where [Z.sub.j] are independent standard

normal random variables.

The null hypothesis will be rejected if the hypothesis-testing statistic's value falls below the critical value of the standardised

normal random variable [z.sub.0,05] = 1.64 .

(1998) found that heteroscedasticity in the

normal random variable introduces substantial biases into Maximum Likelihood (ML) estimator of the parameters in frontier models.

Moreover, if the relative price changes in each holding period are independently and identically distributed

normal random variables, the empirical volatilities, [S.sub.23], are related to the chi-square random variable [[chi].sub.k.sup.2] = by [[chi].sub.k.sup.2] = k h [S.sub.23.sup.2] where k = n -1 and n is the number of trading days in the holding period, in this case 23.

"Excessively different from 0," in employment discrimination matters at least, usually means "more than about two standard deviations."(6) Since that rule arose in connection with

normal random variables, one must convert the above t-statistic making use of the fact that it has N -2 degrees of freedom.

The integral in [A4] is one minus the cumulative distribution function (c.d.f.) of a unit

normal random variable, so [W.sub.1] = [R.sub.0] [1 - N([a.sub.'])], where N(.) is the unit normal c.d.f.

with i = 1,2, where [S.sup.i.sub.t] is the price of the ith asset corresponding to the standard Brown motion, [q.sup.i] is the dividend rate for the ith asset, [h.sup.i.sub.t] is the volatility of the ith asset price, [v.sup.i.sub.t+1], conditional on information at time t, is a standard

normal random variable, [r.sub.j] is the riskless rate of return over the period, and [[lambda] is the unit risk premium for the ith asset.

It is a straightforward exercise to show that, for a

normal random variable Z

For Normal distributions, Stein's lemma states that for a bivariate

normal random variable (X, Y), we have

where [z.sub.[alpha]/2] is the positive value that the standard

normal random variable exceeds with a probability of [alpha]/2.

For

normal random variables, X can be transformed into standard

normal random variables u through a linear transformation, as follows:

A normal or non-normal random variable Z can be approximated using one-dimensional Hermite orthogonal polynomials [[PSI].sub.i] of the standard

normal random variables [xi] as [17,18]