### 统计代写|linear regression代写线性回归代考|Unbiasedness and Efficiency

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|linear regression代写线性回归代考|Unbiasedness and Efficiency

As noted earlier, assembling a good sample is key to obtaining suitable estimates of parameters. This raises the general issue of what makes a good statistical estimator, or formula for finding an estimate such as a mean, a median, or, as discussed in later chapters, a regression coefficient. Developing estimates that are accurate and that do not fluctuate too much from one sample to the next is important. Two properties of estimators that are vital for obtaining such estimates are unbiasedness and efficiency.

Unbiasedness refers to whether the mean of the sampling distribution of a statistic equals the population parameter it estimates. For example, is the arithmetic mean estimated from the sample a good estimate of the corresponding mean in the population? Recall that the formula for the sample standard deviation includes the term ${n-1}$ in the denominator. This is necessary to obtain an unbiased estimate of the sample standard deviation, but it presents a slight degree of bias when estimating the population standard deviation.
Efficiency refers to how stable a statistic is from one sample to the next. A more efficient statistic has less variability across samples and is thus, on average, more precise. The estimators for the mean of the normal distribution and probabilities from binomial distributions are considered efficient. Finally, consistency refers to whether the statistic converges to the population

parameter as the sample size increases. Thus, it combines characteristics of both unbiasedness and efficiency. ${ }^{29}$

A common way to represent unbiasedness and efficiency is with an archery target. As shown in Figure 2.3, estimators from statistical models can be visualized as trying to “hit” the parameter in the population. Estimators can be unbiased and efficient, biased but efficient, unbiased but inefficient, or neither. The benefits of having unbiased and efficient statistics should be clear.

## 统计代写|linear regression代写线性回归代考|The Standard Normal Distribution and Z-Scores

Recall that we mentioned z-values in the discussion of CIs. These values are drawn from a standard normal distribution-also called a z-distributionwhich has a mean of zero and a standard deviation of one. The standard normal distribution is useful in a couple of situations. First, as discussed earlier, the formula for the large-sample CI utilizes z-values.

Second, they provide a useful transformation for continuous variables that are measured in different units. For instance, suppose we wish to compare the distributions of weights of two litters of puppies, but one is from the U.S. and the weights are measured in ounces and the other is from Germany and the weights are measured in grams. Converting ounces into grams is simple

(1 ounce $=28.35$ grams), but we may also transform the different measurement units using z-scores. This results in a comparable measurement scale. A $z$-score transformation is based on Equation $2.9$.
$$z \text {-score }=\frac{\left(x_{i}-\bar{x}\right)}{s}$$
Each observation of a variable is entered into this formula to yield its z-score, or what are sometimes called standardized values. The unit of measurement for $z$-scores is standard deviations. In $R$, the scale function computes them for each observation of a variable (the function may also be used to transform variables into other units in addition to z-scores). Let’s see how to use it on one of the samples of puppy weights along with a new sample of weights measured in grams.

## 统计代写|linear regression代写线性回归代考|Covariance and Correlation

We’ve seen a couple of examples of comparing variables from different sources (e.g., puppy weights from different litters); we now assess whether two variables shift or change together. For instance, is it fair to say that the length and the weight of puppies shift together? Are longer puppies, on average, heavier than shorter puppies? The answer is, on average, most likely yes. In statistical language, we say that length and weight covary or are correlated. The two measures used most often to assess the association between two continuous variables are, not surprisingly, called the covariance and the correlation. To be precise, the most common type of correlation is the Pearson’s product-moment correlation. ${ }^{30}$

A covariance is a measure of the joint variation of two continuous variables. Two variables covary when large values of one are accompanied by large or small values of the other. For instance, puppy length and weight covary because large values of one tend to accompany large values of the other in a population or in most samples, though the association is not uniform because of the substantial variation in the lengths and weights of puppies. Equation $2.10$ furnishes the formula for the covariance.
$$\operatorname{cov}(x, y)=\frac{\sum\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{n-1}$$
The covariance formula multiplies deviations from the means of both variables, adds them across the observations, and then divides the sum by the sample size minus one. Don’t forget that this implies the $x$ s and $y$ s come from the same unit, whether a puppy, person, place, or thing.

A limitation of the covariance is its dependence on the measurement units of both variables, so its interpretation is not intuitive. It would be helpful to have a measure of association that offered a way to compare various associations of different combinations of variables. The Pearson’s product-moment correlation-often shortened to Pearson’s $r$-accomplishes this task. Among several formulas for the correlation, Equations $2.11$ and $2.12$ are the easiest to understand.
$$\begin{gathered} \operatorname{corr}(x, y)=r=\frac{\operatorname{cov}(x, y)}{\sqrt{\operatorname{var}(x) \times \operatorname{var}(y)}} \ \operatorname{corr}(x, y)=r=\frac{\sum\left(z_{x}\right)\left(z_{y}\right)}{n-1} \end{gathered}$$
Equation $2.11$ shows that the correlation is the covariance divided by the pooled standard deviation. Equation $2.12$ displays the relationship between z-scores and correlations. It shows that the correlation may be interpreted as a standardized measure of association. Some characteristics of correlations include:

1. Correlations range from $-1$ and $+1$, with positive numbers indicating a positive association and negative numbers indicating a negative association (as one variable increases the other tends to decrease).
2. A correlation of zero implies no statistical association, at least not one that can be measured assuming a straight-line association, between the two variables.
3. The correlation does not change if we add a constant to the values of the variables or if we multiply the values by some constant number. However, these constants must have the same sign, negative or positive.

## 统计代写|linear regression代写线性回归代考|The Standard Normal Distribution and Z-Scores

（1盎司=28.35克），但我们也可以使用 z 分数来转换不同的测量单位。这导致了可比较的测量规模。一种和- 分数转换基于方程式2.9.

## 统计代写|linear regression代写线性回归代考|Covariance and Correlation

1. 相关范围从−1和+1，正数表示正关联，负数表示负关联（随着一个变量的增加，另一个变量趋于减少）。
2. 相关性为零意味着两个变量之间没有统计关联，至少不是可以假设为直线关联来测量的关联。
3. 如果我们将一个常数添加到变量的值或将这些值乘以某个常数，则相关性不会改变。但是，这些常数必须具有相同的符号，无论是负号还是正号。

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## MATLAB代写

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