Jeff Miller
Department of Psychology
University of Otago
Dunedin, New Zealand

CUPID: A Program for Computations
with Univariate Probability Distributions
Version 2.13

Jan, 2006

Contents

Part 1
Introduction

1  Overview of the Package

2  Introduction

• Here are the kinds of things it does:

  Computation:

  It computes numerical values of many quantities associated with probability distributions, including probability densities, cumulative probabilities, inverse cumulative probabilities, mean, variance, skewness, etc. Such computations are often needed to determine the quantitative predictions of stochastic models, and CUPID can often be used instead of a special-purpose program to do such computations. In addition, because cumulative and inverse cumulative probabilities are available, the program provides a handy set of on-line statistical tables for researchers doing statistical hypothesis testing (i.e., what is the critical F value with 7 and 143 degrees of freedom? what is the p level associated with this F value? etc.).

  Random Number Generation:

  CUPID generates random numbers from any desired distribution. This can be an easy way to generate random numbers for computer simulations (but see the program "RandGen", based on CUPID, for a more power random-number generation tool).

  Estimation:

  CUPID can often find the best estimates of the parameters of a distribution, with "best" defined in terms of either (a) maximizing the likelihood of a given set of observations, (b) matching certain observed percentile values, (c) matching certain observed moments, (d) minimizing the chi-square fit to a set of observed bin proportions, (e) maximizing the likelihood of a set of probit-type data, or (f) minimizing the chi-square fit to a set of probit-type data.

  Graphing:

  CUPID can make some fairly crude plots of the PDF, CDF, MGF, and Hazard functions. In addition, for users wanting more sophisticated graphs or graphs that can be printed, it has a "Columns" command to generate columns of X and Y values that can be used as input to your own graphing program.
• Here are some of the kinds of distributions it knows about:

  Standard distributions:

  CUPID knows about many standard distributions (e.g., Normal or Gaussian, Uniform, Gamma, F, t, ...), which I refer to as "primitive" distributions. The program also knows how to work with various kinds of distributions that can be derived from these primitive distributions, as indicated by the remaining distributions in this list.

  Transformations:

  A very simple derived distribution is a linear transformation of a primitive one (e.g., an exponential plus a constant). More complicated transformations include the exponential transform (e.g., Y = e^X, where X has one of the primitive distributions), the natural log (Y = ln(X) and inverse (1/X) transformations, and power function transformations. More complicated transformations can be constructed by combining several of these.

  Truncated distributions:

  New random variables may be defined by restricting other variables to part of their usual range (e.g., a standard normal restricted to the range from -2 to +2).

  Mixture distributions:

  Random variables may be formed by randomly selecting one of a set of independent random variables (e.g., an RV that is a standard normal with probability p & a uniform from -2 to +2 with probability 1-p).

  Infinite mixture distributions:

  Random variables may be formed by letting one of their parameters vary randomly according to some (usually different) distribution. For example, Y might be distributed following a normal distribution with m = X and s = 1, where X follows a uniform distribution between 0 and 1.

  Convolutions:

  Random variables may be formed by summing values of independent random variables (e.g., sum of independent standard normal and uniform from -2 to +2).

  Order distributions:

  Random variables may be formed by choosing the i'th order statistic from a sample of n independent random variables (e.g., the minimum of 10 standard normals).

  Approximation distributions:

  For complicated distributions that require slow numerical integration, CUPID has some approximations available to speed things up.

3  Installation Instructions

1. Unzip the distribution file CUPID.ZIP.
2. Open the file 00Readme.txt and follow the instructions in it for a brief demo.
3. If you want to be able to run the programs from a command window opened within any directory, move all of the unzipped EXE files to a directory in your path.

4  User Interface

5  Distributions Available in CUPID

5.1  Continuous Distributions

Beta(A,B)

The Beta distribution is defined over the interval from zero to one, and its shape is determined by its two parameters A and B. Its PDF is

f(x) = 1 b(A,B) xA-1 (1-x)B-1,    0 < x < 1

The mean is A/(A+B), and the variance is AB(A+B)^-2(A+B+1)^-1.

Cauchy(L,S)

This distribution is defined in terms of location and scale parameters L and S > 0, respectively. Its PDF is

f(x) = 1 pS é ë 1+ ì í î x-L S ü ý þ 2   ù û

ChiSquare(df)

This is a generalization of the distribution of the sum of df independent squared standard normals. Its parameter is df - a positive real number.

Chi(df)

This is the distribution of the positive square root of a ChiSquare random variable. Its parameter is df - a positive real number.

Cosine

For 0 £ x £ P/2, f(x)=cos(x) and F(x)=sin(x).

ExGaussian(m,s,l)

This is the distribution of the sum of independent Normal and Exponential random variables. Its parameters are the m and s of the Normal, and the rate l of the Exponential. You can change to specifying the exponential in terms of its mean instead of its rate by first issuing the command UseExpMean, described in Section 10.

ExGaussMn(m,s,m_e)

This is a reparameterization of the ExGaussian. Its parameters are the m and s of the Normal, and the mean of the exponential, m_e.

ExGaussRat(m,s,r)

This is just a reparameterization of the ExGaussian. Its parameters are the m and s of the Normal, and the ratio, r, of the mean of the exponential to the sigma of the normal.

Exponential(l)

This distribution is well-known. By default, the parameter is the rate l; the mean is 1/l. You can change to specifying Mean instead of Rate with the UseExpMean, described near the end of Section 10.

ExpSum(r1,r2)

This is the sum (convolution) of two exponentials with different rates. The two parameters are the two rates, which must be different enough to avoid numerical errors. For the convolution of exponentials with the same rates, of course, you should use the Gamma.

ExpSumT(r1,r2,Cutoff)

This is the sum (convolution) of two exponentials with different rates truncated at a given cutoff value. The first two parameters are the two rates, which must be different enough to avoid numerical errors; the third parameter is the upper truncation point. For the convolution of exponentials with the same rates, of course, you should use the Gamma.

ExpoNo(m,s)

I just invented this as an ad-hoc solution for a problem I was working on one time. I don't know whether it has ever been considered before or will ever be useful again, and I certainly don't know whether I gave it a reasonable name. In any case, it is a transformation of a normal random variable X. Specifically, it is the distribution of

Y = eX 1+eX

where X has a normal distribution with mean m and standard deviation s. The two parameters of this distribution are the m and s of the underlying normal X.

ExtremeVal(a,b)

Extreme-value Type I distribution (a.k.a. Fisher-Tippett distribution, Gumbel distribution, sometimes also called the double exponential distribution, to be confused with the Laplace distribution), with parameters a and b > 0. The CDF is

F(x) = exp[-e-(x-a)/b]

ExWald(m,s,a,l)

This is the distribution of the sum of two independent random variables: one from a three-parameter Wald distribution with parameters (m,s,a); and one from an exponential distribution with rate l. Schwarz (2001, 2002) describes the ex-Wald distribution in detail.

ExWaldMn(m,s,a,m_e)

This is a reparameterization of the ExWald. The first three parameters are the same as in the plain ExWald, and the fourth parameter is the mean of the exponential, m_e, instead of the rate.

ExWaldMSM(m,sd,m_e)

This is a further reparameterization of the ExWald. The first two parameters are the mean and standard deviation (not s!) of the Wald component, and the third parameter is the mean of the exponential, m_e. The Wald parameter a is set to 1.0.

F(dfNumer,dfDenom)

This is Fisher's distribution of the ratio of two independent normed Chi-square distributions, as commonly used in linear models (e.g., analysis of variance). The two integer parameters are the degrees of freedom of the numerator and denominator, respectively.

Gamma(N,l)

This is the distribution of the sum of k exponentials, each with rate l. In this distribution, k must be a positive integer. In the RNGamma distribution (see below), k is any positive real. You can change to specifying Mean instead of Rate with the UseExpMean, described near the end of Section 10.

Geary(SampleSize)

The Geary statistic arises in testing to see whether a set of observations come from a normal distribution (D'Agostino, 1970). CUPID will not compute the test for you, but it will give you significance levels of values computed by some other program.

GenErr(Mu,Scale,Shape)

This is the general (a.k.a. "generalized") error distribution (e.g., Evans, Hasting, & Peacock, 1993, p. 57), also known as Subbotin's distribution (e.g., Johnson, Kotz, & Balakrishnan, 1995, 2nd Ed., Vol 2, p. 195). The correct PDF is

f(x) = exp[-|x-Mu|Shape/(2·Scale)] Scale1/Shape ·2(1+1/Shape) ·G(1+1/Shape)

although both EHP and JKB give it with incorrect exponents of the Scale parameter in the denominator. This version uses the shape parameter denoted l by Evans et al. Note that the Laplace, normal, and uniform distributions are special cases of this distribution with this shape parameter equal to 1, 2, and approaching ¥, respectively. In practice, lots of combinations of parameter values give overflow errors, especially if the shape parameter is more than approximately 3.

HypTan(Scale)

This is the Hyperbolic Tangent distribution, whose PDF and CDF are

f(x) = 4·b [ebx + e-bx]2 F(x) = ebx - e-bx ebx + e-bx

where b is the scale parameter. This distribution arises as a model of psychometric functions (e.g., Strasburger, 2001).

Inverse Gaussian

See the Wald3 distribution.

Laplace(L,S)

Also known as the double exponential. In terms of location and scale parameters, L and S > 0, respectively, the PDF is

f(x) = 1 2 S e-|x-L|/S

Lilliefors(SampleSize)

The Lilliefors statistic arises in testing whether a set of observations come from a normal distribution (Dallal & Wilkinson, 1986). CUPID will not compute the test for you, but it will give you the critical values for the test. The approximation given by Dallal and Wilkinson is only accurate for upper tail probabilities (i.e., CDF values of 0.90 or greater), so this distribution really should really be used only for the computation of critical values.

Logistic(m,b)

This distribution is defined in terms of a location parameter m and a scale parameter b. The cumulative form of the distribution is

F(x) = 1 1 + e[(-(x-m))/(b)]

LogNormal(m_n,s_n)

This is the distribution of X such that ln(X) is normally distributed. The parameters are the m_n and s_n of the underlying normal.

LogNormalMS(m_l,s_l)

This is the distribution of X such that ln(X) is normally distributed. The parameters are the m_l and s_l of the overall lognormal. Comparing parameters of the two lognormal distributions,

ml = exp(mn) ·exp(sn2/2) sl2 = [exp(mn)]2 ·exp(s2n) ·[exp(s2n) - 1] sn2 = ln æ è sl2 ml2 +1 ö ø mn = ln æ è ml exp(sl2/2) ö ø

Naka-Rushton(Scale)

This is the distribution of X ³ 0 such that

f(x) = 2 ·x ·a2 (1+(a·x)2)2 F(x) = (a·x)2 1+(a·x)2

where a is the scale parameter. In the actual distribution, moments above the first do not exist; they do exist in CUPID's truncated version of the distribution, however.

NoncentralF(dfNumer,dfDenom,Noncentrality)

This is the distribution of the ratio of independent noncentral and central chi-squares, with the former in the numerator. It is most often used in the computation of power of the F test. The noncentrality parameter is defined in terms of the dfNumer normal random variables whose sum of squares is yields the chi-square in the numerator. Specifically,

l = dfNumer å i=1  Li2

where L_i is the expected value of the ith random variable contributing to this sum of squares.

Normal(m,s)

I'll bet you know this one already. Parameters are m and s, not s².

Pareto1(K,A)

This is a Pareto distribution of the first kind, as defined by Johnson, Kotz, and Balakrishnan (1994, vol 1, p 574), with PDF and CDF

f(x) = A ·KA ·x-(A+1) F(x) = 1 - æ è K A ö ø A

where K > 0, A > 0, and x ³ K. This is a model for income, where K is some minimum income and F(x) is the probability that a randomly selected income is less than or equal to x.

Quantal(Threshold)

This distribution is related to the Poisson. This is the distribution of X ³ 0 such that

F(x) = 1 - T-1 å t=0  xt t! e-x

This distribution arises as a model of psychometric functions in visual psychophysics (e.g., Gescheider, 1997, p. 85). The threshold parameter, T ³ 1, represents an observer's fixed threshold for the number of quanta of light that must be detected before saying "Yes, I saw the stimulus." Quanta are assumed to be emitted from the stimulus according to a Poisson distribution with parameter x. Then, F(x) is the psychometric function for the probability of saying "Yes" as a function of the mean number of quanta, x, emitted by the stimulus. Note that it makes no real sense to think of x as a random variable in this example, but the probability distribution provides a useful model anyway.

Quick(Scale,Shape)

This is the distribution of X ³ 0 with PDF and CDF

f(x) = 2-([x/(a)])b · æ è x a ö ø b   ·b·ln(2) x F(x) = 1 - 2-([x/(a)])b

where a is the scale parameter and b is the shape parameter. This distribution arises as a model of psychometric functions (e.g., Quick, 1974; Strasburger, 2001).

Rayleigh(s)

If Y₁ and Y₂ are independent normal random variables with mean 0 and standard deviation s, then X = Ö{Y₁²+Y₂²} has a Rayleigh distribution with scale parameter s. The PDF is

f(x) = e-x2/(2s2) x s2

RNGamma(k,l)

See "Gamma". In this version, the shape parameter k is a real number rather than an integer. The PDF is

f(x) = xk-1 exp(-x/l) G(k)lk     for x > 0

where G(k) = ò₀^¥ s^k-1 exp(-s) ds.

Rosin(D_m,P)

This distribution is generally known as the Rosin-Rammler, and it arises in analyses of particle sizes. Its CDF is

F(x) = 1 - exp é ë - æ è x Dm ö ø P   ù û

rPearson(SampleSize)

This is the sampling distribution of Pearson's r (correlation coefficient) under the null hypothesis that the true correlation is zero (and assuming the usual bivariate normality). The parameter is SampleSize, the number of pairs of observations across which the correlation is computed.

StudRng(df,NSamples)

Distribution of Studentized range statistic with df degrees of freedom for error and NSamples samples. Because both parameters are integers, automatic program-based estimation of these parameters is rarely successful.

t(df)

Student's t-distribution, with degrees of freedom equal to df.

Triangular(B,T)

In this distribution the density function has the shape of an equilateral triangle across some range. The parameters are the bottom (B) and the top of the range (T). The PDF is then:

f(x) = ì ï ï í ï ï î (x - B) ×Hp if B £ x £ B+T 2 (T - x) ×Hp if B+T 2 £ x £ T

where H_p is the height of the PDF at its peak, adjusted to so that the total area of the triangle is 1.0.

TriangularG(B,P,T)

In this (more general triangular) distribution, the density function has the shape of a not-necessarily-equilateral triangle across some range. The parameters are the bottom of the range (B), the point at which the triangle reaches its maximum (P), and the top of the range (T). The PDF is then:

f(x) = ì ï ï í ï ï î (x - B) ×Hp P-B if B £ x £ P (T - x) ×Hp T-P if P £ x £ T

where H_p is the height of the PDF at its peak, adjusted to so that the total area of the triangle is 1.0.

Uniform(B,T)

This is the distribution in which all values are equally likely within some range. The parameters are the bottom and the top of the range, B and T.

UniGap(T)

This is an equal-probability mixture of two uniform distributions, one extending from -T to 0 and the other extending from T to 2·T. It is "model 4" of Sternberg and Knoll (1973). The median is somewhat arbitrarily defined as T/2.

VonMises(L,S)

This is the Von Mises distribution with location and scale parameters L and S, respectively. It is defined on the range of 0 to 2p, and L must be in this range. Typically, the Von Mises distribution is regarded as an analog of the normal distribution on a circle instead of on the real number line.

Wald3(m, s, a)

This is the general, three-parameter version of the Wald distribution. Specifically, assume a one-dimensional Wiener diffusion process starting at position 0 at time 0 and drifting with average rate m and variance s², and consider X to be the first passage time through position a. The PDF of X is

f(x) = a s Ö 2 px3 · exp é ë - (a - mx)2 2 s2 x ù û

where all three parameters and x must be positive. Note: By default, the sigma parameter is fixed in all parameter searches.

Weibull(Scale,Power,Origin)

As defined by Johnson & Kotz (1970, p. 250): "X has a Weibull distribution if there are values of the parameters c( > 0), a( > 0), and n₀ such that

Y = é ë (X-n0) a ù û c

has the exponential distribution with rate = 1". Here, the parameters c, a, and n₀ are referred to as the "scale," "power," and "origin" parameters, respectively.

5.2  Discrete Distributions

Binomial(N,p)

The distribution of the number of successes in N Bernoulli trials, with probability p of success on each trial.

NegativeBinomial(N,p)

The distribution of the number of failures before reaching N successes in a sequence of Bernoulli trials, with probability p of success on each trial.

NeymanA(m₁,m₂)

This is the Neyman type A distribution, with mass on the nonnegative integers. It has mean m₁·m₂ and variance m₁·m₂·(1+m₂). The PDF is defined by:

Pr (X=0) = exp[ -m1 {1 - exp(-m2)}] Pr (X=k) = m1 k e-m2 k å j=1  m2j Pr (X=k-j) (j-1)!                 for k > 0

Constant(C)

This is a degenerate distribution that always takes on the same value. Its parameter is that value. Perhaps surprisingly, it can be convenient to have this distribution available. Warning: For technical reasons, the constant distribution does not work well in many of the derived distributions discussed in the next section. Thus, it should be avoided whenever possible. For example, you should always use: LinearTrans(Gamma(2,.01),1,100) rather than the equivalent Convolution(Gamma(2,.01),Constant(100)) If the constant distribution does not work where you need it, try instead a normal distribution with a really small standard deviation.

Geometric(P)

The distribution of the trial number of the first success in a sequence of Bernoulli trials, where P is the probability of success on each try.

List(filename)

This random variable allows you to define any discrete set of X values, each with its own arbitrary probability. The command List(FileName) tells CUPID to read the distribution from the indicated file. The first line in the file contains the number of X values in the distribution. After that, there should be one line for each X value, with the first number on the line being the X value itself and the second number being the probability of that X value. These values need not be sorted, and in fact the same X value can appear on several different lines, in which case the associated probabilities will be summed. (If X's do appear on more than one line, then the first line in the file should actually contain the number of X-containing lines rather than the number of distinct X's.)

Poisson(U)

X has a Poisson distribution with parameter U if

Pr (X=x) = e-UUx x! ,    x=0,1,2,¼, U > 0

The mean and variance both equal U.

UniformInt(Low,High)

This is the distribution of equally likely integer values between the two integer parameters, Low and High, inclusive.

5.3  Transformation Distributions

5.4  Derived Distributions

Convolution(RV1,RV2)

This is the distribution of a sum of independent random variables, RV1 and RV2, where RV1 and RV2 are each legal distributions in their own right. For example, Convolution(Normal(0,1),Uniform(0,1)) specifies the convolution of these normal and uniform distributions, and Convolution(Normal(0,1),Uniform(0,1),Gamma(3,0.01)) specifies the convolution of the three indicated distributions.

In general, to define a convolution, the user types something of the form: Convolution(BasisDist1(Parms),...,BasisDistK(Parms)) where Parms stands for the parameters associated with each of the distributions. There are K random variables summed together, and the distributions of these summed variables are simply listed, separated by commas.

CUPID is not very smart about convolutions. At this point, it only knows how to compute means, variances, and random numbers in an intelligent way. Everything else is computed using (recursive) numerical integration, which tends to be pretty slow. Also, CUPID does not "realize" that some convolutions result in a new distribution about which it already knows (e.g., convolution of two normals is normal). Thus, computations involving these convolutions proceed via numerical integration even though direct computation would be possible.

The current version can handle convolutions where all distributions are discrete, all are continuous, or some are is discrete and some continuous, but it cannot handle convolutions in which one or more distributions are mixed (i.e., partly discrete and partly continuous).

I would be very happy for suggestions on how to augment CUPID's handling of convolutions, especially those accompanied by Pascal code.

ConvolutionIID(N,RV)

This is just an easier way to specify a convolution when all N summed random variables have the same distribution, RV. ConvolutionIID(3,Uniform(0,1)) is the same as Convolution(Uniform(0,1),Uniform(0,1),Uniform(0,1))

Difference(RV1,RV2)

This is the distribution of the difference of two independent random variables, RV1 minus RV2, where RV1 and RV2 are each legal distributions in their own right. For example, Difference(Uniform(0,1),Uniform(0,1)) specifies a difference between two standard uniform distributions, which ranges from -1 to 1 (not uniformly). CUPID handles difference distributions dumbly, like convolutions. The current version can handle differences where both distributions are discrete, both are continuous, or one is discrete and one continuous, but it cannot handle differences in which one or both distributions are mixed (i.e., partly discrete and partly continuous).

Mixture(p1,RV1,p2,RV2,...,pk,RVk)

Mixtures are distributions formed by randomly selecting one of a number of random variables. For example, Mixture(0.5,Normal(0,1),0.5,Uniform(0,1)) defines a random variable that comes from a standard normal half the time and a standard uniform the other half of the time. In general, the format of this distribution is: Mixture(p₁,BasisDist1(Parms),p₂,BasisDist1(Parms),...,p_k,BasisDistK(Parms)) and the p_i's must sum to one (it is also legal to omit p_k).

InfMix(RV1,MixParm,RV2(Parms))

The InfMix distribution is an infinite mixture, formed when a parameter of one distribution is itself randomly distributed according to another distribution. For example, InfMix(Normal(0,5),1,Uniform(10,20)) defines a random variable that comes from a normal distribution with standard deviation 5. The first parameter of that distribution (as signified by the "1" between the two distribution names) follows a uniform distribution from 10 to 20. As another example, InfMix(Normal(0,5),2,Uniform(10,20)) defines a random variable that comes from a normal distribution with mean zero and standard deviation varying uniformly from 10 to 20. In general, the format of this distribution is: InfMix(ParentDist(Parms),MixParm,ParmDist(Parms)) where ParentDist is a distribution, MixParm is an integer indicating whether the first, second, ..., parameter of the ParentDist varies randomly, and ParmDist is the distribution of that parameter.

InfMix may be used recursively. For example, InfMix(InfMix(Normal(0,5),1,Uniform(0,2)),2,Uniform(4,6)) defines a normal distribution in which the mean is uniform(0,2) and the standard deviation is uniform(4,6).

Limitations: (1) At present, computations of the upper and lower bounds of InfMix distributions assume that the largest and smallest values of the random variable are obtained when the underlying ParmDist is at its two extremes. (2) Extreme caution is needed with these distributions because problems often arise in numerical integration. I have found it helpful to increase the IntegralMinSteps to 10, which was enough in most of the cases I've looked at, but you may need to adjust this up (for precision) or down (for speed) in your cases.

Truncated(RV,Min,Max)

A truncated distribution is a conditional distribution, conditioning on the random variable RV falling within the interval from Min to Max. For example, Truncated(Normal(0,1),-1,1) defines a random variable that is always between -1 and 1, and which within that interval has relative probabilities defined by the PDF of the standard normal. In general, the format of this distribution is: Truncated(BasisDistribution(Parms),Min,Max)

It is sometimes convenient to specify the truncation boundaries in terms the probabilities you want to cut off rather than the scores themselves. For example, you might want to look at the middle 90% of a normal distribution but might not immediately know which scores cut off the top and bottom 5%. For this reason, there is a variant of the command that takes probabilities instead of values for min and max, like this: TruncatedP(BasisDistribution(Parms),0.05,0.95) With TruncatedP, CUPID will use its InverseCDF function to find the score values that correspond to the cumulative probabilities that you specify, and then truncate at those score values.

Bounded(RV,Min,Max)

I do not know if this is a standard type of distribution or not, and would appreciate any comments on it from those in the know. A bounded distribution is similar to a truncated distribution in that the random variable must fall within the range of Min to Max. The difference is that all values less than Min are converted to Min, and all values less than Max are converted to Max. Thus, there are discrete masses of probability at Min and Max, and the probability density function between Min and Max is not conditionalized.

For example, consider the distribution Bounded(Normal(0,1),-1,1). This is really a mixture of these three distributions:

Distribution	Mixture Probability
Constant(-1)	0.1587
Truncated(Normal(0,1),-1,1)	0.6826
Constant(1)	0.1587

Note that 0.1587 is the probability that a normal(0,1) score is less than -1, and also the probability that it is greater than 1. Bounding the distribution thus means taking all of the probability density higher than the upper value and massing it at that value.

As with the truncated distribution, there is a form of the Bounded distribution based on probabilities, as in: BoundedP(Normal(0,1),0.1,0.9) .

Order(k,RV1,RV2,RV2,...,RVn)

The distribution of this order statistic is the distribution of the k'th largest observation in a sample of n independent observations from the n indicated random variables. For example, Order(2,Normal(0,1),Uniform(0,1),Exponential(1)) defines a random variable that is the median (2nd largest) in a sample containing one score from the standard normal, one from the uniform from 0-1, and one from the exponential with rate 1. In general, the format of this distribution is: Order(k,BasisDist1(Parms),...,BasisDistN(Parms)) In the special case where the basis distributions are all identical, it is more convenient to use the OrderIID distribution, described next.

OrderIID(k,n,RV)

This is the special case of the order distribution in which the basis distributions are identical as well as independent. In general, the format of this distribution is: OrderIID(k,N,BasisDist(Parms)) It is only necessary to specify the basis distribution once, since all are identical; instead, you have to specify how many there are (N).

OrdExp(i,n,l)

This is the special case of OrderIID in which the basis distribution is an exponential with rate l, and you want the i'th order statistic in a sample of n (1 £ i £ n). For this case there are nice fast closed forms for the mean and variance that were given to me by Rolf Ulrich.

OrdBinary(i,n1,RV1,n2,RV2)

This is an order distribution with two types of underlying RVs. For example, OrdBinary(2,5,Normal(0,1),7,Uniform(0,1)) specifies the distribution of the second order statistic in samples of 12 made up of five standard normals and seven standard uniforms.

MinBound(RV1,RV2)

Consider two arbitrary random variables X and Y, which may or may not be independent, and let Z º min(X,Y). The CDFs of these three random variables must obey the inequality

Fz(t) £ Fx(t) + Fy(t)     for all  t

because

Fz(t) = Fx(t) + Fy(t) - Pr (X £ t & Y £ t)

Thus, for any two basis RVs X and Y, we can construct the random variable Z which is a lower bound on the distribution of min(X,Y):

Fz(t) = ì ï í ï î Fx(t) + Fy(t) if  Fx(t) + Fy(t) < 1 1 if  Fx(t) + Fy(t) ³ 1

MinBound implements this lower bound distribution for any two arbitrary random variables X and Y.

5.5  Bin-Based Distributions

Histogram

The histogram is a continuous distribution with a flat PDF within each bin.

Polygon

The polygon is a continuous distribution with a linear but not necessarily flat PDF within each bin. More specifically, the PDF of the polygon distribution is defined by a series of NBins+1 points; the first point gives the height of the PDF at the lower bound of the distribution, and the remaining NBins points give the heights of the PDF at the top of each bin (the top of the top bin showing the PDF at the upper bound of the distribution). Within each bin, the PDF is a straight line going from the height at the bottom of the bin to the height at the top of the bin.

FreqPolygon

The frequency polygon is also a continuous distribution with a linear but not necessarily flat PDF within each bin. Unlike the polygon, though, it keys on the PDF in the middle of each bin rather than the upper edge. More specifically, the PDF of this distribution is defined by a series of NBins+2 points; the first point gives the height of the PDF at the lower bound of the distribution, the last point gives the height of the PDF at the upper bound of the distribution, and the remaining NBins points give the heights of the PDF at the center of each bin. Within each bin, the PDF is formed by straight lines going from the height at the bin's center to the heights at the centers of the adjacent bins.

BinCen

The "BinCenters" random variable is a discrete distribution with all of the probability mass associated with each bin assigned to the midpoint of the bin.

1. The minimum value in the distribution (i.e., the lower edge of the lowest bin).
2. The maximum value in the distribution (i.e., the upper edge of the highest bin).
3. The number of bins, NBins.

5.6  Approximation Distributions

5.6.1  The automatic approximation

5.6.2  Bin-based approximations

ApprPolygon(RV,NBins)

This is a continuous approximation that can be used only for a continuous basis distribution, RV. In brief, the PDF of the approximation distribution is a set of NBins straight lines, matched to the height of the basis distribution's PDF at the bin's edges (further detail is given below). For example, ApprPolygon(Convolution(Normal(0,1),Beta(2,2)),201) approximates the specified convolution with a set of 201 straight lines.

ApprFreqPolygon(RV,NBins)

This is a continuous approximation that can be used only for a continuous basis distribution, RV. In brief, the PDF of the approximation distribution is a set of NBins straight lines, matched to the height of the basis distribution's PDF at the bin's centers. For example, ApprFreqPolygon(Convolution(Normal(0,1),Beta(2,2)),201) approximates the specified convolution with a set of 201 straight lines.

This is my preferred type of approximation. It is generally quite accurate, and it is often much faster than the other approximations.

Details of construction.

Step 1:

The first line starts at X₁=minimum (of the basis distribution) with height PDF at that point and goes to X₂=minimum+W/2 with height PDF=Basis.PDF(X₂). The second line continues from the end of the first line to the point with X₃=minimum+1.5*W and height PDF=Basis.PDF(X₃). And so on, with the final line segment ending at the maximum of the basis distribution and PDF at the maximum.

Step 2:

The PDF just constructed is integrated, and the heights are scaled up or down appropriately so that the total area is 1.00.

ApprHistogram(RV,NBins)

This is a continuous approximation that can be used for either a discrete or a continuous basis distribution, RV. In brief, the PDF is of the approximation distribution is a set of NBins flat lines, as if the basis distribution were uniform within each bin (like in a histogram). For example, ApprHistogram(Convolution(Normal(0,1),Beta(2,2)),201) approximates the specified convolution with a set of 201 bins with equal probability within each bin.

This approximation is more general than ApprPolygon, because it can be used with discrete distributions, and it is less sensitive to abruptly-changing PDFs. But it is usually slower to construct initially, and it is often much slower to do any computations with. The PDF has discontinuities at the bin boundaries (unlike ApprPolygon), and these make numerical integrations converge more slowly.

Details of construction. The CDF of the basis distribution is computed at the top and bottom of each bin, and from these the bin probability is computed. Then, the height of the uniform approximation PDF within that bin is adjusted to give this bin probability.

ApprBinCen(RV,NBins)

This is a discrete approximation, and it can be used to approximate either a discrete or a continuous basis distribution, RV. In brief, the approximation assumes that all of the probability mass is concentrated in a single point at the center point of each bin; moreover, any value in the bin is treated as if it were that center point.

Details of construction. The CDF of the basis distribution is computed at the top and bottom of each bin, and from these the bin probability is computed. All of this probability mass is assigned to the value at the center of the bin. For purposes of PDF and CDF computations, all values within a bin are treated as equivalent to the center.

5.7  Distributions Arising in Connection with Signal Detection Theory

ZfromP(SampleSize,TrueP,Adjust)

This is the discrete distribution of Z, which is derived from the binomial distribution as follows:

1. For any sample from a Binomial(N,P), convert the number of successes k to the probability of success, p º k/N. If p=0, set p = Adjust/N; if p=1, set p=1-Adjust/N. "Adjust" is a parameter between 0 and 1, specified by the user, to indicate how the extreme data values should be treated.
2. Find Z such that p = Pr(z £ Z), where z is a random variable having the standard normal distribution.

APrime(NSignalTrials,PrHit,NNoiseTrials,PrFalseAlarm)

This is the distribution of the sample A¢ computed from an experiment with NSignalTrials signal trials each having the specified true probability of a hit, and NNoiseTrials noise trials each having the specified true probability of a false alarm. Specifically, A¢ is the distribution-free estimate of the area under the ROC curve computed using Equations 2 and 9 of Aaronson and Watts (1987).

APrimeSym(NTrials,PC)

This is a shortcut for the previous distribution that can be used when there are equal numbers of signal and noise trials and when the probability of a correct response (hit or correct rejection) is the same for both signal and noise trials.

YNdPrime(NSignalTrials,PrHit,NNoiseTrials,PrFalseAlarm,Adjust)

This is the distribution of the sample [^d]¢ computed from an experiment with NSignalTrials signal trials each having the specified true probability of a hit, NNoiseTrials noise trials each having the specified true probability of a false alarm, and using the Adjust factor (between 0 and 1) to correct cases with 0% or 100% hits or false alarms (e.g., replace 0 hits with Adjust hits, and replace NSignalTrials hits with [NSignalTrials - Adjust] hits). Programming note: If PDFs are requested, this distribution is implemented using the List (smaller samples) and AppApprCen (larger samples) random variables.

YNdPrimeSym(NTrials,TrueDP,Adjust)

This is the special case of YNdPrime in which NSignalTrials = NNoiseTrials and Pr(Hit) = 1 - Pr(FA). Note that the second parameter is the true d¢ rather than the hit probability.

6  Functions That CUPID Can Compute

CDF

"CDF(x)" requests the cumulative probability at x.

CV

"CV" requests the coefficient of variation, defined as the ratio of the standard deviation to the mean.

ConditionalRawMoment

"ConditionalRawMoment(Min,Max,Power)" requests the expected value of the Power'th raw moment of X conditionalized on X falling within the range of Min to Max.

GofFChiSq

This function almost computes the goodness-of-fit ChiSquare between the current distribution and a set of observed proportions. The syntax is GofFChiSq(B0,O1,B1,O2,B2,...,Ok,Bk) O1, O2, ... are the observed proportions in each bin (e.g., 0.1, 0.9). B0 is the lower boundary of the lowest bin, and B1, B2, ... are the upper boundaries of the various bins, and they must be in ascending order.

CUPID computes the predicted proportion in each bin, P_i, and returns

k å i=1  (Oi - Pi)2 Pi

The actual value of the chi-square statistic is the sample size times this value; since the sample size is not known to CUPID, you must do this computation yourself.

You may specify as many proportion/bin pairs as you like. Implicitly, there may be one extra bin above the last one you specify. The observed proportion in this bin is 1 minus the sum of the specified O_i values, and the predicted value in this bin is of course determined by the current distribution and its parameters.

The number of degrees of freedom associated with this chi-square is the number of specified bins (plus one, if there is any predicted or observed probability in the implicit extra bin) minus the number of free parameters adjusted.

Hazard

"Hazard(x)" requests the value of the hazard function at x: PDF(x) / (1 - CDF(x)).

IntegralXtoNxPDF

"IntegralXtoNxPDF(Min,Max,Power)" requests

ó õ Max Min  xPower f(x) dx

where f(x) is the PDF of x.

CDFIntegral

"CDFIntegral(Min,Max,Power)" requests

ó õ Max Min  xPower F(x) dx

where F(x) is the CDF of x.

InverseCDF

"InverseCDF(P)" requests X such that Pr(x £ X) = P.

Kurtosis

"Kurtosis" requests E[(x-m)⁴] / E[(x-m)²]².

LnLikelihood

"LnLikelihood(X₁,X₂,¼,X_N)" requests

ln é ë N Õ i=1  f(Xi) ù û

Mean

This requests the expected value of the random variable.

Median

x such that F(x)=0.5.

MGF

"MGF(Theta)" requests the value of the moment generating function at Theta (q), defined here as

ó õ Max Min  eqx f(x) dx

where f(x) is the PDF of x and Min and Max are the lower and upper limits of the distribution.

MMMSD

I don't know what to call this, but it is (Mean - Median) / SD. I like it as a measure of skewness. If anyone knows a real name for it, please let me know.

PctileSkew(P)

Here is another measure of skewness. It is

InverseCDF(P)+InverseCDF(1-P)-2×Median InverseCDF(P)-InverseCDF(1-P)

If anyone knows a real name for it, please let me know.

PDF

"PDF(x)" requests the value of f(x).

PDFBin

"PDFBin(x,Bin)" requests the value of F(x+Bin)-F(x). This is particularly useful with discrete distributions because f(x) is zero at lots of intermediate points.

Random

"Random" requests a random number from the distribution. "Random(N)" requests N random numbers from the distribution, and these are printed out as a list with one per line. N can be any integer from 1 to 32767.

RawMoment

"RawMoment(Power)" requests the Power'th raw moment of the distribution.

SIQR

This is the semi-interquartile range, also known as the "DL" in psychophysical work:

InverseCDF(.75)-InverseCDF(.25) 2

Skewness

"Skewness' requests E[(x-m)³] / E[(x-m)²]^3/2. This seems to be the most standard measure of skewness, although many other measures are used too.

SD

Standard deviation.

RawSkewness

"RawSkewness" requests E[(x-m)³]^1/3, which is another potentially useful measure of skewness.

Variance

E[X²]-(E[X])².

YNProbitLikelihood and mAFCProbitLikelihood

These two incredibly special-purpose functions compute the likelihood of a set of probit-type data under the current distribution (see section 11 for a brief description of probit analysis). The first is appropriate when the data come from a yes-no task, and the second is appropriate when the data come from an m-alternative forced-choice task. For example, YNProbitLnLikelihood(N,Cs,Ns,NGs) N is the number of stimulus constants used; the Cs are the constants themselves (there will be a list of these); the Ns are the number of tests at each constant (there will be a list of these); and the NGs are the number of positive test outcomes at each constant (there will be a list of these).

The command for the m-alternative forced-choice task is almost the same mAFCProbitLnLikelihood(m,N,Cs,Ns,NGs) except the first parameter is the number of alternative responses, m.

7  Functions That CUPID Can Plot

Inspect Graph Window, then type ENTER to continue:

8  Generating Columns of Function Values

9  Parameter Estimation

Estimate From Moments

This function tells CUPID to alter the parameter values of the current distribution so as to optimize the fit to a given set of moments. The syntax is: EstFromMoments(M1,M2,...[,ParmCodes]) M1 is the mean, and M2, M3, ... are the observed central moments for which you want to find the best-fitting parameters. You may specify as many moments as you like. Currently, the program minimizes the sum of squared errors between predicted and observed moments. ParmCodes is an optional parameter described below.

For example, if the current distribution is Lognormal(3,.4) and you want the parameters changed so that the mean and variance are 100 and 40000, type: EstFromMoments(100,40000) After a brief search, CUPID will change to a lognormal(3.8,1.269), which has the desired mean and variance.

Note that after the search CUPID reports a "simplex minimum". If this value is greater than zero, then CUPID could not find parameter values giving the exact moments desired. Maybe they don't exist, or maybe the search algorithm can't find them. You might be able to improve the fit by repeating the same command again from different starting parameter values.

Warning: This and the other estimation processes are in real danger of getting caught in local minima. Use with caution! Indeed, in some cases CUPID will bomb during the estimation process due to a numerical error. When this happens, try harder to find starting parameter values in the general neighborhood of the best parameter estimates.

Estimate From ChiSquare

This function tells CUPID to alter the parameter values of the current distribution so as to optimize the fit to a given set of bin proportions. The syntax is: EstFromChiSquare(B0,O1,B1,O2,B2,...[,ParmCodes]) O1, O2, ... are the observed proportions in each bin (e.g., 0.1). B0 is the lower boundary of the lowest bin, and B1, B2, ... are the upper boundaries of the various bins, and they must be in ascending order. ParmCodes is an optional parameter described below. See the function "GofFChiSq" for further details.

Estimate From Percentiles

This function tells CUPID to alter the parameter values of the current distribution so as to optimize the fit to a given set of observed percentile values. The syntax is: EstFromPercentiles(P1,O1,P2,O2,...[,ParmCodes]) P1, P2, ... are the percentile values (e.g., 0.1, 0.9) for which you want to find the best-fitting parameters, and O1, O2, ... are the observed values of those percentiles. You may specify as many pairs as you like. Currently, the program minimizes the sum of squared errors between predicted and observed percentile values. ParmCodes is an optional parameter described below.

Estimate From Observations

This function tells CUPID to alter the parameter values of the current distribution so as to give the maximum likelihood fit to a given set of data points (observations). The syntax is either: EstFromObs(Obs1,Obs2,...[,ParmCodes]) or EstFromObs(FileName[,ParmCodes]) In the first version, the data values used for the likelihood calculation are given as Obs1, Obs2, Obs3, and so on. In the second version, FileName specifies the name of an ASCII file with data points listed one per line (maximum of 1000 data points). ParmCodes is an optional parameter described below.

Estimate From Probit Data

Eight functions tell CUPID to alter the parameter values of the current distribution so as to give the optimal fit-in either a maximum likelihood sense or a minimum chi-square sense, as described further below-to a given set of Probit-type data (see section 11 for a brief description of probit analysis). The syntax is one of the following four options: EstFromYNProbitLnLike(C1,N1,G1,C2,N2,G2,C3,N3,G3,...[,ParmCodes]) EstFromYNProbitChiSq(C1,N1,G1,C2,N2,G2,C3,N3,G3,...[,ParmCodes]) EstFromYNProbitLnLike(FileName[,ParmCodes]) EstFromYNProbitChiSq(FileName[,ParmCodes]) In the first two versions, the data values used for the calculations are given as C1,N1,G1, and so on. In the third and fourth versions, FileName specifies the name of an ASCII file with each line containing one Ci,Ni,Gi triple separated by spaces (e.g., "1.25 40 25"), with a maximum of 1000 triples. ParmCodes is an optional described below.

There are also four options parallel to the above for estimation from data obtained in an m-alternative forced-choice task. Here is one example: EstFrommAFCProbitLnLike(m,C1,N1,G1,C2,N2,G2,C3,N3,G3,...[,ParmCodes]) The first parameter, m, indicates the number of options in the forced-choice task. The other three options are exactly parallel to the options for the yes-no task, with the additional m parameter first.

ParmCodes

This is an optional parameter that can be specified for any of the estimation procedures just described, and you use it to tell CUPID how you want it to handle each of the parameters. Specifically, ParmCodes is a sequence of letters-one letter for each free numerical parameter in the current distribution. Each letter should be F, I, or R, with these meanings:

F

The parameter is fixed at the value used in the current distribution, so the parameter search routine is not allowed to adjust it.

I

The parameter can be adjusted, but it can only be set to integer values.

R

The parameter can be adjusted to any real value.

Here are some examples:

• Suppose you want to work with a beta distribution with mean 0.3 and variance 0.1. Not knowing immediately what parameters to use, you guess to enter a Beta(0.5,0.5). Then, type EstFromMoments(0.3,0.1,RR) to tell CUPID to adjust the two parameters to produce the desired mean and variance. CUPID will search briefly and then change the current distribution to Beta(0.33,0.77), which has the desired moments.
• Suppose the current distribution is a standard normal distribution truncated at -1 and 1: Truncated(Normal(0,1),-1,1) The command EstFromMoments(0.3,0.8,RRFF) tells CUPID to hold the truncation limits fixed at -1 and 1, but adjust the parameters of the normal to yield a truncated distribution with a mean of 0.3 and a variance of 0.22. It takes about 10 seconds on my 486, but CUPID eventually finds that setting the normal mean and standard deviation to 1.039 and 0.9717, respectively, yields a truncated distribution with the desired mean and variance. Note that the truncated distribution has four parameters, corresponding (from left to right) to the normal mean and sd and the lower and upper truncation cutoffs. The ParmCodes of RRFF says, "vary the first two but leave the last two alone." If you used IIFF instead, CUPID would try to find the best solution in which the normal mean and sd were integers.
• Suppose the current distribution is set to: Mixture(0.6,Normal(0,1),0.4,Exponential(1)) In this case, the command EstFromObs(4,1.1,3.2,-0.3,0.8,RFFF) tells CUPID to look for different mixture probabilities maximizing the likelihood of these four observations.

• The bounds of truncated and bounded distributions.
• The multiplier and addend of linear transformation distributions.
• The mixture probabilities of mixture distributions.
• The order and number for Order, OrderIID, OrdBin, and OrdExp distributions.
• The power for Power transformation distributions.

10  Program Control

10.1  Getting Help

10.2  Saving Output to a File

10.3  Exiting the Program

10.4  Continuation Lines

10.5  Color and Mono

10.6  Notes & Comments

10.7  Aliases: User-defined Macros & Abbreviations

• Save an alias file CUPID.ALI in the current directory if you want that alias file loaded automatically any time you start CUPID in that directory.
• Save an alias file CUPID.ALI in the CUPID path (see section 10.19) if you want that alias file loaded automatically any time you start CUPID from any directory.

• The Alias command cannot be aliased.
• The maximum number of aliases is 100.
• An alias file is a regular ASCII file-one line per alias-with a very obvious format. You can edit it with an ASCII editor to make changes.
• If you ever want to clear all the currently defined aliases, just type AliasFromFile without specifying a file name.

10.8  @Self

10.9  Getting Debugging Information

10.10  Iteration - the "Repeat" Command

• Processing of variable names is case sensitive, so Repeat(X,0.1,0.1,0.9)InverseCDF(x) will not work.
• The variable name must be a unique string occurring no where else in the command, or else CUPID will get confused. For example, Repeat(C,0.1,0.1,0.9)InverseCDF(C) will not work because the variable name "C" occurs within the string "InverseCDF". Just use CC instead.

10.11  SkipTo

10.12  IntegralPrecision

10.13  InversePrecisionP and InversePrecisionX

10.14  ZExtreme

10.15  StepSize and MinStepSize

10.16  A Shortcut for Changing Parameter Values

10.17  UseExpRate / UseExpMean

10.18  Seeding the Random Number Generator

• When it starts up, CUPID checks the current directory for a file called CUPID.MT. If this file is found, the initial state of the random number generator (i.e., at the start of CUPID) is retrieved from this file.
• If no file CUPID.MT is found in the current directory, then CUPID checks for the file CUPID.MT in the CUPID path (see section 10.19), and reads the initial state of the random number generator from this file if it is found.
• If no CUPID.MT file is found, then CUPID always starts the random number generator by seeding it randomly when the program begins. Obviously, this means that the random number generator starts in a different state each time the program starts up.
• If a CUPID.MT file was found at startup, then by default CUPID will save the new state of the random number generator in this same file (same directory) when it quits. The net effect is that CUPID's random number generator starts up from where it left off each time, insuring a fresh sequence of random numbers for each run (by default).

Randomize

This command generates a new random state for the random number generator. This command takes no parameters.

RandRdState

The command RandRdState(Filename) reads the state of the random number generator from the file "filename", which should previously have been written using the command described next. If the file name is omitted, CUPID rereads the same CUPID.MT file that it read when it started.

RandWrState

The command RandWrState(Filename) writes the current state of the random number generator to the file "Filename", so that this same state can later be reinstated using the RandRdState command. If the file name is omitted, CUPID writes over the same CUPID.MT file that it read when it started.

RandSave

This command is a toggle that controls whether CUPID  does or does not save the final state of its random number generator. It is here in case you want to turn off CUPID's default procedure of updating its random number generator seed at the end of each run. In other words, if you want to start CUPID with the random number generator in the same state on two successive runs, issue the command RandSave during the first run so that CUPID will not update its seed file.

10.19  The CUPID Path

10.20  Control of CUPID from a command file

10.21  Control of CUPID from the Command Line

10.22  Table of Known Functions

11  Probit Analysis

1. A researcher selects an ordered set of k constants C₁, C₂, C₃,¼,C_k (e.g., 5, 10, 15, 20,¼,50) roughly spanning a probability distribution.
2. For each constant C_i, the researcher takes N_i independent random samples X_ij from the distribution, j=1,¼N_i. The value of X_ij cannot be observed directly, however. Instead, the researcher observes only Y_ij, where
   

   Yij = ì ï í ï î 0 if Xij £ Ci 1 if Xij > Ci
3. The data are summarized by counting the number of observations greater than each C_i:
   

   Gi = Ni å j=1  Yij
4. The problem is to estimate the probability distribution of the X_ij values from the k observed proportions, G_i/N_i, i=1... k.

12  Disclaimer and Warnings

13  Algorithms

13.1  Numerical Integration

13.2  Simplex

13.3  Random Number Generation

14  CupiTest

15  Contributing New Routines to CUPID

16  Release History

• The "columns" command was added, eliminating the need for CupCols.
• The automatic approximation feature was added (i.e., via the asterisk).
• The Rosin-Rammler distribution was added.

• Plots shown in graphics windows.
• Basic facilities for entering and editing the input lines typed by the user, including a history buffer.
• CupiTest was added.
• Some further new distributions.

• These new distributions were added: Bounded, Cosine, ExGaussRat, ExpSum, ExpSumT, InfMix, MinBound, OrdExp, and UniformInt, Wald.
• All of the approximation distributions were added.
• The programs CupCols and CupCols2 were added.
• Command-completion was added, so that it was only necessary to type enough of a command or distribution name to identify it uniquely.
• The automatic @Self abbreviation was added.
• A bug in parsing large integer values was fixed.
• PDFBin was added.
• EstFromChiSquare was added.
• Fixed famous Pascal "CRT bug" that causes run-time errors on fast machines.

17  Related Programs

Cupid

Interactive computations (pdf, cdf, moments, etc) with probability distributions. Can be used (for example) as an on-line table for distributions.

RandGen

Generates random values from a specified probability distribution.

FitDist

Estimates best-fitting parameters of a given probability distribution for a given data set.

MixTest

Computes a likelihood ratio test to see whether the difference between two conditions (say "experimental" versus "control") is a "uniform effect" or a "mixture effect". With a uniform effect, all of the scores in the experimental condition are increased relative to what they would have been in the control condition. With a mixture effect, however, only some of the scores in the experimental condition are affected; the rest of the scores in this condition are the same as they would have been without the manipulation (i.e., the same as they would have been in the control condition).

Pmetric

Estimates the parameters of a probability distribution from a data set relating the proportion of a certain (binary) response to a physical quantity. This type of analysis is often called "probit" analysis, and it is used (for example) in bioassay (analysis of dose/response curves) and psychophysics (analysis of psychometric functions).

18  Author Contact Address

19  Acknowledgements

20  References

Aaronson, D., & Watts, B. (1987). Extensions of Grier's computational formulas for A' and B" to below-chance performance. Psychological Bulletin, 102, 439-442.

D'Agostino, R. B. (1970). Simple compact portable test of normality: Geary's test revisited. Psychological Bulletin, 74, 138-140.

Dallal, G. E., & Wilkinson, L. (1986). An analytic approximation to the distribution of Lilliefors's test statistic for normality. The American Statistician, 40, 294-296.

Devroye, L. (1986). Non-uniform random variate generation. Berlin: Springer-Verlag.

Evans, M., Hastings, N., & Peacock, B. (1993). Statistical distributions. (2nd ed.) New York: Wiley.

Finney, D. J. (1971). Probit analysis: A statistical treatment of the sigmoid response curve. [3rd ed.]. Cambridge: Cambridge University Press.

Finney, D. J. (1978). Statistical method in biological assay. London: Griffin.

Gescheider, G. A. (1997). Psychophysics: The fundamentals. [3rd ed.]. Hillsdale, NJ: Erlbaum.

Guilford, J. P. (1936). Psychometric methods. New York: McGraw-Hill.

Johnson, N. L., & Kotz, S. (1970). Continuous univariate distributions. New York: Houghton Mifflin.

Matsumoto, M., & Nishimura, T. (1998). Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Transactions on Modeling and Computer Simulation, 8, 3-30.

Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. (1986). Numerical recipes: The art of scientific computing. Cambridge: Cambridge University Press.

Quick, R. F. (1974). A vector magnitude model of contrast detection. Kybernetik, 16, 65-67.

Rosenbrock, H. H. (1960). An automatic method for finding the greatest or least value of a function. Computer Journal, 3, 175-184.

Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33, 457-469.

Schwarz, W. (2002). On the convolution of inverse Gaussian and exponential random variables. Communications in Statistics: Theory and Methods, 31, 2113-2121.

Sternberg, S., & Knoll, R. L. (1973). The perception of temporal order: Fundamental issues and a general model. In S. Kornblum (Ed.), Attention and performance IV (pp. 629-685). New York: Academic Press.

Strasburger, H. (2001). Converting between measures of slope of the psychometric function. Perception & Psychophysics, 63, 1348-1355.

21  Software License

21.1  Preamble

21.2  Terms of License

a) You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change.

b) You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.

c) If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.)

a) Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

b) Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or,

c) Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.)

Footnotes: