Be Pro in T-Statistic in No Time: Tips & Tricks

$X_{1}, . . . X_{n}$ iid with $f (x; θ) = \frac{1}{θ} (1 + x)^{\frac{- θ + 1}{θ}}$
For the statistic $T (x) = (X_{1}, . . ., X_{n})$ (which I guess just means compute the MLE in a general sense for the sample)

Evelyn Freeman 2022-10-23

Say that given an estimator $\hat{T}$ , of a statistic $T$ , we have that $\sqrt{n} (\hat{T} - T) \overset{D}{\to} N (0, V a r)$ , where $n$ is the sample size. Consider now $- \hat{T}$ , does the asymptotic result hold also in this case? If yes/no, under which conditions?

Nayeli Osborne 2022-10-22

Let $θ^{'}$ , $θ \in Θ$ such that $θ^{'} \neq θ$ . I want to prove that $T$ is a sufficient statistic if and only if
$\frac{f (x, θ^{'})}{f (x, θ)}$
is a function dependent only on $T (x)$ .

Raiden Barr 2022-10-22

Showing that $\sum_{i = 1}^{n} X_{i}^{2}$ is a sufficient statistic for $θ$ from an $N (0, θ)$ population

Ignacio Riggs 2022-10-21

Why is an “unbiased estimator” called as such?

propappeale00 2022-10-18

The concept of a minimal sufficient statistic, which captures nothing more than the essential.
Is it true that any random variable $X$ as above has a sufficient statistic?

duandaTed05 2022-10-13

Given $X_{1}, X_{2}, \dots, X_{n}$ are i.i.d with $f (x) = \frac{1}{π} \frac{1}{1 + x^{2}}$ . Find the minimal sufficient statistic for $\frac{1}{π^{n} (\prod_{i = 1}^{n} 1 + (x_{i} - θ)^{2})}$ .
If we are given the entire sample set, we could easily obtain the order statistic of that sample, so how could $T (X) = (x_{1}, \dots, x_{n})$ is wrong?

unjulpild9b 2022-09-26

Whether to use $Z$ or $T$ test statistic for a sample of $50$ , known sample variance and unknown population variance

gaby131o 2022-09-25

The $t$ -statistic for the sample mean is
$t = \frac{\bar{x} - μ}{s}$
How is the sample estimate of the standard error computed in this context? In particular, does it make use of the population mean or the sample mean?

Joyce Sharp 2022-09-24

Find the complete statistic for Uniform $θ, θ + 1$

zaviknuogg 2022-09-23

Suppose we have $X_{1}, \dots, X_{n}$ i.i.d, $X_{i} \sim Exp (1, μ)$ (pdf is $f_{μ} (x) = e^{- (x - μ)}$ for $x \geq μ$ and $0$ for $x < μ$ ). Is there any one dimensional (i.e. $T : R^{n} \to R$ ) sufficient statistic for parameter $μ$ ? Obvious two dimensional sufficient statistic is $T (x_{1}, \dots, x_{n}) = (\sum x_{i}, min x_{i})$

cieloeventosm4 2022-08-25

So, assuming that $W_{0}$ is the intercept and $W_{1}$ is the coefficient in a simple linear regression model, the way to calculate a t statistic for $W_{1}$ is
( $W_{1 - 0}$ ) /std error of $W_{1}$
how do I calculate the t statistic for $W_{0}$ ?

Zaiden Soto 2022-08-20

The concept of a minimal sufficient statistic, which captures nothing more than the essential.
Suppose $X$ has a sufficient statistic $T$ . Must it also have a minimal sufficient statistic?

balafiavatv 2022-08-19

How to rigorously justify using a $T$ -test rather than just $\bar{X} - \bar{Y}$ ?

Shyla Odom 2022-08-19

A random variable $X$ is distributed with pdf $f (x, θ)$ . $T (x)$ is a sufficient statistic for $θ,$ , and $S (x)$ is a minimal sufficient statistic for $θ .$ . The following is stated in my notes without explanation:
$E_{θ} (T ∣ S) = g (S)$
for some function $g$ (independent of $θ .$ ). $E_{θ}$ here refers to the fact that the expectation is a function of $θ .$ .
how can we more formally show this is true, using the definition of sufficiency?

June Mejia 2022-08-17

Exponential family:
$f (x | θ) = c (x) d (θ) \exp [a (θ) b (x)]$
$T = \sum b (x_{i})$
Show that $T$ is a sufficient statistic

balafiavatv 2022-08-10

If we know the value of a sufficient statistic, but not the sample that generated it, am I right to suspect that the conditional distribution of any other statistic given the sufficient statistic will not depend on the parameter of interest?

metodystap9 2022-08-09

How to show $T (x)$ isn't a sufficient statistic?

Gifty Addae2022-07-30

Damien Horton 2022-07-23

Given a sample $X_{1}, . . ., X_{n} \sim N (θ, θ^{2})$ show, using the definition of completeness, that the statistic $T = (\sum_{i} X_{i}, \sum_{i} X_{i}^{2})$ is not complete for $n \geq 2$ . Use the fact that $E_{θ} [2 (\sum_{i} X_{i})^{2} - (n + 1) \sum_{i} X_{i}^{2}] = 0$
The statistic $T (\vec{X})$ is said to be complete for the distribution of $\vec{X}$ if, for every misurable function $g$ , $E_{θ} [g (T)] = 0 \forall θ ⟹ P_{θ} (g (T) = 0) = 1 \forall θ$

T-Statistics can help you answer questions about the mean of a population given a sample. This is done by using equations to calculate if the mean of the population is significantly different from a specific value, or to compare two distinct samples to each other. T-Statistics can provide powerful insights and answers, and are a great tool for anyone who wants to analyze a sample to draw conclusions. With the help of a t-statistic, you can answer questions such as "is the average of this sample significantly different from this given value?". T-Statistics can provide helpful information and give you the tools necessary to make effective decisions.

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