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
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]