Tan
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How to estimate $P(x\le0)$ from $n$ samples of $x$?
5 votes

You can do better. In the sense that among all the unbiased estimators, you can find the one with the smallest variance. Our goal is to estimate $\mathbb{P}(X<0)$. An unbiased estimator is $$\...

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minimal sufficient statistic for $U(\theta, \theta+c)$. $(\theta,c)$ unknown
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3 votes

(Nobody answers so I post this answer which I am not sure is correct) We show $(Y_{(1)},Y_{(n)})$ is a sufficient complete statistic, which implies $(Y_{(1)},Y_{(n)})$ is minimal sufficient. We first ...

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MLE estimator for the second parameter of binominal distribution
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1 votes

We have the likelihood of $m$ by $$f(m)=\frac{m!}{(m-X)!X!}p^X(1-p)^{m-X}$$ Here $m\geq X$. Now, let's consider when $\frac{f(m+1)}{f(m)}$ is less than 1. When it starts to be less than 1, we know ...

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$X\sim\frac{1}{b}\exp\{-\frac{1}{b}(x-a)\},x>a$. Find UMVUE of $\frac{a}{b}$
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1 votes

Since nobody answered, I'll try to answer my own question. Please let me know if anything is not correct. Denote $S=\frac{1}{n-1}\sum_{i=1}^n[X_i-X_{(1)}]$. We have $$S\sim \frac{b}{n-1}\text{Gamma}(n-...

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How to find an unbiased estimator?
1 votes

The answer is simple, any statistic $\delta(X)$ satisfying $$\delta(\theta-1)+\delta(\theta)+\delta(\theta+1)=0, \forall\theta\in\mathbb{Z}$$ is an unbiased estimator of zero. Thus, the totality of ...

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On the existence of UMVUE and choice of estimator of $\theta$ in $\mathcal N(\theta,\theta^2)$ population
1 votes

Let me directly find the UMVUE of $\theta$. However, I cannot answer "can the BLUE be the UMVUE", as I do not have enough knowledge of BLUE. It can be shown that $$\mathbb{E}(\bar{X})=\...

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Jointly Complete Sufficient Statistics: Uniform(a, b)
1 votes

Following @whuber's answer, we have the joint density of $(Y_{(1)},Y_{(n)})$ $$f(y_1, y_n,a,b)=\frac{n(n-1)}{(b-a)^n}(y_n-y_1)^{n-1},\quad\forall a\leq y_1\leq y_n\leq b\text{ and }a,b\in\mathbb{R}$$ ...

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R package to solve Gaussian MLE under conditional independence constraints
0 votes

I finished the implementation myself. Here is the code. gaussian_MLE = function(S, omega_structure, initial_omega = initial_omega, maxit = 2e9, tol = 1e-15){ omegas = list() # S is sample ...

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confidence interval of $\beta$, where $X$'s are from exponential distribution
0 votes

I am answering my own question. Please correct me if I make any mistakes. Continuing with $$T=n[-\log(\hat{\beta})+\log(\beta) - 1 +\frac{\hat{\beta}}{\beta}],$$ we can write $$T=n[\log\frac{\hat{\...

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moment generating function of gamma distribution through log-partition function
0 votes

I got it. The correct formula for Moment generating function should be $$M_{T}(u)=\frac{\exp{A(\eta+u)}}{\exp{A(\eta)}},$$ not $$M_{X}(u)=\frac{\exp{A(\eta+u)}}{\exp{A(\eta)}}$$ Here, $T(X)=(\log X,X)$...

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Multiplication of chi-square distribution by constant
0 votes

It is not true that multiplication of a chi-square random variable by a real constant remains chi-square. Chi-square is sum of square of independent standard normal distribution. Multiplying a non-one ...

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Finding complete sufficient statistic
0 votes

Method 1 $(X_{(1)},X_{(n)})$ is not complete because we can find $g\neq0$ but $\mathbb{E}\left[g(X_{(1)},X_{(n)})\right]=0,\forall\theta$. $g$ is $(t_1,t_2)\rightarrow\frac{n+1}{n-1}t_2-\frac{n+1}{1-n}...

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