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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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Sufficient statistic for Poi($i\lambda$)

Let $X_1, X_2,...,X_n$ be n independent random variables, where $X_i$~Poi($i\lambda$), $i=1,2,...,n$. Is $\sum_{i=1}^n X_i$ sufficient for $\lambda$? I have no idea how to solve this problem. How do ...
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Sufficient statistic for a Uniform distribution uniform($i\theta$)

Let $X_1, X_2, ..., X_n$ be $n$ independent random variables, where $X_i$~uniform($i\theta$), $i=1,2,...,n$. Find a sufficient statistics for $\theta$. My Attempt The conditional distribution : $$ ...
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Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family. I know that the beta distribution is$f(x; \alpha, \beta)={1\over B(\alpha, \beta)}x^{\...
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Describe the statistical model for the observed data ($T$) [on hold]

A random sample of $6$ observations $(X_1, X_2, \cdots, X_6)$ is generated from a Geometric($\theta$), where $\theta \in (0, 1)$ unknown, but only $T = \sum_{i=1}^{6} X_i$ is observed by the ...
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How to validate my statistical model

I established some Bayesian model. Then reviewer said that I should validate this model. So, I replicate data $D_1,D_2,..D_n$ from known distributions whose parameter $\theta ^*$ (truth), and ...
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36 views

UMVUE of $\theta$ when $X_1,\ldots,X_n$ are i.i.d with pdf $f(x)=\frac{(\ln\theta)\theta^x }{\theta -1}$

I'm having some trouble finding the UMVU estimator of $\theta$ for the following distribution $$f(x)=\frac{(\ln\theta)\theta^x }{\theta -1} \text{ for } x \in (0,1)$$ Specifically, I know $T_n=\sum ...
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Show $\frac{X_{(i)}-X_{(1)}}{X_{(n)}-X_{(1)}}$ is independent of $(X_{(1)},X_{(n)})$ if $X_i$'s are i.i.d $U(\theta_1,\theta_2)$

Suppose $X_i$'s are i.i.d $U(\theta_1,\theta_2)$. Show that $\frac{X_{(i)} - X_{(1)}}{X_{(n)}-X_{(1)}}$ is independent of $(X_{(1)},X_{(n)})$ for all $2\le i\le n-1$. Specifically I'm trying to ...
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Skewness and Kurtosis of the Degenerate Distribution

If a random variable $X$ has a degenerate distribution, that is it takes a given value $k$ with probability $1$ and every other value with probability $0$, what is the skewness and kurtosis of $X$? ...
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How to create data out of sample?

I have a device, which measures events 1 out of R, where R is $O(10000)$. I started measurements at time $0$ up to time $T$, where $T$ is large. Now I have a sampled data set, recording event and its ...
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Responses to the study questions from Causal Inference in Statistics (by Judea Pearl).

Does anybody know a source where the correct answers to study questions from the mentioned book are described? I would like to validate whether my way of thinking is correct. I have found only two ...
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67 views

Is there any meaning of this “Median-mean”

Given a data set $\{a_1,\cdots,a_n\}$ with median M Define the medimean to be the value of $x$ s.t. $$\left(\frac{1}{n}\sum_{n}{}{a_n}^x \right)^{1/x}=M$$ Is this $x$ value useful / used at all in ...
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How to account/penalize for low sample size when computing performance?

I am interested in analyzing user performance at a per user level. Users have the opportunity to engage with an app by answering trivia questions and can choose how many questions they want to answer. ...
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1answer
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Is $T(X)$ a sufficient statistic for $\lambda$?

Let $X$ be a sample (size n = 1) from the exponential distribution, which has the pdf $$f(x;\lambda) = \lambda \exp(-\lambda x)$$ where $\lambda$ is an unknown parameter. Let's define a statistic as ...
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23 views

What is the sufficient statistic for a beta distribution?

Let {$X_1,\ldots,X_n$} be a random sample from the $beta(\alpha,\beta)$ distribution. Below is the beta distribution with the parameters referred to: $$f_X(x;\alpha,\beta)=\frac{\Gamma(\alpha + \...
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Reference request for “deep understanding” of generalized linear model theory

I'm currently studying in a generalized linear models course and I bought Nelder because I thought it was a "classic" in the field, which it may be, but it is not providing me with what I'm looking ...
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Modelling conditional distribution based on multiple variables of various types?

I have a looking basic statistics problem: basing on a large sample of multivariate data, model conditional probability distribution (continuous) of one variable based on the remaining ones: a few ...
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Posterior Distribution Function and Derivative

Here is my question. Let us consider a function $f(x)$ and its posterior distribution function (derived from certain datasets $d$) $P(f(x)|d)$. It is possible to obtain $P(f'(x)|d)$? I believe that it ...
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Why the confidence interval for proportion does not depend on the population size? [duplicate]

The formula for a 95% confidence interval for proportion is $\hat{p}\pm 1.96 \sqrt{p(1-p)}$ Suposing a sample of 200, calculating a proportion of 1.8%, the confidence interval is 1.67% - 1.93%. But, ...
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Support of Variance of Random IID Sample (Bounded)

Let $X_i$, $i\in\{1, 2, ..., n\}$ be independent and identically distributed random variables with bounded support $[\alpha, \beta]$, with $\alpha,\beta \in \mathbb{R}$. What is the support of ...
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Derivation for marginal effect sizes' distribution in linear model

Model: $$Y_{N\times 1} = X_{N\times M} \beta_{M\times 1} + \epsilon_{N\times 1}$$ the design matrix $X_{N\times M}$ is known, each column of the design matrix has been standardized to have mean 0 and ...
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Which of the following statements are correct regarding MLE

MLE here implies Maximum likelihood estimator. Statements are : $1$. MLEs are always consistent $2$. MLEs are always unbiased $ 3$. MLEs follow normal distribution asymptotically ...
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How do I calculate the F value from the Null Hypothesis of equality of means

This question comes from a practice exam, the following contains the question and the answer provided. However my teachers for this exam are non-respondent and I really need someone to please help ...
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Intuition behind convergence of MCMC inference methods

I'm studying Gibbs Sampling for inference, a popular MCMC algorithm and I was stunned by its ability to fit a Gaussian Mixture just by sampling. I would like to know the intuition behind it, and why ...
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Is there any relationship between efficiency and correlation coefficient?

Let $t_1$ be the most efficient estimator and $t_2$ be the less efficient estimator with efficiency $e$ and let $r$ be correlation coefficient between the two estimator $t_1$ and $t_2$.Define ...
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2answers
32 views

Is estimator $\dfrac{\bar X}{1+\bar X}$ of $\theta$ is consistent?

Let $X_1,X_2,X_3.....X_n$ be a random sample from a population X having the probability density function $$ f(x;\theta) = \begin{cases} \theta x^{\theta -1} & \text{if $0 \le x\le$ 1} \\0 &...
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33 views

MLE of $\lambda$ Given $f(x;\lambda)=1-\dfrac{2}{3}\lambda+\lambda\sqrt{x}$

$f(x;\lambda)=1-\dfrac{2}{3}\lambda+\lambda\sqrt{x}\ \ \ ; 0\le x\le1 $ $0$ otherwise What is the maximum likelihood estimate of the parameter $\lambda$ based on two independent observations $...
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28 views

Minimum Sample size between two samples for a specified confidence interval/level with sigma known

Hi math stack exchange, I came across the following question and found it quite interesting and am struggling to solve it. I haven't seen anything like it because it is both two sample and has sigma ...
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Hypothesis testing using the 95% Confidence Interval of Sample Mean

I have a population mean of 120 (mu). I have a sample distribution with a mean of 131.05 and a standard-deviation of 11.00945. I have a sample size of 20, 19 degrees of freedom (n-1). I am performing ...
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Is it appropriate to use Mann-Whitney U test for each of four outcomes when outcomes are mutually exclusive?

I have an experiment in which we randomize primate subjects into high condition (n=9) and low condition (n=8) groups, and document each subject's behavioral response to 15 consecutive food exposure ...
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Does a zero conditional expectation imply pairwise covariance is 0?

Suppose in econometrics, $$ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + ... + \beta_{k}x_{k} + u$$ In Gujarati's book, it says that the following equation (1) $$ E[u | x_{1}, x_{2},..., x_{k}] = ...
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How do I calculate SS of a Variable and Error provided data

Yields are noted for tree samples from four different varieties in crops in Argentina. The following varieties are: Variety A = 15, 14, 12, 13 Variety B = 11, 18, 13 Variety C = 18, 25, 19, 20 ...
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What is the basic difference between interpolation & inference?

In mathematics we've studied interpolation as predicting the structure of a function from it's given finitely many values from which consequently we use to construct certain function which nearly ...
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1answer
33 views

Find optimal Bayesian decision based on a criterion for the posterior mean

I recently ran into the following exercise when practicing for an exam I have about Statistical Inference. The question looks very large and complex and I'm wondering if this is actually true or if it ...
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1answer
32 views

Compute the posterior density for r

I'm trying to understand the posterior distribution. For a simple example if i consider the beta function with parameters $\alpha=1=\beta$. Then the prior would be uniform in the range of 0 to 1 so $p(...
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Derive the Hajek pojection of $T_n$.

Let $X_1, \dots , X_n$ i.i.d. copies of $X$ with distribution $F$ and density $f$. Let $(X_{1:n}, \dots , X_{i:n}, \dots , X_{n:n})$ be the order statistic. For a given $p \in (0, 1)$ consider the ...
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least square failure as classifier

I was reading pattern recognition and machine learning by Christopher bishop in chapter 4.1.3 page 186 about least square classification failure I stumbled on this phrase "The failure of least ...
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Std Deviation of a point estimate which is the sum of two normally and independently distributed random variables

The problem States: Given $\bar x= 41$ and $\bar y= 40.7$. $σ_x= 0.1$ and $σ_y= 0.19$ $X \sim N[\mu_X; \sigma^2]; Y \sim N[\mu_Y ; \sigma^2]$; with $\mu_X > 0,\; \mu_Y > 0, \sigma > 0$ ...
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Find the MLE of $p$ where $f(y;p)=2p^2y^{-3}$

Find the MLE of $p$ where $f(y;p)=2p^2y^{-3}$. Attempt: Method: find the likelihood function, differentiate with respect to $p$ then set to zero and solve for $p$. $L(p;y)=\prod\limits_{i=1}^n [2p^...
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62 views

Likelihood Ratio Test Variance of Normal Distribution

Let $X_1,...,X_n$ be a random sample from $N(0,\sigma_X^2)$ and let $Y_1,...,Y_m$ be a random sample from $N(0,\sigma_Y^2)$. Define $\alpha := \sigma_Y^2/\sigma_X^2$. Find the level $\alpha$ LRT of $...
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How to find a one dimensional sufficient statistic.

Please could you help check my approach to finding a one dimensional sufficient statistic. Here is the problem. $X_1,...,X_n$ is a random sample with each $X_i$ having the pdf $f(x;\theta)=2\theta^...
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Using change of variables to transform density functions

I'm was working on some exercises on statistical inference and came across a question I could not solve. After a while I decided to take a look at the solution to hopefully understand the problem ...
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67 views

Existence of MLE

I have a problem with MLE's definition: Casella Berger in Statistical Inference and Nitis Mukhopadhyay in Probability and Statistics said that MLE for a parameter $\theta\in\Theta$ is respectively $...
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Using the delta method, is there any clear procedure?

I am currently enrolled in a course in theoretical statistics, exams are comming up and I am trying to study. The course litterature we are using is Van der Vaart's Asymptotic Statistics. I am having ...
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Question on non-parametric statistics

How to estimate quartile function Q(p)=inf{x∣F(x)>p} and survival function F(x):=1−F(x) by non-parametric estimation method
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Standardize binomial variable for non-constant meta-population - binomial z-scores

Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test ...
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How to prove that the following estimators are biased and consistent?

Given a random variable $X$ following a geometric distribution with parameter $p.$ Then one estimator that can be obtained by considering the second moment $E[X^{2}]=\frac{2-p}{p^2}$, which is $$\hat{...
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Binomial distribution/conjugate posterior

Let's consider some experiment with tossing a coin. NOTE: my question is given at the very last paragraph. Observation $y=0$ or $y=1$ [tails (T) or heads (H)], $p \in [0, 1]$ (probability of heads) ...
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Parameter estimation derivation of equations for lower bound in LDA with EP

I am working on the derivations of EP for LDA, and I don't understand how the authors derived the last equations. Basically, they get the following expression (the lower bound in eq 29): $L=\int(\...
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How do convergence rates and small o notation relate when it comes to the asymptotic behavior of estimators?

I am reading a paper on how to estimate causal effects using tools usually associated with the machine learning literature (I provide the link, if you want to give it a look): https://www.econstor.eu/...
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statistical test of normal and non-normal distributed data

I have 2 samples of data. The first sample is the precision of the first algorithm A for 25 test cases and the second sample is precision for 25 test cases of second algorithm named B. I test both ...