# Questions tagged [factorisation-theorem]

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### Finding UMVUE for a function of a Bernoulli parameter

Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ I'm interested in finding the UMVUE of $(1-\theta)^{1/k}$, when $k$ is a positive integer. . I know $\sum X_{i}$ is a ...
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### Problem on sufficient statistics

Let the distribution of $X_1,X_2,...X_n$ depend on two parameters $a, b$ such that there exists a single sufficient statistic, for either parameter when the other is fixed/known. Show that there is ...
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### Showing the sample mean is a sufficient statistics from an exponential distribution

Suppose that the lifelengths ( in thousands of hours) of light bulbs are distributed Exponential($\theta$), where $\theta>0$ is unknown. If we observe $\overline x = 5.2$ for a sample of $20$ light ...
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### Sufficiency of $|X|$ when $X\sim N(0,\sigma^2)$ without using Factorization theorem

Question: Given, $X\sim N(0,\sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $\sigma^2$. My Attempt: This problem is very easy if we use Fisher–Neyman ...
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### When a function of sufficient statistic is itself sufficient?

I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
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### Puzzled by definition of sufficient statistics

I am learning about sufficient statistic from Mood, Graybill, and Boes's Introduction to the Theory of Statistics. I am slightly confused by the book's definition of a sufficient statistic for ...
Let $X_1,...,X_n$ be iid random variables with densities given by $$f_{x_i}(x|\theta)=e^{i\theta - x}\mathbb{I}_{(i\theta,\infty)}(x),$$ when $x>i\theta$ and $x=0$ otherwise. Let $T$ be the ...