Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
19 views

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 \...
0
votes
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?
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 $...
0
votes
0answers
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 ...
0
votes
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 = (...
0
votes
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: $$...
0
votes
0answers
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 ...
0
votes
0answers
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^...
0
votes
0answers
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 ...
0
votes
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)...
2
votes
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 $q_a$...
2
votes
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(...
0
votes
0answers
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 ...
0
votes
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
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 (...
0
votes
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 ...
0
votes
0answers
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
vote
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
vote
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.
0
votes
0answers
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?
0
votes
0answers
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
votes
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 ...
1
vote
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}{\...
0
votes
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 ...
5
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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 ...
0
votes
0answers
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{\...
1
vote
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
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....
0
votes
0answers
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
votes
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(...
0
votes
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
votes
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
votes
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
votes
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
votes
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 ...