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Questions tagged [statistical-inference]

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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Two aspects of randomness

Consider a random sequence of integers 1, 4, 3, 8, 2, 5, 3, 8 ... The only sufficient condition for the sequence to be random is its unpredictability ie. probability of any number coming next ...
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Deriving the UMVUE for Rayleigh scale parameter

Let $X_1,...,X_n$ be iid with the pdf given by $f(x|\theta)=2\theta^{-1}xe^{-x^2/\theta}$ for $x>0$. My task is to find the UMVUE for $\theta$, and I’m given the following hint: “$U(X)=\sum_{i=1}^...
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Deriving Rao-Blackwellized version of unbiased estimator

Let $X_1,...,X_n$ be iid Poisson($\lambda$) with $n\geq 4$. We are given the unbiased estimator $T(X)=I(X_1=0 \cap X_2=0 \cap X_3=0)$ for $f(\lambda)=e^{-3\lambda}$, and my task is to derive the Rao-...
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p-value of a test statistic on a two-sided test

For coursework, I am doing a two-sided test ($H_0 \beta = 0, H_a \beta \neq 0$). The test itself is a generalized likelihood ratio and the test statistic, LR follows a $\chi_1^2$. But I think my ...
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Generalised Linear Models: Binary data

I am currently working on GLM problem. My response variable is binary as are some of my explanatory variable,others are categorical i.e. 1-1day, 2- 2-3days, 3-5+days and so forth. I have coded it ...
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Distribution of the sum of n independent variables of the exponential family.

Suppose you have $n$ random and independent variables $Y_{1},...,Y_{n}$ whose distribution belongs to the uniparametric exponential family. How do I find the distribution of $\sum_{i=1}^{n} Yi$ ?
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Joint mass function of two perfectly correlated categorical variables

Is it possible to derive the joint probability mass function of two discrete random variables (categorical in my specific case) if we know that they are perfectly correlated? Let's assume for ...
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Importance Sampling question from Casella and Berger

The question is extracted from Casella and Berger exercise 5.61 A technique similar to Accept/Reject is $importance\quad sampling$, which is quite useful for calculating features of a distribution. ...
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Probability of an indexing random variable

Assume we have $n$ cards, indexed successively by the integers $1$ to $n$. Now each card is either marked or not. Suppose that card $i$ is marked with probability $p_i$, independently of the others. ...
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Bayesian Inference and Posteriors

I obtained the following posterior distribution function for a parameter, $K_0$. $K_0$ is a parameter that I derived from 3 other parameters, whose posteriors where obtained using emcee. In general ...
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Help with gaussian algebra for bayesian inference

I want to better understand the step for calculating the message from the game factor $h_{g}$ down to the difference variable $d_g$ on the TrueSkill factor. Such message is shown in the Rasmussen's ...
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Multiple Random Variable for a uniform distribution

A random point $(X,Y)$ is distributed uniformly on the square with vertices $(1,1)$, $(1,-1)$, $(-1,1)$, and $(-1,-1)$. That is, the joint pdf is $f(x,y)=\frac{1}{4}$ on the square. Determine the ...
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Show $E\left[\left(\frac{\partial}{\partial\theta} \ln f(X)\right)^2\right]=-E\left[\frac{\partial^2}{\partial\theta^2} \ln f(X)\right]$

I encountered a question given $\theta$ in a random sample of size $n$ and also $$-n\times E\left[\frac{\partial^2}{\partial \theta^2}\times \ln f(X)\right]$$ , where $f(x)$ is the p.d.f. at $x$, ...
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3answers
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Type-I vs. type-II error in statistical hypotheses testing

Let us consider standard statistical hypotheses testing: $$\alpha=P\{\text{type}-I \text{ error}\}=P\{\text{Rejecting } H_0 \text{ when }H_0\text{ is true}\}$$ and $$\beta=P\{\text{type}-II \...
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Division of two population variances

Why do we divide variances of two samples / population while estimating while for mean and proportion we take difference of two population .What is the reason behind division of variance?
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p-value, hypotheses testing, type II error

In hypotheses testing, the $p$-value is the smallest significance level $\alpha$ at which the test rejects $H_0$ if $H_0$ is in fact true. Is there a similar concept as is $p$-value also for the type-...
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What is the reason that Student-t Distribution is used when the number of samples is small

Let $\bar{X}$ be the distribution of sample mean for $n$ identical and independent distributed as Normal distributions $N(\mu, \sigma^2)$. The random variable $$ \frac{\bar{X} - \mu}{\frac{\sigma}{\...
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Posterior cdf of bernoulli and uniform prior

Let X1, . . . , Xn i.i.d. Bernoulli(θ) with a uniform prior. Show that the posterior density of ψ = log(θ/(1 − θ)) is $$h(ψ|x) = \frac{Γ(n + 2)}{Γ(s + 1)Γ(n − s + 1)}(\frac{e^ψ}{1 + e^ ψ})...
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When should we include the “Replicate” in an ANOVA table for Factorial Design?

I already asked on stat.stackexchange, but I thought it wouldn't hurt to ask here as well. The tags may not be the best options but on here they are the closest to an ANOVA tag I could find When I ...
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If $X_i\sim N(0,\frac{1}{\theta})$, find $E\left(\frac{1}{\sum_{i=1}^n X_i^2 +2}\right)$

The initial question states that the $X \sim \mathcal{N}(0,\frac{1}{\theta})$, where $\theta$ follows an exponential distribution with parameter equal to 1. We are asked to derive the Bayesian ...
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solution verification to exercises on hypothesis testing and confidence intervals

The two-sided 95% confidence interval of µX based on 10 independent (inaccurate) measures X of the distance (in metres) between a point A and a point B with a certain measuring instrument is $$[110; ...
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Bayesian Linear Regression

Currently I am trying to understand Bayesian linear regression and there are several things I dont understand. First of all we have $$ p(\beta,\sigma^2|y,X) = \frac{p(y|\beta, \sigma^2, X)p(\beta,...
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Finding BLUE of $\theta$ where $X_1,\ldots,X_n$ have common pdf $f(x)=\frac{1}{2\theta}e^{-|x|/\theta}$

Let $X_1,...,X_n$ have the common pdf $$f(x)=\frac{1}{2\theta}\exp\left(-\frac{|x|}{\theta}\right)$$, where $x$ can be any real number and $\theta$ is positive. I’m trying to construct the best ...
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sampling from an existing sample

I have an existing data set with 8 000 observations. I want to check an association between two parameters in the data set. I also want to see if maybe for this purpose I do not need 8 000 ...
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Hypothesis test between two coins (Please Help!)

Alice has two coins. The probability of Heads for the first coin is 1/4, and the probability of Heads for the second is 3/4. Other than this difference, the coins are indistinguishable. Alice chooses ...
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1answer
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Rate of change in a logarithmic model

Given the following model: $$\ln{\left(W\right)}=\beta_0+\beta_1e+\beta_2e\ast x+\beta_3e\ast\ln{\left(P\right)} +\beta_4\ln{\left(P\right)}\ast x+ error.$$ Setting $x = 4$, and $e =3$, a $1$% ...
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Estimate median of Cauchy distribution

Motivated by this question, assume we have independent samples $(X_i)_{i=1}^{\infty}$ from a Cauchy distribution with unknown median $a\in\mathbb{R}$ and scale parameter $b$. What is the best way to ...
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1answer
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Gamma Distribution Moments

Show that for X ~ Gamma($\alpha$, $\beta$), for positive constant $\nu$, $E[X^\nu] = \dfrac{\beta^\nu*\Gamma(\nu + \alpha)}{\Gamma(\alpha)}$. I have the following solution: Solution However, I don'...
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Estimate signal-to-noise ratios by regression.

Suppose we need to measure $y^*_i$ where $i=1,...,n$. We have two independent noisy measurements, $y_{1i}=y_i^*+\xi_i$ and $y_{2i}=y_i^*+\eta_i$, where $\textrm{CoV}(\eta_i,y_i^*)=\textrm{CoV}(\xi_i,...
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P-value vs. Bayesian statistics

Are there any theoretical considerations between $p$-value and the Bayesian statistics? I mean say, any theorem regarding both of these two concepts at a time.
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Probability and Statistic

I have a question about how to calculate the confidence interval. The problem is: I have a model which gives the probabilities of 4 genotypes AB, Ab, aB and ab as (1/4)(2 + theta), (1/4)(1 - theta), (...
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Rational choices payment plans

I am trying to research what a rational choice constitutes; a financially sound decision based on critically examining a set of data and concluding that the expected value for a given choice is higher ...
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Bayes (optimal) classifier for binary classification with asymmetric loss function

We tweak the usual 0/1 loss function and define the following : $$L_{a,b}(g) = aP[g(X)=1,Y=0] + bP[g(X)=0,Y=1]$$ where $g$ is the classifier, and $(X,Y)$ a random pair modeling the data with labels ...
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To sum uniform (0,1) random variables and to show the natural logarithm.

(The question is extracted from Casella and Berger, Statistics Inference exercise $5.58$) Suppose that $U_1,U_2,...U_n$ are iid uniform $(0,1)$ random variables, and let $S_n=\sum_{i=1}^nU_i$. Define ...
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Hypothesis testing - maximum significance level

I need a clarification of the sentence included in bold. Is that the error type II? The exercise is: The weight of a plate is a continuous random variable, normally distributed with $\mu=330$ gr. ...
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Definition of BLUE

I am tasked with finding the best linear unbiased estimator (BLUE) for the population mean based on $X_1,...,X_n$ iid $Poisson(\lambda)$. My question is, am I supposed to find the best linear unbiased ...
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1answer
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UMVUE of $e^\theta$ when $X_1, X_2, \dots, X_n$ are i.i.d. Uniform[$0, \theta$]

$X_1, X_2, \dots, X_n$ are i.i.d. Uniform[$0, \theta$]. I was able to get that the order statistic $Y_n$ is sufficient and complete. How do I get the UMVUE of $e^\theta$? I was thinking of doing ...
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Minimal sufficiency and completeness

Let $X_1,X_2... X_n$ be i.i.d $N(\theta,\theta^2)$.Then why is $(X_{bar}, S^2)$ not complete despite the fact that $f(x,\theta)$ belongs to $k$-parameter exponential family and jointly minimal ...
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1answer
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Proving the uniqueness of an unbiased estimator

Let $X$ be a random variable having pmf: \begin{array}{ll} p(x)=2 \theta \ \ \ \text{if} \ x=-1 \\ p(x)=\theta^2 \ \ \text{if} \ x=0 \\ p(x)=1-2\theta-\theta^2 \ \ \text{if} \ x=1\\ ...
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1answer
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Expectation of sample mean, given maximum and minimum order statistics [duplicate]

Let $X_1, · · · , X_n$ be i.i.d. $\mathrm{Uniform}[\alpha, \beta]$, where $\alpha$ and $\beta$ are unknown. Show that $$ E(X_\ast|X(1), X(n)) = \frac{X(1) + X(n)}{2} $$ where $X_\ast$ is the ...
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Is a $t$-statistic still $t$-distributed under linearly transforming the original OLS model?

For an OLS model $y=X\beta +\epsilon$ it is well known that any of the estimated coefficients, say $\widehat\beta_i$, divided by its estimated standard error $s_{\widehat\beta_i}$, must follow a $t$-...
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Would this be a binomial probability problem?

Asthma is an important health problem for inner-city children, frequently resulting in hospital admissions if symptoms become exacerbated. A study is proposed in which children will be randomized to ...
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1answer
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Attempt at Finding the Best Linear Unbiased Estimator (BLUE)

Let $X_1,...X_n$ be iid $N(0,\sigma^2)$ where $\sigma$ is unknown. My task is to find the BLUE for $\sigma$ within the set of linear functions of $|X_i|$ for $i=1,...,n$. Here is my work thus far: ...
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Linear Regression.

I have been going through the book called elements of statistical learning, in which I came across the given below equation for the linear regression solution. $$RSS(β)=(Y−Xβ)^T(Y−Xβ)$$ on expanding ...
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How do I choose functions to get a better M-estimator?

Suppose $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}$ are random vector and random variable. Suppose I have $f_1(\cdot)$ and $f_2(\cdot)$ such that $E[Y|X]=f_1(\theta_0^T X)$ and $E[Y|X]=f_2(\theta_0^T X)$...
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Uniform distribution, chi square test

What test or procedure can I use to determine the best estimate $\alpha\in [0,1]$ whether given $N$ numbers come from the uniform distribution in the interval $[0,\theta]$ for a given $\theta>0$? I'...
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Hypothesis testing: two tailed vs one tailed test paradox

Let $H_0$ = population mean is 50 $H_{alt1}$ = population mean is less than 50 $H_{alt2}$ = population mean is not equal to 50 for $H_{alt1}$, we do one tailed test, for $H_{alt2}$ we do two ...
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1answer
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Derivation of Ordinary Least-Squares Slope Estimate

I understand the reasoning behind the fact that $\hat{\beta}_1 \sim N(\beta_1,\frac{\sigma^2}{SSX})$. However, in trying to prove the full formula for estimating $\hat{\beta}_1$, I am having issues/...
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1answer
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Attempt at proving sample variance is unbiased

Assume that $X_1,...X_4$ are idd $N(0,\sigma^2)$ where $\sigma$ is unknown. My task is to show that $T=\frac{1}{3}\sum_{i=1}^4(X_i-\bar X)^2$ is unbiased. I am aware that this question has been asked ...
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1answer
33 views

$p$-value,the significance level

Here in the definition 5.3.11 on the page 249 they write "The $p$-value associated with a test is the smallest significance level $α$ for which the null hypothesis is rejected." My question is what ...