Next: 7.2 Fourier and Related Up: 7. Transforms in Statistics Previous: 7. Transforms in Statistics

# 7.1 Introduction

As an ''appetizer'' we give two simple examples of use of transformations in statistics, Fisher and Box-Cox transformations as well as the empirical Fourier-Stieltjes transform.

Example 1   Assume that we are looking for variance transformation , in the case where is a function of the mean . The first order Taylor expansion of about mean  is

Ignoring quadratic and higher order terms we see that

If is to be , we obtain

resulting in

This is a theoretical basis for the so-called Fisher -transformation.

Let be a sample from bivariate normal distribution , and , .

The Pearson coefficient of linear correlation

has a complicated distribution involving special functions, e.g., Anderson (1984, p. 113)[1]. However, it is well known that the asymptotic distribution for is normal . Since the variance is a function of mean,

is known as Fisher -transformation for the correlation coefficient (usually for and ). Assume that and are mapped to and as

The distribution of is approximately normal and this approximation is quite accurate when is small and when is as low as . The use of Fisher -transformation is illustrated on finding the confidence intervals for and testing hypotheses about .

 (a) (b)

To exemplify the above, we generated pairs of normally distributed random samples with theoretical correlation . This was done by generating two i.i.d. normal samples , and  of length  and taking the transformation , . The sample correlation coefficient is found. This was repeated times. The histogram of  sample correlation coefficients is shown in Fig. 7.1a. The histogram of -transformed 's is shown in Fig. 7.1b with superimposed normal approximation .

(i) For example, confidence interval for is:

where and and stands for the standard normal cumulative distribution function.

If and , and . In terms of the confidence interval is .

(ii) Assume that two samples of size and , respectively, are obtained form two different bivariate normal populations. We are interested in testing against the two sided alternative. After observing and and transforming them to and , we conclude that the -value of the test is .

Example 2   Box and Cox (1964)[4] introduced a family of transformations, indexed by real parameter , applicable to positive data ,

 (7.1)

This transformation is mostly applied to responses in linear models exhibiting non-normality and/or heteroscedasticity. For properly selected , transformed data may look ''more normal'' and amenable to standard modeling techniques. The parameter  is selected by maximizing the log-likelihood,

 (7.2)

where are given in (7.1) and .

As an illustration, we apply the Box-Cox transformation to apparently skewed data of CEO salaries.

Forbes magazine published data on the best small firms in 1993. These were firms with annual sales of more than five and less than million. Firms were ranked by five-year average return on investment. One of the variables extracted is the annual salary of the chief executive officer for the first ranked firms (since one datum is missing, the sample size is ). Figure 7.2a shows the histogram of row data (salaries). The data show moderate skeweness to the right. Figure 7.2b gives the values of likelihood in (7.2) for different values of . Note that (7.2) is maximized for approximately equal to . Figure 7.2c gives the transformed data by Box-Cox transformation with . The histogram of transformed salaries is notably symetrized.

 (a) (b) (c)

Example 3   As an example of transforms utilized in statistics, we provide an application of empirical Fourier-Stieltjes transform (empirical characteristic function) in testing for the independence.

The characteristic function of a probability distribution  is defined as its Fourier-Stieltjes transform,

 (7.3)

where is expectation and random variable has distribution function . It is well known that the correspondence of characteristic functions and distribution functions is -, and that closeness in the domain of characteristic functions corresponds to closeness in the domain of distribution functions. In addition to uniqueness, characteristic functions are bounded. The same does not hold for moment generating functions which are Laplace transforms of distribution functions.

For a sample one defines empirical characteristic function as

The result by Feuerverger and Mureika (1977)[9] establishes the large sample properties of the empirical characteristic function.

Theorem 1   For any

holds. Moreover, when , the stochastic process

converges in distribution to a complex-valued Gaussian zero-mean process satisfying and

where denotes complex conjugate of .

Following Murata (2001)[20] we describe how the empirical characteristic function can be used in testing for the independence of two components in bivariate distributions.

Given the bivariate sample , , we are interested in testing for independence of the components and . The test can be based on the following bivariate process,

where .

Murata (2001)[20] shows that has Gaussian weak limit and that

The statistics

has approximately distribution with  degrees of freedom for any and  finite. Symbols  and  stand for the real and imaginary parts of a complex number. The matrix is matrix with entries

Any fixed pair gives a valid test, and in the numerical example we selected and for calculational convenience.

 (a) (b) (c)

We generated two independent components from the Beta() distribution of size and found statistics and corresponding -values times. Figure 7.3a,b depicts histograms of statistics and values based on simulations. Since the generated components and are independent, the histogram for agrees with asymptotic distribution, and of course, the -values are uniform on . In Fig. 7.3c we show the -values when the components and are not independent. Using two independent Beta() components and , the second component is constructed as . Notice that for majority of simulational runs the independence hypothesis is rejected, i.e., the -values cluster around 0.

Next: 7.2 Fourier and Related Up: 7. Transforms in Statistics Previous: 7. Transforms in Statistics