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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Deriving Joint probability density functions

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 ...
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23 views

$U$ as a random variable? And what's with this integration becoming $1$?

I'm confused about two things. For the set up, I am told that Likelihood function is $L(\theta)=\Pi f(y_i;\theta)$ for a distribution with pdf $f(y;\theta)$. Log likelihood function is $l(\...
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14 views

Horseshoe estimator posterior

Suppose given the Horseshoe estimator: $Y|\beta,\sigma^2 \sim N(X\beta,\sigma^2 I)$ $\beta|\sigma^2,\tau_{1}^2,...,\tau_{p}^2 \sim N(0,\sigma^2 D)$ $\tau_{j} \sim C+(0,1)$ $\sigma^2 \sim \pi (\...
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22 views

how to find AIC values for both models using R software?

I'm studying survival analysis. I estimated both Cox regression model and Buckley&James regression model. In order to determine which model is better for my dataset, I used Akaike Information ...
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43 views

Using head-to-head results and Bayes' Theorem to modify predictions of sport/game contests that are initially derived from Elo-type ratings

I am working on an extension of the Glicko2 rating system to use in predicting the outcome of sport/game contests that uses the actual head-to-head results of previous meetings of competitors to ...
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29 views

Determining the degree of freedom for a $\chi$-squared test

I have read that the degree of freedom is calculated by subtracting $1$ from the number of states a random variable can be in. I am performing a goodness of fit test on a $64\times 32$ matrix where ...
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Help: SPSS and Data Interpretation of Voters. Republican vs. Democrats (1993 election)(Almost finished)

Hello everyone, I am Julieta this time I get stuck in the following exercise. It is a statistical analysis of pools, the statement is quite long I will try to keep it short and put some links. Note: ...
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11 views

Comparing expected counts to observed counts

The question is as follows: A restaurant offers 7 different dishes and predicts the dishes will be ordered in the following proportions: 1 (25%), 2 (20%), 3 (10%), 4 (15%), 5 (5%), 6 (6%), 7 (15%). ...
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$MLE$ of a multinomial. [CORRECTED]

It might be a very silly question, but I just can't figure it out. Let $X_1,...,X_n$ be random variables with pmfs: $$f(k,p)= \begin{cases} p_1, \hspace{0.5cm} \text{if } k=a\\ p_2, \hspace{0.5cm} \...
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Existence of asymptotic variance for an estimator when it doesn't converge to normal distribution.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,\ldots)$, where: $U_i=U_i(X_1,\ldots,X_i)$, if for a sequence of constants $\{k_n\}$: $$...
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How to find n=# of samples

My question is what is the equation to find n if you know standard deviation, and the CI of 95%. Like what is the equation for n?
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Sample Estimate of the Mean of a Population Lying in [a,b]

Suppose I have a population of $N\ge 1$ real numbers, all known to lie in the real interval $[a,b]$, where $a,b\in\mathbb{R}$, $a\le b$, and $a$ and $b$ are known values. I know nothing else about the ...
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24 views

Two sample testing, when all the parameters are unknown

Suppose $(X_1,...,X_p)$ is a sample of independent $\mathcal{N}(μ_1, σ_{1}^2)$ random variables, and $(Y_1, . . . , Y_q)$ is a sample of independent $\mathcal{N}(μ_2, σ_2^2)$ random variables. If all ...
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Existence of asymptotic variance for an estimator when it doesn't converge to normal.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,...)$, where: $U_i=U_i(X_1,...,X_i)$, if for a sequence of constants $\{k_n\}$: $$k_n(U_n-\...
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46 views

Asymptotic variance of MLE of normal distribution.

I am trying to explicitly calculate (without using the theorem that the asymptotic variance of the MLE is equal to CRLB) the asymptotic variance of the MLE of variance of normal distribution, i.e.: $$\...
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Comparing definitions of limiting and asymptotic variances - what is the intuition behind?

In Casella's inference, it says: Definition 10.1.7: For an estimator $T_n$, if $\lim_{n\to \infty} k_n Var T_n = \tau^2 < \infty$, where $\{k_n\}$ is a sequence of constants, then $\tau^2$...
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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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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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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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51 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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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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76 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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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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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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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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19 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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52 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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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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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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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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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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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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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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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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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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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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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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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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1answer
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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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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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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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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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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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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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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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{...