Affine connection

Template:Regression bar In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing. It is a form of a Student's t-statistic, with the estimate of error varying between points.

This is an important technique in the detection of outliers. It is named in honor of William Sealey Gosset, who wrote under the pseudonym Student, and dividing by an estimate of scale is called studentizing, in analogy with standardizing and normalizing: see Studentization.

Motivation

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.

The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions.

Consider the simple linear regression model

$Y={\alpha }_0+{\alpha }_{1}X+\epsilon .$

Given a random sample (X_i, Y_i), i = 1, ..., n, each pair (X_i, Y_i) satisfies

$Y_i={\alpha }_0+{\alpha }_1{X}_i+{\epsilon }_i,$

where the errors ε_i, are independent and all have the same variance σ². The residuals are not the true, and unobservable, errors, but rather are estimates, based on the observable data, of the errors. When the method of least squares is used to estimate α₀ and α₁, then the residuals $\hat{\epsilon }$, unlike the errors $\epsilon$, cannot be independent since they satisfy the two constraints

$\sum _{i=1}^n{\hat{\epsilon }}_i=0$

and

$\sum _{i=1}^n{\hat{\epsilon }}_i{x}_i=0.$

(Here ε_i is the ith error, and ${\hat{\epsilon }}_i$ is the ith residual.)

Moreover, and most importantly, the residuals, unlike the errors, do not all have the same variance: the variance decreases as the corresponding x-value gets farther from the average x-value. This is a feature of the regression better fitting values at the ends of the domain, not the data itself, and is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. This can also be seen because the residuals at endpoints depend greatly on the slope of a fitted line, while the residuals at the middle are relatively insensitive to the slope. The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to each other, is the principal reason for the need for studentization.

It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.

How to studentize

For this simple model, the design matrix is

$X=\left[\begin{array}{cc}1& x_1\\ ⋮& ⋮\\ 1& x_n\end{array}\right]$

and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:

$H=X(X^{T}X{)}^{-1}{X}^T.$

The "leverage" h_ii is the ith diagonal entry in the hat matrix. The variance of the ith residual is

$\mathrm{var}({\hat{\epsilon }}_i)={\sigma }^2(1-h_{ii}).$

In case the design matrix X has only two columns (as in the example above), this is equal to

$\mathrm{var}({\hat{\epsilon }}_i)={\sigma }^2(1-\frac{1}{n}-\frac{(x_i-\overline{x}{)}^2}{\sum _{j=1}^n(x_j-\overline{x}{)}^2}).$

The corresponding studentized residual is then

$\frac{{\hat{\epsilon }}_i}{\hat{\sigma }\sqrt{1-h_{ii}}}$

where $\hat{\sigma }$ is an appropriate estimate of σ (see below).

Internal and external studentization

The usual estimate of σ² is

${\hat{\sigma }}^2=\frac{1}{n-m}\sum _{j=1}^n{\hat{\epsilon }}_j^2.$

where m is the number of parameters in the model (2 in our example). But it is desirable to exclude the ith observation from the process of estimating the variance when one is considering whether the ith case may be an outlier. Consequently one may use the estimate

${\hat{\sigma }}_{(i)}^2=\frac{1}{n-m-1}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{\hat{\epsilon }}_j^2,$

based on all but the ith case. If the latter estimate is used, excluding the ith case, then the residual is said to be externally studentized; if the former is used, including the ith case, then it is internally studentized.

If the errors are independent and normally distributed with expected value 0 and variance σ², then the probability distribution of the ith externally studentized residual is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from $-\mathrm{\infty }$ to $+\mathrm{\infty }$.

On the other hand, the internally studentized residuals are in the range $0\pm \sqrt{\mathrm{r}.\mathrm{d}.\mathrm{f}.}$, where r.d.f. is the number of residual degrees of freedom, namely n − m. If "i.s.r." represents the internally studentized residual, and again assuming that the errors are independent identically distributed Gaussian variables, then

${\mathrm{i}.\mathrm{s}.\mathrm{r}.}^2=\mathrm{r}.\mathrm{d}.\mathrm{f}.\frac{t^2}{t^2+\mathrm{r}.\mathrm{d}.\mathrm{f}.-1}$

where t is a random variable distributed as Student's t-distribution with r.d.f. − 1 degrees of freedom. In fact, this implies that i.s.r.²/r.d.f. follows the beta distribution B(1/2,(r.d.f. − 1)/2). When r.d.f. = 3, the internally studentized residuals are uniformly distributed between $-\sqrt{3}$ and $+\sqrt{3}$.

If there is only one residual degree of freedom, the above formula for the distribution of internally studentized residuals doesn't apply. In this case, the i.s.r.'s are all either +1 or −1, with 50% chance for each.

The standard deviation of the distribution of internally studentized residuals is always 1, but this does not imply that the standard deviation of all the i.s.r.'s of a particular experiment is 1. For instance, the internally studentized residuals when fitting a straight line going through (0, 0) to the points (1, 4), (2, −1), (2, −1) are $\sqrt{2},-\sqrt{5}/5,-\sqrt{5}/5$, and the standard deviation of these is not 1.

See also

• Normalization (statistics)
• Samuelson's inequality
• Standard score

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534