A jackknife estimator creates a series of estimate, from a single data set by generating that statistic repeatedly on the data set leaving one data value out each time. This produces a mean estimate of the parameter and a standard deviation of the estimates of the parameter.

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Resampling -- including the bootstrap, permutation, and other non-parametric tests -- is a method for hypothesis testing, confidence limits, and other applied problems in statistics and probability. It involves no formulas or tables.

Following the first publication of the general technique (and the bootstrap) in 1969 by Julian Simon and subsequent independent development by Bradley Efron, resampling has become an alternative approach for testing hypotheses.

There are other findings: "The bootstrap started out as a good notion in that it presented, in theory, an elegant statistical procedure that was free of distributional conditions. In practice the bootstrap technique doesn't work very well, and the attempts to modify it make it more complicated and more confusing than the parametric procedures that it was meant to replace."

While resampling techniques may reduce the bias, they achieve this at the expense of increase in variance. The two major concerns are:

The loss in accuracy of the estimate as measured by variance can be very large.
The dimension of the data affects drastically the quality of the samples and therefore the estimates.

Further Readings:
Young G., Bootstrap: More than a Stab in the Dark?, Statistical Science, l9, 382-395, 1994. Provides the pros and cons on the bootstrap methods.
Yatracos Y., Assessing the quality of bootstrap samples and of the bootstrap estimates obtained with finite resampling, Statistics and Probability Letters, 59, 281-292, 2002.

Prediction Intervals
In many application of business statistics, such as forecasting, we are interested in construction of a statistical interval for random variable, rather than a parameter of a population distribution.

The Tchebysheff's inequality is often used to put bounds on the probability that a proportion of random variable X will be within k > 1 standard deviation of the mean m for any probability distribution. In other words:

P [|X - m| ł k s] Ł 1/k², for any k greater than 1
The symmetric property of Tchebysheff's inequality is useful; e.g., in constructing control limits in the quality control process. However, the limits are very conservative due to lack of knowledge about the underlying distribution.
The above bounds can be improved (i.e., becomes tighter) if we have some knowledge about the population distribution. For example, if the population is homogeneous; that is, its distribution is unimodal; then,

P [|X - m| ł k s] Ł 1/(2.25k²), for any k greater than 1.
The above inequality is known as the Camp-Meidell inequality.
Now, let X be a random variable distributed normally with estimated mean and standard deviation S, then a prediction interval for the sample mean with 100(1- a)% confidence level is:

ą t_a/2 ´ S ´ (1+1/n)^1/2.

This is the range of a random variable with 100(1- a)% confidence, using t-table. Relaxing the normality condition for sample-mean prediction interval, requires a large sample size, say n over 30.

Further Readings:
Grant E., and R. Leavenworth, Statistical Quality Control, McGraw-Hill, 1996.
Ryan T., Statistical Methods for Quality Improvement, John Wiley & Sons, 2000. A very good book for a starter.

What Is a Standard Error?
For statistical inference, namely statistical testing and estimation, one needs to estimate the population's parameter(s). Estimation involves the determination, with a possible error due to sampling, of the unknown value of a population parameter, such as the proportion having a specific attribute or the average value m of some numerical measurement. To express the accuracy of the estimates of population characteristics, one must also compute the standard errors of the estimates. These are measures of accuracy that determine the possible errors arising from the fact that the estimates are based on random samples from the entire population, and not on a complete population census.
Standard error is a statistic indicating the accuracy of an estimate. That is, it tells us to assess how different the estimate (such as ) is from the population parameter (such as m). It is therefore, the standard deviation of a sampling distribution of the estimator such as . The following is a collection of standard errors for the widely used statistics:

Standard Error for the Mean is: S/n^˝.
As one expects, the standard error decreases as the sample size increases. However the standard deviation of the estimate decreases by a factor of n^˝ not n. For example, if you wish to reduce the error by 50%, the sample size must be 4 times n, which is expensive. Therefore, as an alternative to increasing sample size, one may reduce the error by obtaining"quality" data that provide a more accurate estimate.
For a finite population of size N, the standard error of the sample mean of size n, is:
S ´ [(N -n)/(nN)]^˝.

Standard Error for sample Variance S² is: S²/[(n-1)/2]^˝
Standard Error for the Multiplication of Two Independent Means ₁ ´ ₂ is:
{}^˝.
Standard Error for Two Dependent Means ₁ ± ₂ is:
{}^˝.
Standard Error for the Proportion P is:
[P(1-P)/n]^˝

Standard Error for P₁ ± P₂, Two Dependent Proportions is:
{}^˝.

Standard Error of the Proportion (P) from a finite population is:
[P(1-P)(N -n)/(nN)]^˝.
The last two formulas for finite population are frequently used when we wish to compare a sub-sample of size n with a larger sample of size N, which contains the sub-sample. In such a comparison, it would be wrong to treat the two samples"as if" there were two independent samples. For example, in comparing the two means one may use the t-statistic but with the standard error:
S_N [(N -n)/(nN)]^˝
as its denominator. Similar treatment is needed for proportions.
Standard Error of the Slope (m) in Linear Regression is
S_res / S_xx^˝, where S_res is the residual' standard deviation.
Standard Error of the Intercept (b) in Linear Regression is:
S_res[(S_xx + n ´ ²) /(n ´ S_xx] ^˝.

Standard Error of the Predicted Value using a Linear Regression is:
S_y(1 - r²)^˝.
The term (1 - r²)^˝ is called the coefficient of alienation. Therefore if r = 0, the error of prediction is S_y as expected.
Standard Error of the Linear Regression is:
S_y (1 - r²)^˝.
Note that if r = 0, then the standard error reaches its maximum possible value, which is standard deviation in Y.

Stability of an estimator: An estimator is stable if, by taking two different samples of the same size, they produce two estimates having"small" absolute difference. The stability of an estimator is measured by its reliability:

Reliability of an estimator = 1 / (its standard error)²
The larger the standard error, the less reliable is the estimate. Reliability of estimators is often used to select the"best" estimator among all unbiased estimators.

Sample Size Determination
At the planning stage of a statistical investigation, the question of sample size (n) is critical. This is an important question therefore it should not be taken lightly. To take a larger sample than is needed to achieve the desired results is wasteful of resources, whereas very small samples often lead to what are no practical use of making good decisions. The main objective is to obtain both a desirable accuracy and a desirable confidence level with minimum cost.
Students sometimes ask me, what fraction of the population do you need for good estimation? I answer,"It's irrelevant; accuracy is determined by sample size." This answer has to be modified if the sample is a sizable fraction of the population.