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.

2,478 questions
27 views

27 views

Logistic Regression Explanation

I have two questions regarding logistic regression. 1) I understand that the results of a logistic regression model yield a table stating coefficients together with a p-statistic for each variable . ...
47 views

Poisson distribution test using index of dispersion

I have a data set which Im trying to check if it is poisson distributed. I read some posts here and online but they are a bit "heavy" on the statistics for a newbie in statistics like me, so I would ...
74 views

What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$? [closed]

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?
40 views

Statistics and Confidence Intervals

Given the following set of values: 10,11,14,95,73,30,29,9,97,94,70 How do I calculate a 99% confidence interval for the sample mean? I am assuming that the variance is 10 Well, the idea I have is ...
80 views

Existence of a joint distribution given the conditional and marginal distribution

Can anyone point me a book where it has a proof of Theorem 1.7 (ii) of Jun Shao's book - Mathematical Statistics? I need this to show that given a distribution on one space and a collection of ...
44 views

Sufficient statistic for class of distributions

For the class $\{F_{\theta_1}, F_{\theta_2}\}$ of two DFs where $F_{\theta_1}$ is $N(0,1)$ and $F_{\theta_2}$ is $C(0,1)$, find a sufficient statistic. Let, $X_1, X_2, \dots, X_n$ is a random sample ...
11 views

48 views

Bayesian Statistics exercise?

I am having issues trying to solve this exercise in Bayesian analysis. The waiting time in minutes until being serviced by a phone call center follows an Exponential(λ) model, with E[y|λ] = 1/λ. Out ...
39 views

Ex: $X_1 , X_2 , ... , X_n$ ~ $U(-\theta, \theta); f(x; \theta) = \frac{1}{2\theta}; -\theta \leq X \leq \theta; \theta > 0$ I believe this is the correct approach to finding the MLE in this ...
21 views

Prove the equality of the following

How do i prove this? Can anyone help please. I have no idea how to start. $$\sum_{k=1}^N (x_k-μ)^2 = N(x̄-μ)^2+V$$ where $$x̄=μ_0=\sum_{k=1}^N\frac{x_k}{N}$$ and $$V=\sum_{k=1}^N (x_k-x̄)^2$$
Let $S$ be a strictly proper scoring rule for probability functions. Define $EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$ and let $D_{S}(P, Q) = EXP_{S}(Q|P) - EXP_{S}(P|P)$. Is it true that $D_{... 1answer 56 views How to find a confidence interval of a binomial distribution using a simulated random sample? I have a random sample of 1000 values of deviates from binomial distribution with n = 52 and p^ So I have 1000 values from the distribution. How can I find a 95% confidence interval for the true ... 0answers 25 views Binomial distribution confidence interval using a random sample? I am working with the binomial distribution Bin(52, 0.82) I am looking to find the confidence interval for the observed test statistic of 44 successes. I have generated a random sample of length ... 0answers 12 views In the context of the Cramer-Rao Lower Bound Theorem, how can I prove that$-E[\frac{d^2L(\theta)}{d\theta^2}] = {E[(\frac{dL(\theta)}{d\theta})^2]}$I think this is called Fisher's Information and it is the final piece I prove Cramer-Rao Lower Bound Theorem. 0answers 8 views wilcoxon signed rank test interpretation I try to understand the Wilcoxon signed rank test, but I have not been profoundly successful. I have studied various articles and educational videos, but I have not succeeded yet. The concept I'm ... 1answer 34 views Finding out what data is useful to use… I'm trying to determine the best method for narrowing down data to use in a program I'm writing. I could use some help. Here is a use case... Sometimes we have bad data in our system and it can lead ... 1answer 21 views Margin of error and average for a sample. In a sample of Petri dishes the number of possible infectious microorganisms was counted, obtaining the following results after 11 counts. 3,6,7,2,4,7,8,9,10,2,5 I have to prove that. I) The ... 0answers 12 views If our goal is to minimize$L_{2}$, why do we evaluate the accuracy of multivariate linear regression models with residual standard error? If we're using$L_{2}$as our criterion for choosing$f(x)$, why not use that for assessing fitted model accuracy? Or is that equivalent to using RSE for the purposes of minimizing our regression ... 1answer 17 views Sufficient statistic equivalence Let$\theta'$,$\theta \in \Theta$such that$\theta' \neq \theta$. I want to prove that$T$is a sufficient statistic if and only if $$\frac{f(x,\theta')}{f(x,\theta)}$$ is a function dependent only ... 1answer 39 views Implication of Law of Large Numbers I'm reading through a proof given for the consistency of the maximum likelihood estimator (MLE) of some parameter$\theta$. The begins as follows, Consider maximising $$\frac{1}{n}l(\theta) = \... 0answers 36 views Exercise 2.13 from “Mathematical Statistics - Jun Shao” I'm trying to solve this exercise, but I think some information is missing. It is very vague. Anyone have any tips? Let f be a function from \Omega to \Delta. Show that a) f^{-1}(B^c) =... 1answer 22 views Why is it valid to not fully expand the inequality in the convergence in probability definition? Let X_1,\ldots,X_n be iid random variables with common pdf$$f(x) = e^{−(x−\theta)}\quad, x > \theta ,\, − \infty < \theta < \infty\,; \quad 0 \quad\text{elsewhere}$$Let Y_n = \min({... 0answers 86 views Minimum sufficient statistic for logistic regression model For the question in the link below, I am seeking the minimal sufficient statistic for \theta={\beta_1,\beta_2} in the linear regression model given. I have taken the ratio of likelihoods ... 2answers 76 views Poisson Conditional Expectation ( searching best estimator for h(λ) ) Suppose X_1,X_2,X_3,.....,X_n are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) (t = a value) E [$$X_2$ - I{$x_1$=1}|$\sum_{i=1}^n X_i=t$$] =E ... 0answers 25 views Mathematical Statistics (Significance level) Let X_1, X_2 be a random sample of size n=2 from the distribution having pdf$$f(x;\theta)=\left( \dfrac 1{\theta} \right)e^{-\frac x{\theta}}, 0 \lt x \lt \infty$$We reject H_0: \theta=... 0answers 11 views Mathematical Statistics Question (Power Function) Can someone explain to me why we would want to maximize the power function (the probability our parameter is part of our alternative hypothesis) if that minimizes Type II Error when Type I Error is ... 2answers 48 views Find the Unbiased Estimator (Poisson) Suppose x_1,x_2,x_3,.....,x_n are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) Find an unbiased estimator for λ^2 E [$$\left(\frac{2 }{e^{-λ}}\... 1answer 47 views Most likely value of k A while ago I got this question on my exam, anyone got an idea how to solve this? Smarties are a chocolate candy that come in k different colors. Suppose that we do not know k. a. We draw three ... 1answer 65 views Consistent estimator for the variance of a normal distribution So I have to show that$\hat{\sigma}_n^2=\frac{1}{n}\cdot (\sum_{i=1}^n(X_i-\bar{X})^2)$is a consistent estimator for the variance$\sigma^2$when$X_1,X_2,...,X$are i.i.d. from a normal ... 0answers 29 views Point estimator as a vector I am given the following definition of a point estimator. Definition:$\hat{\theta}$is point estimator of$\theta$if$\hat{\theta} = g(X_1,...,X_n)$where$X_1,...,X_n$are iid distributed with ... 1answer 22 views Sensor and Random Variable Please, let us imagine that I have got a sensor D that measures the temperature in my room. My issue and questions are: some papers claims to model it as a Random Variable (RV) X. What does it mean? ... 1answer 39 views What exactly are sample elements from a population? If there is a sample with sequence$ x_1 , x_2 , ... , x_n$for example, that is taken randomly from a population, what exactly are these elements$ x_1 , x_2 , ... , x_n$? What do they represent? ... 0answers 21 views Distribution of General Pivotal Quantity Let$f(x; \theta) = g(\theta)h(x)$for$ a(\theta) \leq x \leq b(\theta)$where$ a(\theta)$decreases and$b(\theta)$increases with$\theta$. I'm trying to show that \begin{equation} P(S ;\theta) ... 1answer 30 views Optimal Number of Realizations for a Discrete Stochastic Process I have a curiosity concerning discrete stochastic processes. Let us say we have a discrete stochastic process$X_{i} = \left(x_1,x_2,...x_i,...,x_N \right)$, hence we have N random variables with an ... 0answers 30 views Proportion Confidence Interval [duplicate] I understand this: There is a 95% chance any sample from a binomial distribution of samples of size n will have a sample proportion of success between: $$p\pm 1.96*\sqrt{\frac{p(1-p)}{n}}$$ I don't ... 0answers 14 views Prove$XY^{1/r}$~Gamma(r,1) Given X~Gamma(r+1,1), Y~Uniform(0,1) How to prove the distribution of$XY^{1/r}$is Gamma(r,1) Thanks! 0answers 24 views The natural sufficient statistic is minimal sufficient We say the distribution of a r.v$X$belongs to the exponential family with parameter$\theta$if there exists functions$c,h,q_i,T_i$independent of$\theta$such that$$P_\theta(X=x)= c(\theta)h(x)e^... 0answers 21 views Neyman-Pearson Lemma for two hypothesis pairs and three parameters Let$\Theta = \{\theta_0, \theta_1, \theta_2\}$. I want to test the hypotheses$H_0$:$\theta = \theta_0$vs.$H_1$:$\theta = \theta_1$. Put$\Lambda_0(x) = \frac{\mathcal L(\theta_0|x)}{\mathcal L(\...
Let $X_1,\cdots,X_n$ be a sample from the density \begin{align*} f(x|\theta_1,\theta_1)=\frac{1}{\theta_1+\theta_2} \begin{cases} e^{-\frac{x}{\theta_1}}, x>0,\\ e^{\frac{x}{\...