O pdf de

Suponhamos que $X_1, X_2,...,X_n$ seja iid de $N(\mu,\sigma^2)$ com desconhecido $\mu \in \mathcal R$ e $\sigma^2>0$

Seja $Z=\frac{X_1-\bar{X}}{S},$ S é o desvio padrão aqui.

Pode-se demonstrar que $Z$ possui o pdf de Lebesgue

f (z) = \frac{\sqrt{n} Γ (\frac{n - 1}{2})}{\sqrt{π} (n - 1) Γ (\frac{n - 2}{2})} {[1 - \frac{n z^{2}}{(n - 1)^{2}}]}^{n / 2 - 2} I_{(0, (n - 1) / \sqrt{n})} (| Z |)

$f(z)=\frac{\sqrt{n} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}(n-1)\Gamma\left(\frac{n-2}{2}\right)}\left[1-\frac{nz^2}{(n-1)^2}\right]^{n/2-2}I_{(0,(n-1)/\sqrt{n})}(|Z|)$

Minha pergunta é como obter este pdf?

A questão é daqui em diante no exemplo 3.3.4 para encontrar o UMVUE de $P(X_1 \le c)$ . Entendo a lógica e os procedimentos para encontrar o UMVUE, mas não sei como obter o pdf.

Eu acho que esta questão também se relacionam com este um

Muito obrigado pela ajuda ou aponte para referências relacionadas também serão apropriadas.

self-study umvue

— Norte profundo
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Respostas:

O que é tão intrigante sobre esse resultado é o quanto ele se parece com a distribuição de um coeficiente de correlação. Há uma razão.

Suponha que $(X,Y)$ seja bivariado normal com correlação zero e variância comum $\sigma^2$ para ambas as variáveis. Desenhe uma amostra iid $(x_1,y_1), \ldots, (x_n,y_n)$ . É sabido e prontamente estabelecido geometricamente (como Fisher fez um século atrás) que a distribuição do coeficiente de correlação da amostra

r = \frac{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{(n - 1) S_{x} S_{y}}

$r = \frac{\sum_{i=1}^n(x_i - \bar x)(y_i - \bar y)}{(n-1) S_x S_y}$

f (r) = \frac{1}{B (\frac{1}{2}, \frac{n}{2} - 1)} {(1 - r^{2})}^{n / 2 - 2}, - 1 \leq r \leq 1.

$f(r) = \frac{1}{B\left(\frac{1}{2}, \frac{n}{2}-1\right)}\left(1-r^2\right)^{n/2-2},\ -1 \le r \le 1.$

(Aqui, como de costume, e são médias das amostras e e são as raízes quadradas de estimadores de variância imparcial.) é a função beta , para os quais $\bar x$ $\bar y$ $S_x$ $S_y$ $B$

\begin{matrix} (1) & \frac{1}{B (\frac{1}{2}, \frac{n}{2} - 1)} = \frac{Γ (\frac{n - 1}{2})}{Γ (\frac{1}{2}) Γ (\frac{n}{2} - 1)} = \frac{Γ (\frac{n - 1}{2})}{\sqrt{π} Γ (\frac{n}{2} - 1)} . \end{matrix}

$\frac{1}{B\left(\frac{1}{2}, \frac{n}{2}-1\right)} = \frac{\Gamma\left(\frac{n-1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{n}{2}-1\right)} = \frac{\Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{n}{2}-1\right)} . \tag{1}$

To compute $r$ , we may exploit its invariance under rotations in $\mathbb{R}^n$ around the line generated by $(1,1,\ldots, 1)$ , along with the invariance of the distribution of the sample under the same rotations, and choose $y_i/S_y$ to be any unit vector whose components sum to zero. One such vector is proportional to $v = (n-1, -1, \ldots, -1)$ . Its standard deviation is

S_{v} = \sqrt{\frac{1}{n - 1} ((n - 1)^{2} + (- 1)^{2} + \dots + (- 1)^{2})} = \sqrt{n} .

$S_v = \sqrt{\frac{1}{n-1}\left((n-1)^2 + (-1)^2 + \cdots + (-1)^2\right)} = \sqrt{n}.$

Consequently, $r$ must have the same distribution as

\frac{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (v_{i} - \bar{v})}{(n - 1) S_{x} S_{v}} = \frac{(n - 1) x_{1} - x_{2} - \dots - x_{n}}{(n - 1) S_{x} \sqrt{n}} = \frac{n (x_{1} - \bar{x})}{(n - 1) S_{x} \sqrt{n}} = \frac{\sqrt{n}}{n - 1} Z .

$\frac{\sum_{i=1}^n(x_i - \bar x)(v_i - \bar v)}{(n-1) S_x S_v} = \frac{(n-1)x_1 - x_2-\cdots-x_n}{(n-1) S_x \sqrt{n}} = \frac{n(x_1 - \bar x)}{(n-1) S_x \sqrt{n}} = \frac{\sqrt{n}}{n-1}Z.$

Therefore all we need to is rescale $r$ to find the distribution of $Z$ :

f_{Z} (z) = | \frac{\sqrt{n}}{n - 1} | f (\frac{\sqrt{n}}{n - 1} z) = \frac{1}{B (\frac{1}{2}, \frac{n}{2} - 1)} \frac{\sqrt{n}}{n - 1} {(1 - \frac{n}{(n - 1)^{2}} z^{2})}^{n / 2 - 2}

$f_Z(z) = \big|\frac{\sqrt{n}}{n-1}\big| f\left(\frac{\sqrt{n}}{n-1}z\right) = \frac{1}{B\left(\frac{1}{2}, \frac{n}{2}-1\right)} \frac{\sqrt{n}}{n-1}\left(1- \frac{n}{(n-1)^2}z^2\right)^{n/2-2}$

for $|z| \le \frac{n-1}{\sqrt{n}}$ . Formula (1) shows this is identical to that of the question.

Not entirely convinced? Here is the result of simulating this situation 100,000 times (with $n=4$ , where the distribution is uniform).

The first histogram plots the correlation coefficients of $(x_i,y_i),i=1,\ldots,4$ while the second histogram plots the correlation coefficients of $(x_i,v_i),i=1,\ldots,4)$ for a randomly chosen vector $v_i$ that remains fixed for all iterations. They are both uniform. The QQ-plot on the right confirms these distributions are essentially identical.

Here's the R code that produced the plot.

n <- 4
n.sim <- 1e5
set.seed(17)
par(mfrow=c(1,3))
#
# Simulate spherical bivariate normal samples of size n each.
#
x <- matrix(rnorm(n.sim*n), n)
y <- matrix(rnorm(n.sim*n), n)
#
# Look at the distribution of the correlation of `x` and `y`.
#
sim <- sapply(1:n.sim, function(i) cor(x[,i], y[,i]))
hist(sim)
#
# Specify *any* fixed vector in place of `y`.
#
v <- c(n-1, rep(-1, n-1)) # The case in question
v <- rnorm(n)             # Can use anything you want
#
# Look at the distribution of the correlation of `x` with `v`.
#
sim2 <- sapply(1:n.sim, function(i) cor(x[,i], v))
hist(sim2)
#
# Compare the two distributions.
#
qqplot(sim, sim2, main="QQ Plot")

Reference

R. A. Fisher, Frequency-distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507. See Section 3. (Quoted in Kendall's Advanced Theory of Statistics, 5th Ed., section 16.24.)

— whuber
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The link to the reference is broken.

— Sextus Empiricus

@Martijn Thank you for checking. I see what you mean--the link works, but it doesn't go to anything relevant! I have fixed it up.

— whuber

I'd like to suggest this way to get the pdf of Z by directly calculating the MVUE of $P(X\leq c)$ using Bayes' theorem although it's handful and complex.

Since $E[I_{(-\infty,c)}(X_1)]=P(X_1\leq c)$ and $Z_1=\bar X$ , $Z_2=S^2$ are joint complete sufficient statistic, MVUE of $P(X\leq c)$ would be like this:

ψ (z_{1}, z_{2}) = E [I_{(- \infty, c)} (X_{1}) | z_{1}, z_{2}] = \int_{- \infty}^{\infty} I_{(- \infty, c)} f_{X | Z_{1}, Z_{2}} (x_{1} | z_{1}, z_{2}) d x_{1}

$\psi(z_1,z_2)=E[I_{(-\infty,c)}(X_1)|z_1,z_2]=\int_{-\infty}^{\infty}I_{(-\infty,c)}f_{X|Z_1,Z_2}(x_1|z_1,z_2)dx_1$

Now using Bayes' theorem, we get

f_{X | Z_{1}, Z_{2}} (x_{1} | z_{1}, z_{2}) = \frac{f_{Z_{1}, Z_{2} | X_{1}} (z_{1}, z_{2} | x_{1}) f_{X_{1}} (x_{1})}{f_{Z_{1}, Z_{2}} (z_{1}, z_{2})}

$f_{X|Z_1,Z_2}(x_1|z_1,z_2)={{f_{Z_1,Z_2|X_1}(z_1,z_2|x_1)f_{X_1}(x_1)}\over{f_{Z_1,Z_2}(z_1,z_2)}}$

The denominator $f_{Z_1,Z_2}(z_1,z_2)=f_{Z_1}(z_1)f_{Z_2}(z_2)$ can be written in closed form because $Z_1 \sim N(\mu,\frac{\sigma^2}{n})$ , $Z_2 \sim \Gamma({n-1\over 2},{2 \sigma^2\over n-1})$ are independent of each other.

To get the closed form of numerator, we can adopt these statistics:

W_{1} = \frac{\sum_{i = 2}^{n} X_{i}}{n - 1}

$W_1 = {\sum_{i=2}^n X_i \over n-1}$

W_{2} = \frac{\sum_{i = 2}^{n} X_{i}^{2} - (n - 1) W_{1}^{2}}{(n - 1) - 1}

$W_2 = {\sum_{i=2}^n X_i^2 -(n-1) W_1^2 \over (n-1)-1}$

which is the mean and the sample variance of $X_2, X_3, ..., X_n$ and they are independent of each other and also independent of $X_1$ . We can express these in terms of $Z_1, Z_2$ .

$W_1={n Z_1 - X_1\over n-1}$ , $W_2={(n-1)Z_2+nZ_1^2-X_1^2-(n-1)W_1^2 \over n-2}$

We can use transformation while $X_1=x_1$ ,

f_{Z_{1}, Z_{2} | X_{1}} (z_{1}, z_{2} | x_{1}) = \frac{n}{n - 2} f_{W_{1}, W_{2}} (w_{1}, w_{2}) = \frac{n}{n - 2} f_{W_{1}} (w_{1}) f_{W_{2}} (w_{2})

$f_{Z_1,Z_2|X_1}(z_1,z_2|x_1)={n \over n-2}f_{W_1,W_2}(w_1,w_2)={n \over n-2}f_{W_1}(w_1)f_{W_2}(w_2)$

Since $W_1 \sim N(\mu,\frac{\sigma^2}{n-1})$ , $W_2 \sim \Gamma({n-2\over 2},{2 \sigma^2\over n-2})$ we can get the closed form of this. Note that this holds only for $w_2 \geq 0$ which restricts $x_1$ to $z_1-{n-1 \over \sqrt n}\sqrt{z_2} \leq x_1 \leq z_1+{n-1 \over \sqrt n}\sqrt{z_2}$ .

So put them all together, exponential terms would disappear and you'd get,

f_{X | Z_{1}, Z_{2}} (x_{1} | z_{1}, z_{2}) = \frac{Γ (\frac{n - 1}{2})}{\sqrt{π} Γ (\frac{n - 2}{2})} \frac{\sqrt{n}}{\sqrt{z_{2}} (n - 1)} (1 - {(\frac{\sqrt{n} (x_{1} - z_{1})}{\sqrt{z_{2}} (n - 1)})}^{2})

$f_{X|Z_1,Z_2}(x_1|z_1,z_2)={\Gamma({n-1 \over 2}) \over \sqrt{\pi} \Gamma({n-2 \over 2})} {\sqrt{n} \over \sqrt{z_2} (n-1)} (1-{({\sqrt{n} (x_1 -z_1) \over \sqrt{z_2} (n-1) })}^2)$ where

z_{1} - \frac{n - 1}{\sqrt{n}} \sqrt{z_{2}} \leq x_{1} \leq z_{1} + \frac{n - 1}{\sqrt{n}} \sqrt{z_{2}}

$z_1-{n-1 \over \sqrt n}\sqrt{z_2} \leq x_1 \leq z_1+{n-1 \over \sqrt n}\sqrt{z_2}$ and zero elsewhere.

From this,at this point, we can get the pdf of $Z={X_1- z_1 \over \sqrt{z_2}}$ using transformation.

By the way, the MVUE would be like this :

ψ (z_{1}, z_{2}) = \frac{Γ (\frac{n - 1}{2})}{\sqrt{π} Γ (\frac{n - 2}{2})} \int_{- \frac{π}{2}}^{θ_{c}} c o s^{n - 3} θ d θ

$\psi(z_1,z_2)={\Gamma({n-1 \over 2}) \over \sqrt{\pi} \Gamma({n-2 \over 2})} \int ^{\theta_c} _{-{\pi \over2}} cos^{n-3} \theta d\theta$ while

θ_{c} = s i n^{- 1} (\frac{\sqrt{n} (c - z_{1})}{(n - 1) \sqrt{z_{1}}})

$\theta_c = sin^{-1} ({\sqrt{n}(c-z_1)\over(n-1)\sqrt{z_1}})$ and would be 1 if

c \geq z_{1} + \frac{n - 1}{\sqrt{n} \sqrt{z_{2}}}

$c \geq z_1+{n-1 \over \sqrt{n} \sqrt{z_2} }$

I am not a native English speaker and there could be some awkward sentences. I am studying statistics by myself with text book introduction to mathmatical statistics by Hogg. So there could be some grammatical or mathmatical conceptual mistakes. It would be appreciated if someone correct them.

Thank you for reading.

— KDG
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