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
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Find an unbiased estimator based on sufficient statistic.

Suppose $X_1$ and $X_2$ are independent observations from $$ f(x)=\frac{1}{\beta}e^{-x/\beta}, \quad x>0 $$ We know that a complete sufficient statistic for $\beta$ will be $X_1+X_2$. Now how do I find an unbiased estimator of…
Fireant
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distribution of residuals in linear regression

We consider the "common" linear regression model: $$y=\alpha + \beta x + \epsilon \;\;\;\;\mbox{ with } \;\;\; \epsilon \sim N(0, \sigma^2)$$ In all the textbooks I am using, residuals are estimated by defining: $$S = \frac{y_i - A -…
zeroKnowl
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How to statistically estimate weighted average with some unknown values?

I am stuck in a tricky problem. There are 32 companies. They have their own GHG emission factors and their market shares. This is, $ X_1, X_2, ..., X_n $ and $ Y_1, Y_2, ..., Y_n $. I am looking for its weighted mean. $\sum_{i=0}^n \frac{X_i…
Hanoi
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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 regarded as worse?
Paul Ash
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Proving that a minimal sufficient statistic is not complete

Problem Let $X_1, \ldots, X_n$ be a sample from normal distribution $N(\theta, \theta^2)$, where $\theta > 0$. I am to find minimal sufficient statistic and prove that it is not complete. Finding the minimal sufficient statistic First of all I did…
Hendrra
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Test the hypothesis that $B_1=0$ at the $5 \%$ significance level

I am told to test the hypothesis and this is what I did: $H_{0}:\beta_{1}=0$ $H_{a}:\beta_{1}\not=0$ So then I have $$t^{*}=\dfrac{\hat{\beta_{1}}-\beta_{1}}{\dfrac{s_{\beta_1}}{\sqrt n}}$$ $\beta_{1}$=.5 and $\hat{\beta_{1}}$=.283 n=10 and…
Gamecocks99
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Need help deriving plug-in (functional of distribution) estimator

I need help with homework exercise, have no idea how to approach it. Assume we have i.i.d. observations $x_1,\ldots,x_n$ of a continuous random variable $X$, taking values in $\mathbb R^+$. Define the discrete random variable $Y$…
user25470
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Estimating the variance in the many Normal means model

Is there a Bayesian way to estimate $\sigma^2$ given data $Z_i \sim N(\theta_i, \sigma^2)$, $i = 1, 2, \dots, n$? The MLE is $\hat\sigma^2 = 0$ which is unfortunate, so I'm considering a Bayesian approach. Wasserman's "All of Nonparametric…
nattyp
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Computing confidence interval of a proportion using prop.test() in R

I am using prop.test() to calculate the confidence interval of a proportion. I found that prop.test() result differs from the z-score confidence interval when $\hat p$ is small, but coincide otherwise. In particular, with $\hat p=0.916$ it works…
Freeman
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Should the choice between two sided or one sided test be based on sample's data?

"Suppose the assets price follow a normal distribution with variance 9. The sample mean of 10 assets is equal to 11.15. The manager of the investment fund says that the population mean is 12.5. Using the sample mean, find the p-value. Would you say…
Eduardo Yada
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$p(x\mid \theta)= \theta x^{-2}$, use moment method to estimate $\theta$

Let $X_1,\cdots, X_n$ be a random sample with pdf $$p(x\mid \theta) = \theta x^{-2}, \text{ with } 0<\theta\le x < \infty$$ Use moment method to estimate $\theta$. This problem is 7.6 from Casella's Statistical Inference. Strangely, I found that…
3x89g2
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What is Degrees of freedom and How to derive its formula?

I was looking at an example of difference of two means test where $t$-statistic is calculated as follows: $$t=\frac{(\bar x_1-\bar x_2)-D}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$ and degrees of freedom for the $t$-distribution is obtained…
user429311
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MLE for normal sample with known mean

What is the Maximum Likelihood Estimator of the variance of a sample $X_1\dots X_n$ of i.i.d variables with normal distribution $X_i\sim \mathcal{N}(\mu_0,\sigma^2)$ when the mean $\mu_0$ is known? Show that this estimator is unbiased. I've worked…
Kriss
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Can you give description for the third graph?

How Annual energy reduced while wind speed is increasing?
Vadivel S
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Estimating the Binomial parameter $n$ ( sample size)

What will happen if the sample mean exceeds the sample variance ? I tried to find the answer but all answers gave me the different low between two means. Could anyone help me to find the best answer. Thanks
Hanan
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