# 统计代写|统计推断代写Statistical inference代考|Sample quantiles and order statistics

## 统计代写|统计推断代写Statistical inference代考|Sample quantiles and order statistics

Mean and variance are not the only measures of central tendency and spread. Commonly quoted alternative measures include the median and interquartile range, both of which are calculated from quantiles. Quantile-based statistics are generally more robust than moment-based statistics; in other words, quantile-based statistics are less sensitive to extreme observations.

For a random variable $Y$ with cumulative distribution function $F_Y$, the $\alpha$-quantile is defined as the smallest value, $q_\alpha$, such that $F_Y\left(q_\alpha\right)=\alpha$. The median of a distribution is the point that has half the mass above and half the mass below it, $\operatorname{med}(Y)=q_{0.5}$. The interquartile range is the difference between the upper and lower quartiles, $\operatorname{IQR}(Y)=q_{0.75}-q_{0.25}$. In the context of inference, the median, interquartile range, and other functions of quantiles may be viewed as population parameters to be estimated. An obvious starting point for estimating population quantiles is to define sample quantiles. In turn, sample quantiles are most readily constructed from order statistics.
Definition 7.4.1 (Order statistics)
For a sample, $Y_1, \ldots, Y_n$, the order statistics, $Y_{(1)}, \ldots, Y_{(n)}$, are the sample values placed in ascending order. Thus, $Y_{(i)}$ is the $i^{\text {th }}$ smallest value in our sample.

A simple consequence of our definition is that order statistics are random variables, $Y_{(1)}, \ldots, Y_{(n)}$, such that
$$Y_{(1)} \leq Y_{(2)} \leq \ldots \leq Y_{(n)} .$$
Definition 7.4.1 does odd things with random variables. In particular, we say that
$$Y_{(1)}, \ldots, Y_{(n)}$$
arise from putting
$$Y_1, \ldots, Y_n$$
in ascending order. We can make this more precise using the definition of random variables as functions from the sample space, $\Omega$, to the real line, $\mathbb{R}$. The order statistics, $Y_{(1)}, \ldots, Y_{(n)}$, are the functions such that, for every $\omega \in \Omega$,
$$Y_{(1)}(\omega), \ldots, Y_{(n)}(\omega)$$
is an ordering of $Y_1(\omega), \ldots, Y_n(\omega)$ satisfying
$$Y_{(1)}(\omega) \leq Y_{(2)}(\omega) \leq \ldots \leq Y_{(n)}(\omega)$$

## 统计代写|统计推断代写Statistical inference代考|Functions of a sample

The formal definition of a statistic is surprisingly loose: any function of the sample is a statistic. The function may be scalar-valued or vector-valued. As the sample is a random vector, a statistic is also a random vector.
Definition 8.1.1 (Statistic)
For a sample, $\boldsymbol{Y}=\left(Y_1, \ldots, Y_n\right)^T$, a statistic, $\boldsymbol{U}=\boldsymbol{h}(\boldsymbol{Y})$, is a random vector that is a function of the sample and known constants alone. Given an observed sample, $\boldsymbol{y}=\left(y_1, \ldots, y_n\right)^T$, we can compute the observed value of a statistic, $\boldsymbol{u}=\boldsymbol{h}(\boldsymbol{y})$.

Loose definitions crop up repeatedly in statistical inference. Our approach is to define concepts very broadly, then specify desirable characteristics that will only be found in members of a much smaller subset. Statistics are useful as devices for data reduction when their dimension is smaller than that of the sample. In particular, we will often consider scalar statistics, that is, statistics that reduce the sample to a single random variable.

The distribution of the statistic $\boldsymbol{U}$ is often referred to as the sampling distribution of $\boldsymbol{U}$. Although a statistic is a function of the sample and known constants alone, the distribution of a statistic may depend on unknown parameters. The observed value of a statistic, $\boldsymbol{u}$, is just a vector of real numbers. Just as the observed sample, $\boldsymbol{y}$, is thought of as one instance of the sample, $\boldsymbol{Y}$, the value $\boldsymbol{u}=\boldsymbol{h}(\boldsymbol{y})$ is taken to be one instance of the statistic $\boldsymbol{U}=\boldsymbol{h}(\boldsymbol{Y})$.

In some cases, we can use the distributional results from previous chapters to work out the sampling distributions of statistics of interest. In others, the mathematics involved are not tractable and we need to resolve to simulation-based approaches; these are discussed in Chapter 12.

# 统计推断代考

## 统计代写|统计推断代写Statistical inference代考|Sample quantiles and order statistics

$$Y_{(1)} \leq Y_{(2)} \leq \cdots \leq Y_{(n)}$$

$$Y_{(1)}, \ldots, Y_{(n)}$$

$$Y_1, \ldots, Y_n$$

$$Y_{(1)}(\omega), \ldots, Y_{(n)}(\omega)$$

$$Y_{(1)}(\omega) \leq Y_{(2)}(\omega) \leq \ldots \leq Y_{(n)}(\omega)$$

## 统计代写|统计推断代写Statistical inference代考|Functions of a sample

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