Questions tagged [f-distribution]

The F distribution is a continuous probability distribution which is commonly used in statistical testing procedures.

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Sampling distribution of Coefficient of determination in general

I was studying the properties of multiple linear regression, and stumbled across the F statistics often used for the tests. We are testing for the regression coefficients (let's say there are $p$ ...
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F-distibution only depends on degrees of freedom

I am reading ISLP Chapter 3.2 on multiple linear regressions. They define the F-statistic as $S_1$, the variance of sample 1 from population 1, divided by $S_2$, the variance of sample 2 from ...
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Distribution of $\frac{X^2}{Y^2}$ where $X$ and $Y$ are iid standard normal

Let $X,Y\in \mathcal{N}(0,1)$ with $X$ and $Y$ independent. I am interested in the pdf of $Z=X^2/Y^2$ which has a $\mathcal{F}(1,1)$ Fisher distribution. I know that his can be done by first finding ...
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Distribution of the number of iterations to achieve success

Let $Z=X+jY$ ($j$ is the imaginary unit), with $X\sim\mathcal{N}(\mu,1)$ and $Y\sim\mathcal{N}(0,1)$. I'm running an algorithm that at every iteration $k$ samples a complex number $z_k$ that follows $...
mateusgl's user avatar
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Does Cook's distance follow an $F$-distribution?

The question is pretty much contained in the title: in a linear model $Y=X\beta +\epsilon$, with $\epsilon\sim N_n(0,\sigma^2 I_n)$, does the Cook's distance, defined as $$ D_i:=\frac{\frac{1}{p}||X(\...
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non-central F-statistic confidence intervals seem inconsistent with ANOVA p-values

I have done a series of one-way ANOVA tests. For each test I have calculated the corresponding alternative hypothesis F-statistic and confidence intervals, based on the degrees of freedom involved and ...
treemake's user avatar
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Is it possible to use acritical value from the F distribution with a CI of 95% and obtain the z value with a CI of 99% in the same problem?

I am studying this article "Variance sensitive adaptive threshold-based PCA method for fault detection with experimental application (Alkaya,2011)" in order to apply the adaptive threshold ...
TyDurden's user avatar
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Why large sample size makes ANOVA less affected by normality assumption violation?

One of the assumptions of ANOVA, and basically any parametric hypothesis test, is that the population (or each population group) should be normally distributed. But we learn in most stats classes that ...
Robin311's user avatar
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Sum of two i.i.d R.V having singly non-central F distribution

Noncentral F-distribution is used frequently in communication areas. In one of the applications, I need to do a sum of two i.i.d R.V having non-central F-distribution with parameter 1 (d.o.f for ...
Jaimin Shah's user avatar
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How sample size affect the chance to reject null hypothesis in ANOVA?

I am reading a book about the introduction to statistics. One chapter is about ANOVA, F distribution, and the null hypothesis. In ANOVA, the F value is $$ F = \frac{\text{SSB}/(k-1)}{\text{SSW}/(n-k)...
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Show that the two random variables with F-distribution are independent

The following is an exercise problem from Hogg, McKean and Craig's book titled "Introduction to Mathematical Statistics" that I am working on. 3.6.16. Let $X_1$, $X_2$, and $X_3$ be three ...
TryingHardToBecomeAGoodPrSlvr's user avatar
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F-Distribution theoretical question

Given a sample $X_1,...,X_N$. If $X_i \overset{idd}{\sim} F$ which will be the distribution of sample $Y_1, ..., Y_N$ where $Y_i=F(X_i)$ and why?
George Verouchis's user avatar
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What is the distribution of this ratio?

Suppose $X\sim F(n_1,n_2)$, then, what is the distribution of $Y=\log \frac {\exp(X)}{X}$? I don't even have a clue about this distribution. Any helpful suggestions and answers are greatly appreciated!...
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How to go from this formula to F distribution

I have the following formula. $$\int_{0}^{\infty} x^{\frac{n-1}{2}-1} (a+x)^{\frac{1}{2} \frac{-n}{2} -v} dx.$$ The quantities $d1 ,d2$ appearing in the pdf of the F distribution are the following. $$...
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Derivation of F distribution from quotient distribution

I'm looking for the derivation of the F distribution. There is a fairly understandable (for me) derivation presented on this site: http://www.milefoot.com/math/stat/pdfc-fdist.htm. On this site, the ...
ironwest's user avatar
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Compounding Gamma with Gamma to yield F-distribution?

I am working through some problems from my Bayesian Statistics course and am having trouble understanding a step in the solution to a question. For reference this is the question: And here is the ...
guestaccount798's user avatar
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What is $F_{k,\infty}$, i.e., $F$ distribution when the second degree of freedom approaches infinity?

What is $F_{k,\infty}$, i.e., $F$ distribution when the second degree of freedom approaches infinity? I'm wondering if there is a known distribution(such as $\chi_k^2$) that it converges to.
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Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution

According to Golam Kibria & Joarder (2006, p.7) available here and Kotz & Nadarajah (2004, p. 19) visible in google, the distribution of $X'\Sigma^{-1}X /p$, for a known correlation matrix $\...
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Test for overall significance of regression when the variance of errors is known

Say we want to test the overall model adequacy in a multiple linear regression model: $$H_0: B_1 = B_2 = ... = 0 $$ $$H_1: B_j \neq 0 \text{ for at least one j}$$ the random errors $\epsilon$ are ...
RidgeAllen's user avatar
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1 answer
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When a population is not normal, can F ratio be used for ANOVA analysis?

As it is known a F ratio assumes that the random variables in the numerator and denominator (variances in the case of ANOVA) follow a chi squared distribution, which is true for the sampling ...
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Transforming F-dist or long one tail

I have been going through different kaggle datasets recently to apply different techniques I have learned. I have seen loads of articles related to preprocessing your features with normalization or ...
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Derivation of F-distribution from inverse Chi-square?

I am trying to derive F-distribution from Chi-square and inverse Chi-square. Somewhere in process I make a mistake and result slightly differs from the canonical form of Fisher-Snedecor F distribution....
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Asking about the meaning of abbreviation f(1, 78)

Please I have a thesis defense and I wrote from this article this number f(1, 78) but I don't know what it refers to Can anyone clear it up for ME? THANK YOU
Razan Najee's user avatar
-2 votes
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how to understand the logic of null distribution

In a statistics textbook I read the following about comparing two population variances by using independent samples: If a null hypothesis (stating that the ratio of two population variances is 1) is ...
TooOldToLearn's user avatar
2 votes
2 answers
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F test for linear regression - why only upper tail

In linear regression when considering the f test for slopes be it MLR or SLR, we only consider the upper tail probability for the p value. Here's a video for reference: link. Why is that? EDIT: as ...
user2793618's user avatar
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1 answer
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Prove that $Z = \frac{X_1}{X_2}$, has an F-distribution

Let $X_1, X_2$ be independent random variables following density law $f(x) = e^{-x} , 0 < x < \infty$, Show that $Z = \frac{X_1}{X_2}$, has an F-distribution. I thought of solving this by ...
Aakash Malviya's user avatar
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Prove that $P(X \le a) + P\{Y \le \frac{1}{a}\} = 1$

Prove that if $X$ has the F-distribution with $(m, n)$ d.f. and $Y$ has the F-distribution with $(n, m)$ d.f., then for every $a > 0$, $$ P(X \le a) + P\left\{Y \le \frac{1}{a}\right\} = 1 $$ I ...
Aakash Malviya's user avatar
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Why do we get different f-statistics for the ANOVA of two models in R if we add third model

For example, using the mtcars built-in dataset. A two-model ANOVA: ...
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Why doesn’t a $F$-statistic of 1 result in a $p$-value of about 0.5?

I’ve been presented with the following chain of reasoning. In an ANOVA, if the null hypothesis is true the F-statistic is expected to be about 1. A p-value is the probability of obtaining test ...
user1205901 - Слава Україні's user avatar
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Connection between t and F distributions [duplicate]

We have: $Z\sim N_{0,1}$ (standard normal distribution) , $U\sim X^2_{k}$ ($X^2$ distribution, with k df) with $Z \bot U$ Then $X = \frac{Z}{ \sqrt{U /k} } \sim t (k)$ (t-distribution, with k df) I ...
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In an F distribution, when would the first parameter, d1, ever be greater than the second parameter, d2?

It seems that when running an ANOVA, the degrees of freedom of the mean sum of squares between groups (SS_b) will always be less than or equal to the degrees of freedom of the (summed for all groups) ...
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F-Distribution: difference between the expected value for a random variable and the expected value of its inverse?

It is known that the expected value of the F-Distribution depends only on the degrees of freedom associated with the Chi-square random variable in the denominator. In other words, E[1/X] does not ...
Ryan Rothman's user avatar
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1 answer
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Expected value of the F-Distribution dependent on the degrees of freedom associated with the Chi-square random variable in the denominator?

0 The F-Distribution has a Probability Density Function that can be defined as: with an expected value of: What would be a logical explanation for why the expected value only depends on the ...
Ryan Rothman's user avatar
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F distribution definition with gamma function or square root [closed]

On wikipedia (https://en.wikipedia.org/wiki/F-distribution) I see the pdf of F distribution defined using squared root function, while other places I see the pdf defined with the gamme function. E.g ...
Endre Moen's user avatar
1 vote
1 answer
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t-student or f-distribution?

I am reading 'Applied multivariate statistical analysis' by Richard Johnson and I do not understand that first to explain this test, talks about the t-student distribution. And then out of nowhere he ...
Chicago1988's user avatar
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Asymptotic distribution of intraclass correlation

Consider the random effects ANOVA model below (notation based on Snijders and Bosker, 1999), where $j$ represents a group and $i$ an individual: $$ Y_{ij} = \mu + U_j + R_{ij}, \qquad var(Y_{ij}) = \...
Waldir Leoncio's user avatar
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1 answer
323 views

Ratio of Standard Deviations from a Normal Distribution to an F Distribution

Apologies if the title is confusing, I couldn't think of a more apt title. I have that $W_i$s are iid $N(\mu_a,\sigma_a^2)$ and independent of $Z_i$s which are iid $N(\mu_b,\sigma_b^2)$. This means $...
strwars's user avatar
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ratio of 2 chi square to get f-distribution

How can I use two chi square functions to make a f function? I only see invchisquare functions in some languages when I need to use an inv f-dist to get the exact binomial CI. So, looking to figure ...
user250064's user avatar
2 votes
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$F$-test with a variance being $C$ times as large as the other

I am familiar with $F$-tests in which the alternative hypothesis is defined as $H_a=\{\sigma^2_1/\sigma^2_2>C\}$ (the sign "$>$" can be either "$<$" or "$\ne$" as well), where $C=1$. If I ...
Cromack's user avatar
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How to interpret $F > F_{\rm crit}$, but $p > \alpha$

My F is greater than F critical, while P-Value is 0.99 greater than alpha (0.05) which one should I consider the P value or F statistic, rejecting the Ho: I am confused. Edit: $$ \begin{array}{ccc} \...
Moro's user avatar
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2 answers
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Why is this statistic F-distributed?

A book I'm reading claims that the statistic: $\frac{(RSS_0 - RSS_1) / (p_1 - p_0)}{RSS_1 / (N - p_1 - 1)}$ has an F distribution. Why is this? I know that an F distribution is something like $\frac{\...
serendipity's user avatar
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Does X^2 divided to DF converge to F distribution when X^2 is Rao Scott chi squared statistic?

I'm studying Rao-Scott chi-squared test, kind of survey sampling analysis method. It uses when we want to test independence between two variables under complex sampling data instead of Pearson chi-...
김민호's user avatar
4 votes
1 answer
195 views

From expected value of R2 in Regression to expected value of partial eta squared in ANOVA

In general, $R^2$, the estimator of "multiple correlation coefficient" in regression, is known to be positively biased. Given $K$ predictors, and $N$ total sample size, Johnson, Kotz, & ...
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Derive the distribution of the ANOVA F-statistic under the alternative hypothesis

Say we have $k$ samples of data, where sample $i$ is of size $n_i$ and we write it as $x_{i1}, ... , x_{in_i}$. Let the total sample size be $N$. The ANOVA model is $X_{ij} \sim N(\mu_i, \sigma^2)$ ...
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What is Cov(X,Y) when X,Y is a F-distribution (a,b) and (c,d) ,in case X,Y Not independent [closed]

I want to know Covariance of random variable between $X$ and $Y$ when $X$ is a F-distribution with degree of freedom $a$ and $b$ and $Y$ is a F-distribution with degree of freedom $c$ and $d$,in case $...
Itee Louis's user avatar
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The behaviour of the critical F per F distribution

I noticed that, for increasing degrees of freedom of the MSE (Mean square error / denominator in the F-ratio), the critical F declines. This is also the case for increasing degrees of freedom of the ...
postnubilaphoebus's user avatar
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2 answers
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When should an F-test be one sided vs two sided? [duplicate]

If we increase both numerator and denominator degrees of freedom of the F distribution, then the pdf narrows down on the value 1. This suggests that a two-sided test is reasonable: If we have high ...
user56834's user avatar
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4 votes
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What is the median of the non-central F ratio distribution

I am looking for a simple approximation to the median of the (simply) non-central F distribution with parameters dlnum, dldenominator, and ncp, the non-centrality parameter. Clearly, there is no ...
Denis Cousineau's user avatar
3 votes
0 answers
207 views

Tail bounds for F-distribution (not using $\chi^2$ bounds)

Are there any sharp tail bounds for an $F_{p,q}$ distribution? That is, if $X \sim F_{p,q}$, then for a $t_1,t_2 > 0$, what are the sharpest $\delta_1$ and $\delta_2$ known such that $$P(X > ...
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Link between Fisher and Chi-squared

I am fairly new to statistics, and during practice, i used this link between Chi-squared and F-statistics where F equals to ration between 2 Chi squared distributions. Can someone elaborate more on ...
Nemanja Boskovic's user avatar