Category INTRODUCTION TO STATISTICS AND ECONOMETRICS

Variance-Covariance Matrix Assumed Known

Consider the case of К = 2. We can write 0 = (0b 02)’ and 0O = (0ю, 02o)’ • It is intuitively reasonable that an optimal critical region should be outside some enclosure containing 0O, as depicted in Figure 9.9. What should be the specific shape of the enclosure?

An obvious first choice would be a circle with 0O at its center. That would amount to the test:

Подпись: (9.7.1)

image460

Reject HQ if (01 — 0ioC + (02 — 02oT > c

for some c, where c is chosen so as to make the probability of Type I error equal to a given value a. An undesirable feature of this choice can be demonstrated as follows: Suppose F0i is much larger than V02...

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DURATION MODEL

The duration model purports to explain the distribution function of a duration variable as a function of independent variables. The duration variable may be human life, how long a patient lives after an operation, the life of a machine, or the duration of unemployment. As is evident from these examples, the duration model is useful in many disciplines, including medicine, engineering, and economics. Introductory books on duration analysis emphasizing each of the areas of application mentioned above are Kalbfleisch and Prentice (1980), Miller (1981), and Lancaster (1990).

We shall initially explain the basic facts about the duration model in the setting of the i. i.d. sample, then later introduce the independent variables.

Denoting the duration variable by T, we can completely characterize ...

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WHAT IS AN ESTIMATOR?

In Chapter 1 we stated that statistics is the science of estimating the probability distribution of a random variable on the basis of repeated observations drawn from the same random variable. If we denote the random variable in question by X, the n repeated observations in mathe­matical terms mean a sequence of n mutually independent random vari­ables X, X2, . . . , Xn, each of which has the same distribution as X. (We say that (X;) are i. i.d.)

For example, suppose we want to estimate the probability (p) of heads

for a given coin. We can define X = 1 if a head appears and = 0 if a tail appears. Then Xt represents the outcome of the zth toss of the same coin. If X is the height of a male Stanford student, X* is the height of the ith student randomly chosen.

We call the basic random var...

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DEFINITION OF BASIC TERMS

Matrix. A matrix, here denoted by a boldface capital letter, is a rectan­gular array of real numbers arranged as follows:

A matrix such as A in (11.1.1), which has n rows and m columns, is called an n X m (read “n by m”) matrix. Matrix A may also be denoted by the symbol {fly}, indicating that its i, jth element (the element in the ith row and jth column) is aly

Transpose. Let A be as in (11.1.1). Then the transpose of A, denoted by A’, is defined as an и X и matrix whose i, jth element is equal to a]r For example,

1 Подпись: 1 2 3 4 5 64 ‘

2 5

3 6

Note that the transpose of a matrix is obtained by rewriting its columns as rows.

Square matrix. A matrix which has the same number of rows and col­umns is called a square matrix. Thus, A in (11.1.1) is a square matrix if

n = m.

Symmetric matrix...

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Cramer-Rao Lower Bound

We shall derive a lower bound to the variance of an unbiased estimator and show that in certain cases the variance of the maximum likelihood estimator attains the lower bound.

THEOREM 7.4.1 (Cramer-Rao) Let L(Xb X2, . .., Xn | 0) be the likeli­hood function and let 0(Xlt X2,. . . , Xn) be an unbiased estimator of 0. Then, under general conditions, we have

(7.4.1) V(0) > ——— ^———

log L

d02

The right-hand side is known as the Cramer-Rao lower bound (CRLB).

(In Section 7.3 the likelihood function was always evaluated at the observed values of the sample, because there we were only concerned with the definition and computation of the maximum likelihood estimator...

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Heteroscedasticity

In the classical regression model it is assumed that the variance of the error term is constant (homoscedastic). Here we relax this assumption and specify more generally that

0^3.1.12) Vut = пі t = 1, 2, . . . , T.

This assumption of nonconstant variances is called heteroscedasticity. The other assumptions remain the same. If the variances are known, this model is a special case of the model discussed in Section 13.1.1. In the present case, X is a diagonal matrix whose tth diagonal element is equal to erf. The GLS estimator in this case is given a special name, the weighted least squares estimator.

If the variances are unknown, we must specify them as depending on a finite number of parameters. There are two main methods of parameteri­zation.

In the first method, the variances are assum...

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