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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Relationship between 0-1 Loss and Type I and II error in Neyman Pearson

In the context of hyphotesis test $$H_0:\theta\in \Theta_0$$ $$H_1:\theta\notin \Theta_0$$. Find the relationship between the 0-1 loss defined by $$L(\theta,\delta)=1-\delta \theta\in\Theta_0$$ $...
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38 views

Clarification on tests for independence and homogeneity?

If the null hypothesis for a test of independence is true, what distribution does the test statistic have? Would it still follow a chi-square distribution, or a normal distribution? In a goodness-of-...
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20 views

Can't figure out the correct degrees of freedom for my goodness of fit

I am trying to solve the following question: The number of goals scored by a certain football team was recorded for each of 100 matches and the results are summarised in the following table. $$\...
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1answer
41 views

why equal variance assumption is necessary in T-test

So I generally understand the basis for the t-test: i.e. you take advantage of the fact that you can make $\bar{X}-\bar{Y}$ standard normal: $$Z = \frac{((\bar{X}-\bar{Y})-(\mu_X-\mu_Y))}{\sigma \sqrt{...
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47 views

Desirable properties of statistical estimators?

What are some of the properties that people will consider when designing a statistical estimators? For example, unbiasedness and sufficiency are some of the factors considered. Please give some ...
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11 views

Drawing conclusions from given data as percentiles

For a problem I want to determine the number of errors. I have data from two observers, let's say observer A and B. This data is given as $5\%$, $50\%$ and $95\%$ percentiles. Observer A: $5\...
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1answer
74 views

Difference beteween ANOVA and ANCOVA

In the context of using only experiment data for ANOVA analysis, ANCOVA offers post hoc statistical control. Is this a valid conclusion and why?
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17 views

Understanding the shape of T distributions

I'm trying to understand why a T distribution with a small sample size has fatter tails and what this means. My textbook says "...t distributions have more probability in the tails and less in the ...
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44 views

Calculate calculate $E(\frac{1}{\bar X})$, Var$(\frac{1}{\bar X})$

Consider n i.i.d. observations from a Poisson (λ) distribution. Suppose $\bar X =\frac1n\sum_{i=1}^n X_i.$ How do I calculate Var$(\frac{1}{\bar X})$ or even $E(\frac{1}{\bar X})$ for that matter. ...
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17 views

Multilinear loss in Uniform-Exponential model

Let a prior $\pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta))$ and $f(x\mid\theta)=\theta e^{-\theta x}$. Taking the multilinear loss $$...
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18 views

using the Cramer-Rao bound to find the approximate variance of a bernoulli trial

Given you have an independent random sample $X_1, X_2,..., X_n$ of a Bernoulli random variable with parameter $p$, estimate the variance of the maximum likelihood estimator of $p$ using the Cramer-Rao ...
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23 views

What is the standard deviation of the sample distribution for the sample standard deviation?

I know that for the sample distribution for the sample mean given a large sample or a normal underlying distribution, the mean of the sample distribution is the population mean of the underlying ...
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45 views

rigorous statistics book recommendations

I am learning statistical inference by myself, I have skim through a few books like Casella Hoggs and I find it omitted lots of details, for example, they didn't introduce the conditional expectation, ...
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17 views

What statistical tests should I perform to determine significance in differences between control, treatment, and sham groups?

Say I have 3 groups of scores: one control, one treatment, and one sham. I want to determine whether there is a significant increase (or decrease) in the mean scores with the treatment compared to the ...
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18 views

Hypothesis testing using proportions

Suppose that you interview 100 exiting voters about who they voted for governor. Of the 100 voters, 55 reported that they voted for the Democratic candidate. Is there sufficient evidence to suggest ...
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23 views

Hypothesis testing and the size of critical region

I'm studying stats in order to obtain a certificate and I'm struggling with this question. I've searched a lot and I really didn't understand how to procede in this question. I even don't know how to ...
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85 views

How to solve an integratation involved an unknown function?

Can anyone have any suggestions how to solve this equation for $w_i$, that is, what is the solution of $w_i$? $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ $$ Where, $f(ε_i)$ is the ...
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30 views

How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
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10 views

Bayesian Inferences: Finding Posterior HPD Interval

I am currently working with Beta-Bernoulli and Beta-Binomial models. I have been searching around for the specific steps in obtaining the Posterior HPD intervals for both. Does anyone know how to find ...
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43 views

Finding the Cramer-Rao Lower Bound

Given the probability density function $$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$ and $0$ otherwise, find the Cramer-Rao Lower Bound for $\...
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22 views

Transformation of continuous random variables

Let's say that $Y = \log T = \alpha + \sigma W$. I know that If $W$ has logsitic distribution, the $T$ will have the log-logistic distribution. Also, if $W$ has the standard normal distribution then $...
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24 views

If the conditional expectation of the random variable does follow a linear function, can we show the probability a particular data set happens?

Suppose that $\mathbb{E}[Y\mid X=x]=\beta_0+\beta_1x$ where $X, Y$ are random varibles. Given a set of observations consisting pairs of $X,Y$, is it possible to attach it as probabiltiy density ...
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1answer
16 views

Moment generating function of quadruple-form of Gaussian RVs

Let $X \sim N(0, I_d)$ be a $d$-dimensional Gaussian random variable. Let $\beta_1$ and $\beta_2$ be two $d$-dimensional vectors. I would like to compute expectation \begin{align} \mathbb{E}\Bigl \{ \...
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20 views

Using the normal approximation, what is the $z$-value of a sample difference of $\hat p_1−\hat p_2=−0.18$? [closed]

A few concepts from my textbook that I do not understand: If the Random, Normal, and Independent conditions are met, then is it true that $\hat p_1+2\hat p_2$ is approximately normally ...
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18 views

how to determine the expected value of the first element of an ordered statistic

let $X$ be a random variable with density $$f_x(x) = e^{\alpha-x} \: x>a$$ (a) what is the maximum likelihood estimator for alpha (b) make the estimator in (a) unbiased, and evaluate its ...
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62 views

Estimate of Proportion

An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within two ...
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66 views

Obtain a $(1-\alpha) 100$% confidence interval for $\theta$ using the moment estimator

Suppose $x_1,..x_n$ is a random sample from a distribution with probability density $f(x|\theta )=\theta x^{\theta -1} 0<x<1 \ and \ \theta >0.$ Find the moment estimator of $\theta$ and ...
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1answer
19 views

One-sample t tests for sample mean with outliers

In carrying out a one-sample t test for a sample mean, how should outliers be dealt with? For example, for the data ${110, 110, 110, 118, 122, 150}$ in a sample size of 6, evidently $150$ is an ...
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9 views

How to find the effects of the parameters with the Latin Hypercube Simulation?

We are working with an epidemic model where we have 21 parameters which affect the $R_{0}$-value. We want to study the effect of those parameters on the $R_{0}$-value with the Latin Hypercube ...
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31 views

Why this equation holds, using Bayes formula?

My mysterious equation is: $$p(x|\chi)=\int_{\theta\in\Theta}p(x|\theta)p(\theta|\chi)d\theta$$ where $\chi$ is some samples drawn from sample space parameterized by $\theta\in\Theta$. Follows the ...
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38 views

Correctly calculating the bias of an estimator

I'm currently learning about method of moments and maximum-likelihood estimators and have confused myself with this issue: First, let me estimate the parameter $\lambda$ from the exponential ...
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2answers
28 views

Confidence intervals and significance tests

A sample is used both to construct a 95% confidence interval for a population proportion p and to run a significance test with null hypothesis $H_0:P=0.07$ and significance level $\alpha=0.05$. Is it ...
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Additive model ANOVA with interaction. How to obtain gammas from the graph?

I try to complete this exercise of statistic that said that I need to compute the values of alpha, beta and gammas from the graph. Graph: I allready compute the values algebricaly, and this are: <...
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11 views

How to prove the ancillary statistic problem?

Let $X_1, ...,X_n$ be a random sample from the following pdf: $f(x) = e^{-(x-\theta)} \exp(e^{-(x-\theta)}), -\infty<\theta<\infty$ a. Define $W=\log(\sum_{i=1}^{n}e^{-X_i})$.Shoe that $\bar{...
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Distribution of Subvector Sums

Suppose $X_1,\dots,X_N \sim_{iid} \mathcal{N}(0,1)$ are iid normal, and let $K=N/2$. Let $S$ denote the collection of all subsets of $\left\{1,\dots,N\right\}$ with $K$ elements. For any $s\in S$ ...
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18 views

estimating $\mu$ of a poisson random variable using method of moments or maximum likelihood

A used car salesman is willing to assume the number of sales he makes, per day, is a Poisson random variable with parameter $\mu$. Over the past 30 days, he made $0$ sales on $20$ days and one or more ...
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23 views

Simple random sample of a Bernoulli and probability function of a statistic.

Let $x_1, \ldots, x_n$ an simple random sample of a Bernoulli distribution of parameter p, $0<p<1$. Let $\overline{\textbf{x}}$ be the mean sample and $S^2=\frac{1}{n-1}\sum_{i_1}^n (x_i-\...
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21 views

For $X = (X_1,X_2,…,X_n)$ i.i.d. $N(0,\theta),$ show $ T = \bar X $ is not complete for $\theta$.

For $X = (X_1,X_2,...,X_n)$ i.i.d. $N(0,\theta),$ show $ T = \bar X $ is not complete for $\theta$. Definition: Let f (t, θ), θ ∈ Θ be a family of distributions for a statistic T (X). The family is ...
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68 views

Non-Linear Model Transformation

I want to transform this Non-Linear Model $y= 8-ae^{bx}$ to Linear.And my issue is in this step $lny=ln(8-ae^{bx})$ how can simplify it to reach in a linear model which is like this $y*=b0+b1x$ ...
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31 views

expected value of random variable times sample mean

Assume a population of $M$ numbers, where $M_i$ have values $y_i$, $i = 1,2,...,k$; thus $\sum_{i=1}^k{M_i} = M$. Thus the population mean is $\mu = \sum_{i=1}^k{\frac{M_iy_i}{M}}$ and the ...
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21 views

Can I bound $P[R > x + \epsilon]$ independently of R?

I have this probability distribution: $P[\Theta < \varphi] = \frac{\varphi}{\pi}$ for $\phi \in [0,\pi]$. Now I have $n$ samples of $D = R\Theta$ i.i.d. ($R>0$) and I want to estimate $R$ as $\...
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28 views

Definition of Power and relationship with Type II error

I've seen two definitions of Power: $P(\text{Rej. } H_0|\theta \in \Theta_1)$, from Wiki. $P_{\theta}(\text{Rej. } H_0)$, from Casella and Berger 'Statistical Inference'. Which one is true? If we ...
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18 views

Statistics for Sports League Qualification

How can one quantify and predict the needed points for qualify in a league given an up-to-date results registry? For instance, regarding Basketball Euroleague, there's 8 teams in a league with direct ...
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confidence about range of standard deviation

Suppose $X_1, X_2$ is a random sample of size 2 from a normal population known to have mean $0$ and variance $\sigma^2$; further assume $x_1 = -0.75$, $x_2=0.16$. How sure (or confident) would you be ...
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21 views

How do I show that the two methods of permutation test are both the same?

My main objective is to show the methods described below are really the same. However, I am having difficult both formulating the idea clearly and proving my assertion. Below is my attempt. Suppose ...
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28 views

Why can we simply pool the realized observations in a permutation test?

Let a vector of i.i.d random varibles $(X_1,X_2,X_3,\cdots, X_m)$ and another vector of i.i.d rvs $(Y_1,Y_2,Y_3,\cdots, Y_n)$ be given. Suppose $X_i$ stands for the recovery time using a new treatment ...
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16 views

Singular matrix in GMM based on variational Bayes computation

I have implemented some codes according to the variational inference here. In the code $W_k$ is computed according to $$ W_k^{-1}=W_0^{-1}+N_kS_k+\frac{\beta_0N_k}{\beta_0+N_k}(\bar x_k-\mu_0)(\bar ...
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20 views

Residual plot in the logarithmic model.

We've got some data containing two variables, where $x$ is the predictor and $y$ is the response variable. We make a model of the form of: $$y=\alpha+\beta \cdot x + \epsilon$$ Then we see that in the ...
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31 views

Bayesian inference for sum of random variables

Assume that we have a random variable $Z = X + Y$ for $X$ and $Y$ independent. Then if w use two independent data-sets $D_1$ and $D_2$ to try and approximate the distribution of $Z$, i.e. $$p(Z|D_1,...
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18 views

Why aren't the degrees of freedom included in the confidence interval formula?

I have one question. In the formula for calculating the confidence interval, I don't see the degrees of freedom in the formula. In my book, the formula is : $$\text{Confidence interval:} \qquad \mu ...