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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In Bias-variance decomposition of Mean Squared Error, why must one assume the error $\epsilon$ has $\text{Var}(\epsilon) = \sigma^{2}$?

I'm trying to understand bias-variance decomposition. Proofs of the decomposition often start by stating that the data conforms to the statistical model $y = f(x) + \epsilon $, where $\epsilon$ is an ...
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Maximum Likelihood of a uniform distribution centered at a random point

I have to compute the maximum likelihood estimator for a uniform distribution centered at a random point. I have a random variable $A \sim \mathcal{N}(\mu,\sigma^2) $ and a random variable $B \sim Exp(...
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Central limit theorem application to general functions

Let $X_n$ be a $\mathbb{N}$-valued random value. Suppose we have that \begin{equation} \frac{X_n-g(n)}{\sqrt{g(n)}}\sim N(0,1) \end{equation} for a function $g(n)$. This looks a lot like the Central ...
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Construct a confidence interval for $\theta$

Let $X_{1}, \cdots, X_{n}$ be a random c.i.i.d sample such as, given $\theta$, $X_{1} \sim \mathcal{N}(0,\theta)$. Construct a confidence interval for $\theta$ using asymptotic results. This question ...
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What's a good predictive modeling approach for a very small dataset (50-100 rows) [closed]

I want to model the performance of specific athletes, but instead of taking the big data approach and letting the model generalize (something I've already done), I want to create a high bias model ...
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Estimating integral $\int f(x)pdf(x)^{\alpha} dx$ involving some power of a probability distribution

Suppose a random variable $X$ has an unknown probability distribution $p(x)$ which we can draw samples $X_1,\ldots,X_n$ from. For a known function $f$, under some sufficiently nice conditions, we may ...
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Show that $T = \max_{i} (X_{i}/i)$ is a sufficient statistic. [closed]

Let $X_{1}, \cdots, X_{n}$ c.i.i.d such as, given $\theta$, have the following p.d.f: $$ f_{X_{i}}(x|\theta) = \begin{cases} e^{i\theta - x}, & x \geq i\theta\\ 0, & x < i\theta \end{...
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Physical interpretations of covariance and correlation

From what I know, theoretically, covariance is a measure of the degree to which two variables change together. A positive covariance indicates that the variables increase or decrease together, while a ...
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Non-existence of Lebesgue probability densities

A standard result taught in mathematical statistics courses is that the multivariate gaussian only has a density if the covariance matrix is non singular: i.e. if $X \sim N_p(\mu, \Sigma)$ and $\Sigma$...
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What does "spectral representation of a Gaussian Process" mean?

In a class about statistical learning, the professor was talking about various theorems like Mercer's theorem, Bochner's theorem, and representer theorem in context of Gaussian Processes (GPs). Two ...
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Maximum Likelihood for $\beta_1$ Using Partial Derivatives

Assume $y_i = \beta_0 + \beta_1 x_i + e_i$ and $e_i\sim N(0,\sigma^2)$. One way to estimate $\mathbf{\beta}$ is via maximum likelihood estimate. I saw from elsewhere that $\mathbf{\beta}$ could also ...
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Showing sufficiency of a statistic (the hard way)

Given $X_1,\dots,X_n$ i.i.d. from $P_\theta$, the usual way of showing that a statistic $T$ is sufficient is to use the factorization lemma as opposed to computing the distribution of $(X_1,\dots,X_n)|...
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conditional distribution of sample given maximum

Given an i.i.d. sample $X_1,\dots, X_n$ from the uniform distribution on $[0,\theta]$, and denoting their order statistics by $X_{(1)} < X_{(2)} < \cdots < X_{(n)}$, it is easy to show that $...
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Solved the problem using the echolan form. However, I am currently struggling to find the variables associated with this solution.

The matrix is $$\left( \begin{array}{cccc|c} 4 & -1 & 1 & -1 & 3 \\ 1 & 2 & -3 & 2 & 2 \\ 2 & -5 & 7 & -5 & -1 \\ 7 & -4 & 5 & -4 &...
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How can I get the average value of a class A∩B knowing the average values of classes A and B?

I am trying to model the population density for my metropolitan region, but the published census data does not meet the precision requirements I need. The most accurate data I have avaiable are ...
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George Casella book on Statistical Inference ; question 4.28

In question 4.28, it says let $X$ and $Y$ be independent standard normal random variables. Find the distribution of $X/|Y|$. I can understand the answer is Cauchy(0,1) distribution. Then question (c) ...
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Confidence interval for prediction

If we have the following linear model for the response $y_i$, how does one compute a confidance intervall for the expected value of $y_i$?
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Solutions to Rao's book [closed]

I hope this is the appropriate forum and the questions hasnt been asked already but does anyone know if there is a solution manual for Rao's 'Linear statistical inference and its applications'(2nd ed)?...
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I have trouble solving the following statistics exercise:

A company produces lightbulbs with an average lifetime of 1000 hours and standard deviation of 50 hours. Find the probability that in a sample of 100 bulbs there are at least two which stop working ...
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Is statistical inference just probability applied to the results from descriptive statistics?

I have that doubt due to I've been told that this branch of statistics infers from the data, understanding inferring as theestimate of a result, so I guess you use probability to make the estimates. ...
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Conditional probabilities and conditional validity

In the paper "Conditional validity of inductive conformal predictors" by V. Vovk, the author considers a condition of the form $$P_{X\times Y}(Y\in\Gamma(X)\mid X=x)\geq1-\alpha,$$ where $\...
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Expectation of a MM estimator [duplicate]

So I want to calculate the expectation of my estimator to see if it is biased or not. Data: $\bar{X} = \frac{1}{n}\sum_{i}x_i$ $E[\bar{X}]= \frac{\theta}{\theta + 1}$ $\hat{\theta}_{mm}$ = $\frac{\bar{...
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Find a sufficient statistic for a non parametric model.

So I'm trying to solve this question but I have no clue about how to go about it, since I haven't done any non parametric sufficient statistics. Let $X_1, \ldots, X_n$ be an i.i.d. sample, where each $...
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How to do Hypothesis testing?

I have two time series data. Electric vehicle demand at a station (whole number time series) every hour called "Clstr 175" Weather description (categorical time series) at the station every ...
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DIC, WAIC: non-invariant posterior mean, "negative effective dimension"

I am going through some lecture notes that unfortunatly do not contain any examples. "However, the DIC relies on the posterior mean as a point estimate, is not invariant with respect to ...
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conjugate normal prior for normal distribution: sigma known vs sigma unkown - how to derive posterior

Let $X_1, ..., X_n$ be independent, normally distributed random variables with unknown mean μ and known variance $\sigma^2$. Further, assume the mean has the following prior distribution μ ∼ $N (m, s^...
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Algebraic manipulation question coming from Method of Moments application

So today in my statistical inference class the professor wrote on the board: Using the Method of Moments: $$\sigma^2 = E[Y_1^2|\theta] - E[Y_1|\theta]^2$$ $$= \frac{1}{n}\sum_{j = 1}^{n}{Y_j^2} - \...
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Building Confidence Interval for Population Proportion [closed]

I'm having trouble understanding how the 95% confidence interval equation for population proportions gets simplified. Here is the initial equation: $$ Pr(\overline{X} - 2\hat{SE}(\overline{X}) ≤ p ≤ \...
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Can (1 - p-value) be the weight of a graph's edges?

I understand that the p-value can be translated as the strength of the statistical tests' results. From this perspective, can (1 - p-value) be the weight of an undirected graph's edges (like an ...
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How does the [UK] Office of National Statistics make allowances for incomplete census information?

I entered into a rather ill-tempered argument with a stranger yesterday. It started going downhill when he asserted that there were in fact 90 million people in the United Kingdom, not the 60 that I ...
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Is $(X_1+X_2+...+X_n)/n$ a random variable?

If $X_1,X_2,...,X_n$ are i.i.d random variables and are all discrete/continuous, then is $(X_1+X_2+...X_n)/n$ also a random variable? My attempt: For continuous type, I guess it is a random variable. ...
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How to determine sample size for Generalized Linear Mixed Models?

I want to design a cross-over study (clinical) to assess the performance of different treatments. The response of the study is binary outcome: success or failure. I have two fixed effects and one ...
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1 answer
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Questions about significance level and Type II error

I am learning tests of statistical hypotheses by myself. The book I used is Probability and Statistical Inference by Robert V. Hogg, etc. I want to confirm some of my understandings are right or not. $...
2 votes
1 answer
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Coincise introduction to background for semiparametric statistics

I plan to study the theory behind Targeted Maximum Likelihood Estimation, Doubly Robust Estimation, and Semiparametric Theory. I have a background in bioinformatics: I took courses in basic linear ...
2 votes
2 answers
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Definition about probability mass function by Rober V. Hogg

The following definition of pmf is on page51 from Probability and statistical inference by Robert V. Hogg, etc. The pmf $f(x)$ of a discrete random variable X is a function that satisfies the ...
1 vote
1 answer
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Can Strong LLN and Weak LLN apply to continuous distributions?

Can Strong LLN and Weak LLN apply to continuous distributions? Or it can only apply to discrete distributions? Representation of LLN: $X_1,X_2,\ldots,X_n$ are i.i.d, and their expectation values are ...
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Asymptotic distribution of sample variance

Asymptotic distribution of sample variance $S^2$ for an exponential distribution $\mathcal{E}(\frac{1}{\theta})$ $S^2= \frac{1}{n}\sum (X_i-\bar{X})^2$ $\mathbb{E}(S^2)= \frac{n-1}{n}\theta^2$ $\sqrt{...
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Regression relation to casual relationship

If the correlation coefficient of two variables is 0, can there still be a causal effect between them? And can the causal relationship between these two variables be studied by regression analysis?
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Analytic form of an ROC curve

I'm studying the problem of combining two sensors for anomaly detection. I want to analyze its performance by the ROC curve. Now I have obtained a parametric equation about the ROC curve: $$(x,y) = (...
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1 answer
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Why is the variance used so often in statistics instead of other higher moments? [closed]

I am learning statistics recently. By far, many statistical tests I saw, e.g. F-test, ANOVA, uses variance as their components. Hardly can I find any statistical test that uses third moment or even ...
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How we get the basis vector of the following statistical manifold?

Let me define it first, Let $S=\{p(x;\xi);\xi\in E\subseteq \mathbb{R^n}\}$ be a statistical manifold. We have the basis vector as $\partial_{i}=\frac{\partial}{\partial\xi^{i}}|_{i=1}^{n}$. Now let ...
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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.
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Alternative estimators for variance question

Question Consider another estimator of $\sigma^2$, based on the inter-quartile range of the sample. Let $Y_{i}$ be the $i^{th}$ smallest observation in the sample, or the $i^{th}$ order statistic. ...
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estimator for exponential distribution?

Let $(X_1,\cdots,X_n)$ be a sample iid of law that belongs to the family of law $\varepsilon(\lambda)$ the question is to compare between two estimators. the first is $\frac{1}{n}\sum X_i$ they found ...
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Questions about the relation between convergence in distribution and convergence in probability

I have two sequences of random variables $\{ X_n\}$ and $\{Y_n \}$. I know that $X_n \to^d D, Y_n \to^d D$. Can I conclude that $X_n - Y_n \to^p 0$? If I cannot, what other conditions do I need for ...
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Fisher information matrix of the binomial distribution

The definition of fisher matrix from the book "Methods of information geometry by shun-Ichi-Amari" is as follows Let $S=\{p_{\xi}\}$ be an n-dimensional statistical model. Where $\xi$ is a ...
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Distribution of a sum of normal random variables [closed]

Let $X_1, X_2, ..., X_n$ be independently distributed variables where $X_k \sim N(k\mu, 1)$ for $k = 1, 2, ..., n$ and $\mu\in\mathbb{R}$ unknown. I calculated that the ML estimator for $\mu$ is equal ...
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MLE as a pivot for a sample of normally distributed random variables

Let $X_1, X_2, ..., X_n$ be independently distributed variables where $X_k \sim \mathcal{N}(k\mu, 1)$ for $k = 1, 2, ..., n$ and $\mu\in\mathbb{R}$ unknown. I calculated that the MLE for $\mu$ is ...
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Why do we bother converting test statistics from st. normal distributions to chi-squared?

Examples are the wald estimator which we obtain by converting a standard normal distributed random variable to a $\chi^2$ distributed variable by taking the square. I am wondering why we bother with ...
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Soft: Understanding the difference between machine/statistical learning and parameter estimation

Say we have a known model $M$ with unknown parameters and more specifically, $M$ is a parametric model. Parameter estimation on $M$ is applying an appropriate method for estimating the parameters. My ...
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