Mathematics Stack Exchange
Mathematics 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.

Statistical inference makes propositions about a population using data sampled from the population. To test a hypothesis about a population, a typical workflow is to select a statistical model of the process that generates the data and then deduce propositions from the model.

Statistical propositions include—

  • a point estimate, which is a particular value that best approximates some parameter of interest,

  • an interval estimate, for example, a confidence interval (or set estimate), which is an interval constructed using a data set drawn from a population so that, under repeated sampling of such data sets, such intervals would contain the true parameter value with the probability at the stated confidence level,

  • a credible interval, which is a set of values containing, for example, 95% of posterior belief,

  • rejection of a hypothesis, or

  • clustering or classification of data points into groups.

3930 questions
1
vote
1 answer

MLE coincides asymptotically with true value of parameter

The theorem is that under certain assumptions (regularity conditions): $$ \lim_{n\to \infty} P_{\theta_0} [L(\theta _0, X) > L(\theta, X)]=1, \,\,\forall \theta \neq\theta _0$$ I'm having difficulties understanding the proof. The setup is $L(\theta;…
asahi
  • 209
1
vote
1 answer

Inference Ph in water

This is an example from my book and Iam translating from it so there might be some things that are not all clear. Just let me know so i can edit. Someone is measuring PH-level in a sea and for that she has taken $\textbf{x} =(x_1,...x_n)$ which is a…
Danny
  • 1,547
1
vote
1 answer

Easy Maximum-Likelihood problem

Problem $F(x)=x^{\alpha}$ for $0 \le x \le 1$. Where $\alpha>0$ is unknown. Observed values of x: 0,57 0,81 0,63 0,44 0,31 0,91 0,36 0,65 0,74 0,99 Do an ML estimate of $\alpha$. Attempt: I find $f(x) =\alpha x^{\alpha}$. I build the ML function…
jacob
  • 2,965
1
vote
1 answer

Finding values of x using inverse transform method

A random variable X has probability density function: f(x) = 2e^(2x)/e^2−1 for 0 You want to generate five observations of X by using the inverse transform method and then calculate the mean and standard deviation of this sample. You are given that…
suraj
  • 31
1
vote
2 answers

Type I Error and Type II Error

Rigourously speaking, can I say that the type I error is the probability that I reject the null, when the null is true? I've seen some say that is incorrect to say, since type I error is defined to be the abovementioned event , not a probability.…
user191919
  • 713
1
vote
0 answers

How to test that a a time series follows a Markov process?

I am searching for a statistical test that tells wether a finite-alphabet time series is a Markov process of a given order or not.
user93236
  • 11
  • 1
1
vote
0 answers

Find $\pmb{\widehat{\beta}_1}$ and $\pmb{\widehat{\beta}_2}$, two different solutions to the normal equations $\,\mathbf{X'X\pmb{\beta}= X'Y}$.

This is my attempt at a solution. Please help complete the solution, and I will be grateful to everyone who helps me my try.
Ahmed Ahmed
  • 71
1
vote
0 answers

precision on inference from two distribution

can someone clarify a doubt about this exercise: the literacy rate of a nation measures the proportion of people age 15 and over who can read and write A statistician is interested in the estimation of parameter $\theta$, the literacy rate for woman…
Mattia Marletta
  • 11
1
vote
1 answer

Doubt in deducing probability distribution from given data

I am solving some questions from MIT OCW introduction to probability and statistics (spring 2014) :- https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/exams/MIT18_05S14_Prac_Exam2.pdf On page-3…
Ayush
  • 39
1
vote
1 answer

question on inference

In a population of working couples,the man's weekly income X and the woman's weekly income Y are statistically independent and normally distributed.X has a mean of 250 and variance of 50,while Y has a mean of 200 and a variance of 60.What is the…
clarkson
  • 1,907
1
vote
1 answer

What is the standard error of the sample means and how to calculate it?

A welterweight boxing team has 4 boxers who weigh $68.9kg$, $70.7kg$, $72.5kg$ and $74.3kg$. Determine all possible random samples (with replacement) of $n=2$ and calculate the mean of these samples. Using this information, calculate the…
CountDOOKU
  • 1,065
1
vote
0 answers

Give an approximation for P[Y > 110]

I found the following exercise interesting while reading L. Bain's book on introduction to probability and statistics: Let $X_1, X_2, . . . , X_{100}$ be a random sample from an exponential distribution, $X_i ∼ EXP(1)$, and let $Y = X_1 + X_2 + · ·…
Curious
  • 541
1
vote
1 answer

formal definition statistics model

i can't find rigorous definition about statistic model. Many authors use the definition below let suppose to have a probability space $$(\Omega,\mathcal{F},\mathbb{P})$$ a statistic model is $$( \{X_i \}_{i \in I_n}, F, \Theta)$$ where $\{X_i \}_{i…
anto_zoolander
  • 129
  • 4
1
vote
1 answer

Minimally Sufficient Statistic

In a nutshell, an example of a necessary and sufficient statistic is given in an article, but it is not explained there why the statistic is in fact necessary and sufficient. I'd appreciate help in understanding this. Bahadur ascribes to Lehmann…
Evan Aad
  • 11,422
1
vote
0 answers

Run test statistics

I'm currently implementing the Wald-Wolfowitz runs test on a variable $\epsilon \in \{0;1\} $. According to the original paper, the number of runs, $V_n$ converges to the normal distribution with: $$\mu = \frac{2n_1n_2}{n}+1$$ and $$\sigma^2 =…
thiebaud
  • 31
1 2
3
…
8 9