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.
2,476 questions
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What would be a consistent estimator for the mean in this simple case of INID random variables?
Let there be a set of observations $\{Y_0, Y_1, \dots, Y_n \}$ from a stochastic process $\{ Y_t\}_{t \in \mathbb{N}}$ where $Y_t \sim N( \theta^t \mu, 1)$, $Y_t$ is independent of $Y_s$ for all $s \...
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1answer
41 views
Effect of deviations from a normal distribution on Wilcoxon signed-rank test
Which type of deviations from a normal distribution, skewness or heavy-tailedness,
appears to have a greater effect on the Wilcoxon signed-rank test? Why?
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1answer
18 views
Why must this ratio of these likelihood functions be less than $1$?
let $X_1, \dots, X_n$ be a random sample from a binomial $\text{bin}(k,p)$ population where $p$ is known and $k$ is unknown. We attempt to maximize $L(k | x,p)$ the likelihood function without ...
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1answer
33 views
How do you determine if an occurrence in a subset is significant.
There are roughly $22,000$ genes.
I have $1,200$ genes randomly chosen from the 22K in Group $A$.
I have $80$ genes in Group $B$ randomly chosen from the 22k.
How do I determine the probability of at ...
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2answers
66 views
+50
MLE of Negative Binomial distributions of different sizes
There are two teams that are competing in a series of matches. These matches are a best-of-$x$ format, so after either team wins $\lfloor{\dfrac{x}{2}}\rfloor+1$ the series is over. This is, in ...
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1answer
25 views
Find $c$ such that the test which rejects when $X>c$
Consider a random variable $Z$ having pdf $f(z)=\frac{1}{2} e^{-|z-\mu|}$ , where $z$ is real.We observe $X=max(0,Z)$.
Find $c$ such that the test which rejects when $X>c$ has size $0.05$ under $...
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11 views
Test for equal means
I have a control experiment and an experiment. In these two experiments, I can detect different peptides. The presence of the peptides in the experiment is measured by determining the concentration of ...
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1answer
39 views
How are these order statistics inequalities derived?
Let $\theta \lt x_{(1)} \lt x_{(n)} \lt \theta + 1$ be ordre
statistics for $X_1, \dots, X_n$ iid uniform random variables on
$(\theta, \theta + 1)$.
Let $R = X_{(n)} - X_{(1)}$ and $M = (...
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1answer
31 views
Neyman-Pearson Theorem Question
Find the Neyman-Pearson test with size $\alpha$ to contrast
$$ H_0: \beta = 1$$
$$ H_1: \beta = \beta_1$$
with $\beta_1$ > 1 based in a sample of size 1 of the random variable with density:
$$...
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19 views
Distribution of Wald test
My teacher said, about testing significance of coefficient, that the Wald test has an inverse normal distribution, that is to verify the null hypothesis that the coefficient $\beta_j=0 $ vs the ...
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18 views
Parametrising a sparse orthogonal matrix [migrated]
I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $A A^...
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19 views
Chi-squared test for independence problem
I have the following exercise to do:
The U.S. has bilateral extradition treaties with many countries. ( A
person charged with a crime in his home country may escape to the
U.S.; if he is ...
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1answer
23 views
Factorization theorem proof question
From Statistical Inference by Casella and Berger:
$(1)$ $T$ is a statistic for $X$,
$(2)$ $q(T(x) | \theta)$ is the pmf of $T(X)$ and $\theta$ is a parameter,
$(3) $$A_{T(x)} = \{y : T(y)...
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1answer
37 views
normal test to student test, logic behind it
Let's say we have :
$$ X_i \sim \ N( \mu, \sigma^2 ) $$
iid
I'm constructing this test function, in order to test two hypothesis on $\mu$ : $$ \mathbb { 1} {\{ \sum^n X_i < q_a \} } $$
where $q_a$...
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0answers
42 views
consistency of Neyman Pearson lemma in the case simple vs simple test for exponential families
Basics, jump to section 2 for the question :
I know that in the case of an exponential family with 1 parameter, meaning the distribution function of the sample variables can be written like :
$$ f_X(...
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21 views
Probability of the shaded area is $P(V \le y, U \le \frac{1}{c}f_Y(V))$?
From Statistical Inference by Casella and Berg:
We are trying to generate a random variable $Y \text{ ~ }
\text{beta}(a,b)$ distribution. Let $a = 2.7, b = 6.3$. In the graph
below we have ...
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1answer
37 views
Maximum likelihood estimator of $f(\theta)=e^{-(x-\theta)}$
Let $X_1, ..., X_n$ be a random sample of a random variable with density function:
$$f(x,\theta)=\left\{
\begin{array}{c}
e^{-(x-\theta)}, {\ \ \ }x \gt \theta \\
0 \ , \ \ x \geq \theta
\end{...
2
votes
2answers
43 views
How does $Y_n$ approach $\theta$ in probability here?
From Statistical Inference by Casella and Berger:
$\text{(Delta Method)}$ Let $Y_n$ be a sequence of random variables that
satisfies $\sqrt{n}(Y_n - \theta) \rightarrow n(0,\sigma^2)$ in
...
2
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2answers
43 views
Maximum-likelihood estimator of set of data from Normal Distributions
I have -before- found the MLE of the two parameters of a Normal Distribution but I don't have any idea about how to proceed in this case.
Problem
A sample of size $n$ is drawn from each of four ...
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12 views
What is the difference between manipulation in casual inference and the conditional possibility?
What is the difference between manipulation in casual inference and the conditional possibility?
In cause inference, we manipulate a variable and set it to arbitrary value, while in conditional ...
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1answer
25 views
If $x=[x_1,…,x_n]$ is Multivariate normal, what is the $x_1,…,x_k$ that will maximise $P(x_1,…,x_k , x_{k+1},…x_n| \mu, \Sigma)$?
How would you compute the $x_1,...,x_k$ that will maximise $P(x_1,...,x_k ,x_{k+1},...x_n| \mu, \Sigma)$?
if x was 2D then I think the main eigenvector of the covariance matrix at fixed $x_1$ will ...
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1answer
14 views
Confidence in Zero Defects
A friend of mine sent me a stats question, and since stats is definitely one of my weak points, I struggled a bit with this one, and I'm looking for some help.
The question is
Imagine a production ...
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0answers
23 views
Finding a test using asymptotic theory. for Poisson $(\lambda)$
If we have a sample of Poisson $(\lambda)$
(a) Find a rest for $H_0: \lambda =2$ vs $ H_a: \lambda =\lambda_1> 2$
(b) Find a test using asymptotic theory.
(c) Compare the results in en (a) y (...
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0answers
15 views
Normal distribution and sample distribution question
In a recent year, the distribution of scores of students on the ACT college entrance exam was modelled by a normal distribution with a mean of 20.9 and a standard deviation of 4.7.
1) The mean score ...
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27 views
Gaussian product - posterior probability distribution
I am with Elements of Statistical Learning 8.4 Relationship between the bootstrap and bayesian inference.
We observe a single observation $z$ from a normal distribution $z \sim N(\theta,1)$
We ...
1
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1answer
27 views
Find the critical region and the power when $H_0$ is false.
In a sample $N(0, \sigma ^2)$ we have two hypothesis. $H_0: \sigma ^2 =16$ and $H_1: \sigma ^2 =4$
(a )For a sample of size n, find the form of the best critical region.
(b) If n = 10 and $\alpha ...
1
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1answer
84 views
Is the following always true: $\mbox{Var}[\mbox{Range}(X_1,\cdots,X_n)] = O(n^{-B})$ with $0\leq B \leq 2$?
Here $X_1,\cdots,X_n$ are i.i.d. The two extremes $B=0$ and $B=2$, and the standard case $B = 1$ are illustrated in the picture below. For the reference, see here.
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16 views
Proving independence between Beta estimated and Delta in OLS
I know that in ordinary least squares $b$(beta estimated) and $\delta^2$(variance estimated) are independent, but how do I prove that?
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17 views
Show Consistency for every component
For $j=1,...,k$ let $t_{n,j}:\Omega_n \rightarrow \mathbb{R}$ be an estimator for $h_j(\theta) \in \mathbb{R}$.
Show that $t_n(X)=(t_{n,1}(X),...,t_{n,k}(X))$ is a consistent estimator of $h(\theta)=(...
2
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1answer
36 views
How to argue why one dice is more rigged than the other?
Let $\omega$ be a finite set and $P : \Omega \rightarrow \mathbb{R}$ be a probability measure.
You are given a set of three dices $\{A, B, C\}$. The following table describes the outcome of six ...
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1answer
46 views
Asymptotic Normality lemma (Serfling - 1980)
I'd like some assistant on the proof of the following Lemma:
If $X_n$ is $AN(\mu,\sigma_n^2)$, then also $X_n$ is $AN(\overline\mu,\overline\sigma_n^2)$ if and only if $\frac{\overline\sigma_n}{\...
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0answers
18 views
Why wont the mean height of 70 fall under 99.7 percent when using formula: mean+/-3(SD)
Sample problem: In general, the mean height of women is 65″ with a standard deviation of 3.5″. What is the probability of finding a random sample of 50 women with a mean height of 70″, assuming the ...
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0answers
49 views
How easy is it to create false evidence for a biased coin?
I have a biased coin which comes up heads with probability $p$. I know the value of $p$, but I want to falsely claim that the coin has a different probability of heads, $q$, where $q > p$.
To ...
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0answers
8 views
link function interpretation for models
In the categorial regression using the logit link function, the estimated coefficients are used to calculate the odds ratios or the ratio between two odds, that is, the model is interpreted through ...
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1answer
54 views
Finding the UMVUE of $\frac{1}{\lambda}$
I have been given the pdf:
$$f_X (x; \lambda) = \left(\frac{\lambda}{\pi}\right)^{\frac{1}{2}} x^{-\frac{3}{2}} e^{-\frac{\lambda}{x}} $$
with support $x>0$ and $\lambda>0$.
I am asked to ...
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0answers
5 views
Confusion with Gauss Markov
Consider the the linear regression model
Yi = β xi + ei
,
where the numbers x1, . . . , xn are known, the independent random variables e1, . . . , en
have the N(0, σ 2
) distribution, and the ...
2
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0answers
22 views
Why is the KL divergence the number of bits required to represent the error of an estimator?
I am familiar with several interpretations of the KL divergence, last week I heard of a new one, mentioned in a lecture on probabilistic graphical models. It was stated kind of offhandedly, so I hope ...
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18 views
Conditional Interpretations of Linear Regression
We estimate a linear regressor in the 1 dimensional with x and y random variables with zero mean:
y/x = $\alpha$ x
We can rewrite this using the variance of the variables as:
y/x = $\rho \frac{\...
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1answer
125 views
Can I do statistical analysis over a MILP problem?
I'm trying to solve a delivery problem which involves transportation of goods from a set of sources to a set of destinations ...
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26 views
Rao Blackwell and sufficient statistics
Suppose that X1, . . . , Xn are independent identically distributed random variables
with a B(m, θ) distribution
where m is a known positive integer and θ is unknown.
I have shown that θ* = X1/m is ...
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0answers
28 views
Does $X_1,…,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$?
Does $X_1,...,X_n$ being a random sample from $N(\mu,\sigma^2)$
$\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$?
If so does the above imply that a standard normal divided by the ...
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19 views
Goodness of fit test
I have the following exercise that shows $n=6$ numbers:
$$ 1.40, 1.55, 1.35, 1.50, 1.29, 1.64 $$
Is data normally distributed at the 5% significance level?
Surely $\overline{x} = 1.455$, $s=0....
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28 views
If $X_i \sim U(\theta-\frac{1}{2};\theta+\frac{1}{2})$, show that $[X_{(1)},X_{(n)}]$ is a confidence interval
Let $X_1,...X_n$ random sample from
$f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$.
a) Show that $[X_{(1)},X_{(n)}]$ is a confidence interval for $\theta$.
b) Compute the ...
2
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2answers
37 views
$U$~$N(3,16)$ $V$ ~$\chi_{9}^{2}$ U and V are independent random variables. Find $P(U-3<4.33\sqrt{V})$
$U$~$N(3,16)$
$V$ ~$\chi_{9}^{2}$
U and V are independent random variables.
Find $P(U-3<4.33\sqrt{V})$
(The notes I'm working through don't seem to approach this rigorously...)
The answer is $P(...
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0answers
12 views
Equality regarding the square of the sample mean
Given that $X_1,...,X_n$ is an i.i.d sample and its sample mean is $\overline X_n$, I have to prove the following equation:
\begin{equation*}
\frac{n-1}{n} \sum_{i=1}^n(\overline{X}_{n-1,i}^2 ...
0
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0answers
11 views
Generating random samples from a posterior distribution
Let
$$p(D \mid \mu,\sigma^2) \sim \mathcal{N}(\mu,\sigma^2)$$
where $D=(x_1\ldots x_n)$ is my data.
I imposed a normal prior on the mean as
$$\pi(\mu) \sim \mathcal{N}(\mu_0,\sigma_0^2)$$
Using Bayes, ...
0
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0answers
20 views
Find $k \in R$ such that $P\left(\max\left\{\frac{{S_x}^2}{{S_y}^2}, \frac{{S_y}^2}{{S_x}^2}\right\} > k\right)= 0.05$
Let $\overline{X}$ and $\overline{Y}$ sample means and ${S_x}^2, {S_y}^2$ unbiased estimators for the variance of 2 independent random samples of size 7 with normal distribution with mean unknown and ...
0
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1answer
59 views
Finding a confidence interval for shifted exponential distribution
Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\sigma ,\theta)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}, x\gt \theta$$ where $\sigma \gt 0 $ and $\theta \in R$ .
a) if ...
0
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0answers
10 views
econometrics: weighting frequency over time
I want to measure a model like:
Score = arunning_sum_of_all_exercises + brunning_sum_of_identical_exercise + c*exercise_density_value_in_period_x --other variables --
Lets say I want to test ...
1
vote
1answer
25 views
Finding shortest Confidence Interval for an Exponential Distribution
Let $X$ such that $f_{X}(x\mid\theta) = \theta e^{-\theta x} I_{(0, \infty)}(x)$, where $\theta > 0$. If $[X, 2X]$ is a confidence interval for $\frac{1}{\theta}$:
a)Find the confidence ...
