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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18 views
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 :
$$ ...
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Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.
Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.
I know that the beta distribution is$f(x; \alpha, \beta)={1\over B(\alpha, \beta)}x^{\...
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1answer
30 views
Describe the statistical model for the observed data ($T$) [on hold]
A random sample of $6$ observations $(X_1, X_2, \cdots, X_6)$ is generated from a Geometric($\theta$), where $\theta \in (0, 1)$ unknown, but only $T = \sum_{i=1}^{6} X_i$ is observed by the ...
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How to validate my statistical model
I established some Bayesian model. Then reviewer said that I should validate this model. So, I replicate data $D_1,D_2,..D_n$ from known distributions whose parameter $\theta ^*$ (truth), and ...
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36 views
UMVUE of $\theta$ when $X_1,\ldots,X_n$ are i.i.d with pdf $f(x)=\frac{(\ln\theta)\theta^x }{\theta -1}$
I'm having some trouble finding the UMVU estimator of $\theta$ for the following distribution
$$f(x)=\frac{(\ln\theta)\theta^x }{\theta -1} \text{ for } x \in (0,1)$$
Specifically, I know $T_n=\sum ...
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27 views
Show $\frac{X_{(i)}-X_{(1)}}{X_{(n)}-X_{(1)}}$ is independent of $(X_{(1)},X_{(n)})$ if $X_i$'s are i.i.d $U(\theta_1,\theta_2)$
Suppose $X_i$'s are i.i.d $U(\theta_1,\theta_2)$. Show that $\frac{X_{(i)} - X_{(1)}}{X_{(n)}-X_{(1)}}$ is independent of $(X_{(1)},X_{(n)})$ for all $2\le i\le n-1$.
Specifically I'm trying to ...
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1answer
15 views
Skewness and Kurtosis of the Degenerate Distribution
If a random variable $X$ has a degenerate distribution, that is it takes a given value $k$ with probability $1$ and every other value with probability $0$, what is the skewness and kurtosis of $X$? ...
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How to create data out of sample?
I have a device, which measures events 1 out of R, where R is $O(10000)$.
I started measurements at time $0$ up to time $T$, where $T$ is large.
Now I have a sampled data set, recording event and its ...
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1answer
23 views
Responses to the study questions from Causal Inference in Statistics (by Judea Pearl).
Does anybody know a source where the correct answers to study questions from the mentioned book are described?
I would like to validate whether my way of thinking is correct.
I have found only two ...
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1answer
67 views
Is there any meaning of this “Median-mean”
Given a data set $\{a_1,\cdots,a_n\}$ with median M
Define the medimean to be the value of $x$ s.t.
$$\left(\frac{1}{n}\sum_{n}{}{a_n}^x \right)^{1/x}=M$$
Is this $x$ value useful / used at all in ...
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14 views
How to account/penalize for low sample size when computing performance?
I am interested in analyzing user performance at a per user level. Users have the opportunity to engage with an app by answering trivia questions and can choose how many questions they want to answer. ...
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1answer
62 views
Is $T(X)$ a sufficient statistic for $\lambda$?
Let $X$ be a sample (size n = 1) from the exponential distribution, which has the pdf $$f(x;\lambda) = \lambda \exp(-\lambda x)$$ where $\lambda$ is an unknown parameter. Let's define a statistic as
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1answer
23 views
What is the sufficient statistic for a beta distribution?
Let {$X_1,\ldots,X_n$} be a random sample from the $beta(\alpha,\beta)$ distribution.
Below is the beta distribution with the parameters referred to:
$$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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20 views
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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3 views
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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17 views
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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2answers
22 views
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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5 views
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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52 views
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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1answer
20 views
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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0answers
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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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17 views
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
$$ f(x;\theta) = \begin{cases} \theta x^{\theta -1} & \text{if $0 \le
x\le$ 1} \\0 &...
1
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1answer
33 views
MLE of $\lambda$ Given $f(x;\lambda)=1-\dfrac{2}{3}\lambda+\lambda\sqrt{x}$
$f(x;\lambda)=1-\dfrac{2}{3}\lambda+\lambda\sqrt{x}\ \ \ ; 0\le x\le1 $
$0$ otherwise
What is the maximum likelihood estimate of the parameter $\lambda$ based on two independent observations $...
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Minimum Sample size between two samples for a specified confidence interval/level with sigma known
Hi math stack exchange,
I came across the following question and found it quite interesting and am struggling to solve it. I haven't seen anything like it because it is both two sample and has sigma ...
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2answers
26 views
Hypothesis testing using the 95% Confidence Interval of Sample Mean
I have a population mean of 120 (mu). I have a sample distribution with a mean of 131.05 and a standard-deviation of 11.00945. I have a sample size of 20, 19 degrees of freedom (n-1). I am performing ...
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12 views
Is it appropriate to use Mann-Whitney U test for each of four outcomes when outcomes are mutually exclusive?
I have an experiment in which we randomize primate subjects into high condition (n=9) and low condition (n=8) groups, and document each subject's behavioral response to 15 consecutive food exposure ...
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1answer
30 views
Does a zero conditional expectation imply pairwise covariance is 0?
Suppose in econometrics,
$$ 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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1answer
18 views
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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32 views
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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12 views
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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1answer
22 views
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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23 views
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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1answer
49 views
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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1answer
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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1answer
30 views
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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1answer
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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24 views
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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21 views
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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1answer
43 views
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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28 views
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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0answers
11 views
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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13 views
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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23 views
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 ...
