about the quantified properties ...
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1answer
13 views

How to choose the degree of freedom when calculating the variance var(X)?

I have seen in the text books that sometimes while calculating the Variance of a set of numbers $Var(X)$ = $\sum_{i}^{n} \frac {(x_{i}-\mu)^2}{n}$ also $Var(X)$ = $\sum_{i}^{n} \frac ...
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1answer
18 views

Estimating variance of estimator of bernoulli process

The maximum likelihood estimate of a Bernoulli process is simply given by $\hat{\theta}=\frac{\sum X_i}{N}$, where N is the total number of bernoulli trial and $X_i$ is the outcome of each trial. ...
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1answer
23 views

Find the MLE of $N(\theta,\theta)$

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Find the MLE of $\theta$. I got $\frac{\partial logL(x|\theta)}{\partial ...
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62 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 ...
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1answer
20 views

How to find the maximum likelihood estimators of parameters in the Pareto distribution? [closed]

Here's the Pareto distribution: $$f(x; \theta_1, \theta_2) = 1 - (\theta_1 /x)^{\theta_2}, \theta_1 \le x, \theta_1, \theta_2 > 0$$ Its likelihood function is complicated and so is its ...
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1answer
16 views

Maximum likelihood estimator of $\theta$, $f(x;\theta) = 1/2 e^{-|x - \theta|}$

I'm given $f(x;\theta) = \frac12 e^{-|x - \theta|}$, $-\infty < x < \infty$ and $0 < \theta < \infty$. I want to find the maximum likelihood estimator of $\theta$. I found: $$\ln ...
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0answers
14 views

When can we not use Jensen's Inequality to compare risks?

I have two closely related questions regarding Jensen's Inequality. To show that for any estimator $\delta(X)$, there is another estimator based only on sufficient statistic $T$ that gives the ...
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31 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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2answers
40 views

Existence of complete sufficient statistics

Suppose $X_1,\ldots,X_n$ are iid r.v.'s, each with pdf $f_{\theta}(x)=\frac{1}{\theta}I\{\theta<x<2\theta\}$. I find the minimal sufficient statistics $(X_{(1)},X_{(n)})$. I am trying to prove ...
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1answer
40 views

What is the probability that statistical significance will be achieved in a second test?

Assume I'm thinking of investing in a small biotech company. In a phase 2 study with 120 patients split 80/40 between the new drug and old drug, the progression free survival (PFS) rate of the ...
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36 views

minimum kullback leibler estimator

Suppose that one has independent and identically distributed samples $x_i,i=1,...,n$ from some unknown density and one wants to fit a probability distribution $f_\theta(x)$, where $\theta$ is a ...
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20 views

Bayesian Optimization and Averaging

I read in a statistics book that optimizing the likelihood function (or more generally Quasi-likelihood function) in a Bayesian framework is the same as averaging the posterior means. Why is this ...
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31 views

Positive semi-definite in Linear model

Suppose $Y_{n \times 1} \sim N(X\beta,\sigma^2V)$ where $V_{n\times n}$ is invertible and $X_{n\times p}$ is of rank $p$ and $\beta_{p \times 1}$ is unknown and to be estimated by $Y$ and $X$. Which ...
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2answers
24 views

How to calculate probability with Z score not on table?

According to this Z table in my book, anything with a $z$ of over $3.16$ is probability 1, but this is not right. The textbook also has an example where $\mathbb P(Z\geq3.9) = 0.000048$; can somone ...
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1answer
20 views

Determining the MVUE of $ f(x;\theta) = \theta^x (1-\theta)$.

The Statement of the Problem: Let $X_1, X_2, ... , X_n$ be a random sample from $$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$ (a) Find the ML estimator of $\theta$. (b) Show that $T ...
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21 views

Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$. Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: ...
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1answer
17 views

A question about Fisher–Neyman factorization theorem

$f_{\theta}(x)$, then $T$ is sufficient for $\theta$ if and only if nonnegative functions $g$ and $h$ can be found such that $f_{\theta} = h(x)g_{\theta}(T(x)) $ The statement is: if $F(t)$ is a ...
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1answer
33 views

Determining the efficiency of $2 \overline X$.

Let $X$ be a random variable with pdf $$ f\left(x;\theta\right) = \left\{ \begin{array}{lr} \frac{3\theta^3}{(x+\theta)^4} & \text{if } 0<x<\infty \text{ and } 0 < \theta ...