Let X1, X2 Xn be a random sample from a distribution with mean m and variance a2.
Show that the sample mean X is a consistent estimator of m.
From elementary statistics it is known that E(X) = m andvar(X) = a2/n. Since E(X) = m regardless of the sample size, it is unbiased. Moreover, as n increases indefinitely, var(X) tends toward zero. Hence, X is a consistent estimator of m.
The following rules about probability limits are noteworthy.
1. Invariance (Slutsky property). If 0 is a consistent estimator of 0 and if h(0) is any continuous function of 0, then plim h( 0) = h(0)
What this means is that if 0 is a consistent estimator of 0, then 1 /0 is also a consistent estimator of 1/0 and that log(0) is also a consistent estimator of log (0). Note that this property does not hold true of the expectation operator E; that is, if 0 is an unbiased estimator of 0 [that is, E(0) = 0], it is not true that 1 /0 is an unbiased estimator of 1/0; that is, E(1/0) = 1 /E(0) = 1/0.
That is, the probability limit of a constant is the same constant.
3. If 01 and 02 are consistent estimators, then plim (01 + 02) = plim 01 + plim 02 plim (0102) = plim 01 plim 02
The last two properties, in general, do not hold true of the expectation operator E. Thus, EO9/2) = E(§1)/E(§2). Similarly, §2) = E0)E0). If, however, 01 and 02 are independently distributed, E(0102) = E(01)E(02), as noted previously.
7More technically, E(§n) = 6 and limn^œ var (§n) = 0.
APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS 905
Asymptotic Efficiency. Let 0) be an estimator of 0. The variance of the asymptotic distribution of 0 is called the asymptotic variance of 0. If 0 is consistent and its asymptotic variance is smaller than the asymptotic variance of all other consistent estimators of 0,0 is called asymptotically efficient.
Asymptotic Normality. An estimator 0 is said to be asymptotically normally distributed if its sampling distribution tends to approach the normal distribution as the sample size n increases indefinitely. For example, statistical theory shows that if X1, X2,..., Xn are independent normally distributed variables with the same mean p and the same variance a2, the sample mean X is also normally distributed with mean p and variance a2 /n in small as well as large samples. But if the Xi are independent with mean p and variance a2 but are not necessarily from the normal distribution, then the sample mean X is asymptotically normally distributed with mean p and variance a 2/n; that is, as the sample size n increases indefinitely, the sample mean tends to be normally distributed with mean p and variance a2/n. That is in fact the central limit theorem discussed previously.
A.8 STATISTICAL INFERENCE: HYPOTHESIS TESTING
Estimation and hypothesis testing constitute the twin branches of classical statistical inference. Having examined the problem of estimation, we briefly look at the problem of testing statistical hypotheses.
The problem of hypothesis testing may be stated as follows. Assume that we have an rvX with a known PDF f(x; 0), where 0 is the parameter of the distribution. Having obtained a random sample of size n, we obtain the point estimator 0. Since the true 0 is rarely known, we raise the question: Is the estimator 0 "compatible" with some hypothesized value of 0, say, 0 = 0", where 0" is a specific numerical value of 0? In other words, could our sample have come from the PDF f (x; 0) = 0 ? In the language of hypothesis testing 0 = 0" is called the null (or maintained) hypothesis and is generally denoted by H0. The null hypothesis is tested against an alternative hypothesis, denoted by H1, which, for example, may state that 0 = 0". (Note: In some textbooks, H0 and H1 are designated by H1 and H2, respectively.)
The null hypothesis and the alternative hypothesis can be simple or composite. A hypothesis is called simple if it specifies the value(s) of the para-meter(s) of the distribution; otherwise it is called a composite hypothesis. Thus, if X ~ N(p, a2) and we state that
it is a simple hypothesis, whereas
is a composite hypothesis because here the value of a is not specified.
906 APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS
To test the null hypothesis (i.e., to test its validity), we use the sample information to obtain what is known as the test statistic. Very often this test statistic turns out to be the point estimator of the unknown parameter. Then we try to find out the sampling, or probability, distribution of the test statistic and use the confidence interval or test of significance approach to test the null hypothesis. The mechanics are illustrated below.
To fix the ideas, let us revert to Example 23, which was concerned with the height (X) of men in a population. We are told that
Let us assume that
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