Introduction to Linear Regression Analysis. Douglas C. Montgomery. Читать онлайн. Newlib. NEWLIB.NET

Автор: Douglas C. Montgomery
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Математика
Год издания: 0
isbn: 9781119578758
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Cases Stocked, x 10.15 25 2.96 6 3.00 8 6.88 17 0.28 2 5.06 13 9.14 23 11.86 30 11.69 28 6.04 14 7.57 19 1.74 4 9.38 24 0.16 1 1.84 5 image image

      Figure 2.15 also shows the 95% confidence interval or E(y|x0) computed from Eq. (2.54) and the 95% prediction interval on a single future observation y0 at x = x0 computed from Eq. (2.55). Notice that the length of the confidence interval at x0 = 0 is zero.

      SAS handles the no-intercept case. For this situation, the model statement follows:

      model time = cases/noint

      The method of least squares can be used to estimate the parameters in a linear regression model regardless of the form of the distribution of the errors ε. Least squares produces best linear unbiased estimators of β0 and β1. Other statistical procedures, such as hypothesis testing and CI construction, assume that the errors are normally distributed. If the form of the distribution of the errors is known, an alternative method of parameter estimation, the method of maximum likelihood, can be used.

      Consider the data (yi, xi), i = 1, 2, …, n. If we assume that the errors in the regression model are NID(0, σ2), then the observations yi in this sample are normally and independently distributed random variables with mean β0 + β1xi and variance σ2. The likelihood function is found from the joint distribution of the observations. If we consider this joint distribution with the observations given and the parameters β0, β1, and σ2 unknown constants, we have the likelihood function. For the simple linear regression model with normal errors, the likelihood function is

      (2.56) image

      The maximum-likelihood estimators are the parameter values, say in52-1, in52-2, and in52-3, that maximize L, or equivalently, ln L. Thus,

      (2.57) image

      and the maximum-likelihood estimators in52-4, in52-5, and in52-6 must satisfy

      (2.58b) image

      and

      (2.58c) image

      (2.59a) image

      (2.59c) image

      Notice that the maximum-likelihood estimators of the intercept and slope, in53-1 and in53-2, are identical to the least-squares estimators of these parameters. Also, in53-3 is a biased estimator of σ2. The biased estimator is related to the unbiased estimator in53-4 [Eq. (2.19)] by in53-6. The bias is small if n is moderately large. Generally the unbiased estimator in53-6 is used.

      In general, maximum-likelihood estimators have better statistical properties than least-squares estimators. The maximum-likelihood estimators are unbiased (including in53-7, which is asymptotically unbiased, or unbiased as n becomes large) and have minimum variance when compared to all other unbiased estimators. They are also consistent estimators (consistency is a large-sample property indicating that the estimators differ from the true parameter value by a very small amount as n becomes large), and they are a set of sufficient statistics (this implies that the estimators contain all of the “information” in the original sample of size