支撑依赖参数的二维充分统计量：顺序、完备性及分布表述咨询

Does the order of components in a 2D sufficient statistic matter? #

Nope, the order of components in a multi-dimensional sufficient statistic has no impact on its sufficiency. Here's why: sufficiency hinges on whether the statistic captures all sample information relevant to estimating the parameter(s). Reordering components just rearranges the same set of information—you’re not adding or removing any data content. For example, if $(T_1, T_2)$ is sufficient for a parameter vector $\theta$, then $(T_2, T_1)$ is also sufficient. Both contain exactly the same information about $\theta$; they’re just presented in a different sequence.

Completeness of the 2D statistic #

Completeness is also unaffected by component order. A statistic is complete if no non-zero function of it has an expected value of zero for all parameters in the distribution family. Since reordering components doesn’t change the statistic’s functional relationship to the sample, it doesn’t alter this property. If $(T_1, T_2)$ is complete, then $(T_2, T_1)$ is too.

A quick side note: Completeness always depends on the specific distribution family you’re working with. A statistic might be complete for one family but not another, even if it’s sufficient in both cases.

Professional English Phrasing #

In statistical writing, the most natural and widely accepted ways to state that a sample comes from a specific distribution are:

• "Let $X = (X_1, \dots, X_n)$ be a random sample from the distribution with probability density function (pdf) given by..."
• "Consider a random sample $X = (X_1, \dots, X_n)$ drawn from the distribution defined by the following pdf:"

Breaking down your options:

• "given by": Perfect here—it’s standard when presenting the explicit form of a pdf.
• "defined by": Also acceptable, as the pdf formally defines the distribution.
• "derived from": Not appropriate here. "Derived from" implies the sample originates from a process stemming from the pdf, which isn’t accurate. The sample is drawn directly from the distribution described by the pdf.

Corrected Joint Density Function #

Your joint density was cut off, so here’s the complete, correct version. For the given shifted exponential distribution (a two-parameter exponential family):

Individual pdf:
$$f(x) = \frac{1}{\sigma}\exp\left{-\frac{x - m}{\sigma}\right}\mathbf{1}_{(m, \infty)}(x)$$

Joint pdf of the sample $X = (X_1, \dots, X_n)$:
$$f(X) = \frac{1}{\sigma^n}\exp\left(\frac{nm}{\sigma}\right)\exp\left{-\frac{n\overline{X}}{\sigma}\right}\mathbf{1}{(m, \infty)}(X{(1)})$$

Where $X_{(1)} = \min(X_1, X_2, \dots, X_n)$ (the smallest sample value, or first order statistic). The indicator function $\mathbf{1}{(m, \infty)}(X{(1)})$ ensures all sample observations are greater than $m$, which is equivalent to the smallest observation exceeding $m$.

内容的提问来源于stack exchange，提问作者Kusavil