StatLib---Applied Statistics algorithms

StatLib---Applied Statistics
algorithms
The Royal Statistical Society has been publishing algorithms in its journal Applied Statistics
since 1968. As of 1989, there are over 250 of them. Most are in Fortran, though a few were in
Algol, and some recent ones have been in Pascal. The book - Applied Statistics Algorithms by
Griffiths, P. and Hill, I.D., Ellis Horwood: Chichester (1985) contains translations of several
algorithms from Algol to Fortran. A few of the other algorithms have been supplied in Fortran
translations, though a few are in Algol or Pascal. The index which follows indicates which
algorithms are not in Fortran.
The full source code is published in the journal. Those available here have been transcribed
manually, mainly within CSIRO Division of Mathematics & Statistics, or have been supplied
directly by the authors or by the RSS Algorithms Editor. In some cases, later corrections or
improvements published in Applied Statistics, have been incorporated.
It is the policy of the editors of the algorithms section of Applied Statistics that algorithms do
not use double precision. Many of the algorithms here have been converted to double
precision, though users should be careful to check what precision is used. Many of the
algorithms require the use of other algorithms, particularly functions for the gamma function
and the normal distribution function. Where such functions or subroutines are required, an
appropriate comment has been added to the algorithm.
In a few cases, alternative algorithms from other sources have been added. For instance, three
algorithms are included in the file for as66 for calculating the area under the normal curve, and
an alternative random number generator is provided in the file for as183. The Applied
Statistics algorithm for Nelder-Mead simplex minimization (AS 47) does not include the
fitting of a quadratic surface; the CSIRO/ Rothamsted implementation which does this is
included with AS 47.
Users must consult the original journal articles for details of the calling arguments; they are not
included with these algorithms.
*** Warning. In some cases, there are different arguments (usually more) in the Griffiths &
Hill versions of these algorithms. Users should check this.

It has been assumed that the user will be using a Fortran-77 compiler, so functions such as alog
and amin1 have been converted to their generic forms (log and min). This simplifies
conversion of code between single and double precision. Also all constants in the code, such as
1.0, 0.d0, etc., have been replaced with one, zero, etc., and these are defined in either data or
parameter statements. Many of the algorithms have been entered in lower case, which is not
acceptable in Fortran, though most compilers accept it.
Some of the algorithms need machine-dependant constants. The user should check this. In
most cases, these have been set for compilers which allow a range of floating-point numbers
from about 10**(-37) to 10**(+37), though most modern compilers allow a much wider range
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StatLib---Applied Statistics algorithms

in double precision.
No guarantee is given that the algorithms have been entered correctly, or that they perform as
claimed in the journal.
To obtain an algorithm, send an E-mail request of the form: send index from apstat send 207
from apstat to
The Royal Statistical Society holds the copyright to these routines, but has given its permission
for their distribution provided that no fee is charged.
The full collection of Applied Statistics algorithms is very large. Please only request those
algorithms which you need. Requesting large numbers of algorithms places a great strain on
the StatLib system and the underlying mail networks.
As of the end of 1997 the Applied Statistics journal does NOT accept algorithms

Listing of available Applied Statistics
algorithms
No.
Brief description (volume number/year in brackets)

3
Student's t-distribution. (17/1968) See also AS 27.
5
The non-central t-distribution. (17/1968) See also AS 243.
6
Cholesky decomposition of a symmetric +ve definite matrix. (17/1968)
7
Inversion of a symmetric matrix stored in triangular form. (17/1968)
13
Minimum spanning tree. (18/1969) See also AS 40.
14
Printing the minimum spanning tree. (18/1969)
15
Single-linkage cluster analysis. (18/1969)
22
Calculate treatment effects in a complete factorial experiment for any numbers of levels
of factors using the extended Yate's method. (19/1970)

(2 of 18) [3/3/2002 9:48:19 PM]


StatLib---Applied Statistics algorithms

27
Upper tail area under Student's t-distribution. (19/1970) See also AS 3.
30
Half-normal plotting. (19/1970)
32
The incomplete gamma integral. (19/1970) See also AS 239.
34

Update inverse of symmetric banded matrix. (19/1970)
38
Calculate R-squared for all possible regression subsets using the Gauss-Jordan method.
(20/1971) See also AS 268.
40
Update a minimum spanning tree. (20/1971)
41
Updates corrected sums of squares and products matrices. (20/1971) See also AS 240.
45
Histogram plotting. (20/1971)
46
Gram-Schmidt orthogonalization. (20/1971)
47
Nelder & Mead simplex method of unconstrained minimization without requiring
derivatives. Does not include the quadratic surface fitting part of the algorithm. A
CSIRO/Rothamsted version of the algorithm, which does include fitting a quadratic
surface, is also included. (20/1971) (Updated, 20/Dec/93)
51
Log-linear fit for contingency tables. (21/1972) See also AS 160.
52
Calculation of sums of powers (up to 4) of deviations from the mean. (21/1972)
53
Wishart variate generator. (21/1972)
57
Printing multi-dimensional tables. (22/1973)
58
Allocates observations to clusters to minimize within-cluster sum of squares. (22/1973)
See also AS 136.
60


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StatLib---Applied Statistics algorithms

Eigenvalues/vectors of a symmetric matrix. (22/1973)
62
Distribution of the Mann-Whitney U statistic. (22/1973)
63
Incomplete beta function. (22/1973) See also TOMS algorithm 708. TOMS algorithms
are available from netlib.
64
Inverse of the incomplete beta function ratio. (22/1973) The file here is actually the
Griffiths & Hill version of AS 109.
65
Expands structure formula to a list of binary integers. This is actually remark R82 which
replaces the original AS65. (39/1990)
66
The normal distribution function. Two other algorithms (not from Applied Statistics)
have also been included. (22/1973)
75
Algorithms for least-squares calculation using square-root free planar rotations (Morven
Gentleman's package). (23/1974)
76
An integral useful in calculating noncentral t and bivariate normal probabilities.
(23/1974)
77
Calculate exact null distribution of the largest root of a beta matrix. (23/1974)
78
The mediancentre (i.e. the median in a multi-dimensional space). (23/1974) See also AS

143.
83
Complex discrete fast Fourier transform. (24/1975)
84
Measures of multivariate skewness and kurtosis. (24/1975)
88
Generate all nCr combinations by simulating nested Fortran DO-loops. (Jane
Gentleman's routines). (24/1975) See also AS 172.
89
Tail probabilities for Spearman's rho. (24/1975)
91
Percentage points of the chi-squared distribution. (24/1975)
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StatLib---Applied Statistics algorithms

93
Calculates frequency distribution for the Ansari-Bradley test statistic. (25/1976) A
routine has been added to return the distribution function.
95
Maximum likelihood estimation of scale and location parameters from grouped data.
User's distribution function. (25/1976)
96
Finding `nice' scales for graphs. (25/1976)
97
Real discrete fast Fourier transform. Series length must be a power of 2. (25/1976) See
also AS 117 and AS 176.
99
Fitting Johnson curves by moments. (25/1976)

100
Normal-Johnson and Johnson-Normal transformations. (25/1976)
103
Psi or digamma function. (25/1976)
107
Calculate operating characteristics and average sampling number for a general class of
sequential sampling plans. (26/1977)
108
Multiple linear regression minimizing the sum of absolute errors. (26/1977) See also AS
238 (in Pascal).
109
Inverse of the incomplete beta function. (26/1977)
110
LP-Norm fit of straight line by extension of Schlossmacher. (26/1977)
111
Percentage points of the normal distribution. (26/1977) See also AS 241.
114
Compute the numerator of certain ordinal measures of association (Kendall's tau,
Somer's d, Goodman and Kruskal's gamma) when the data are ordered categories.
(26/1977)
116
Calculate the tetrachoric correlation and its standard errors. (26/1977)
117

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StatLib---Applied Statistics algorithms

Fast Fourier transform for series of any length using the CHIRP algorithm. (26/1977)

121
Trigamma function. (27/1978)
123
Distribution function of mixtures of beta distributions. (27/1978)
125
Maximum likelihood estimation for censored exponential survival data with covariates.
(27/1978)
126
Distribution function of the range for the normal distribution. (27/1978)
127
Generation of random orthogonal matrices. (27/1978)
128
Computes approximate covariance matrix for normal order statistics. (27/1978)
132
Simple regression minimizing the sum of absolute deviations. (27/1978)
133
Finding the global maximum or minimum of a function of 1 variable. (27/1978)
134
Generate random beta variates for alpha < 1 and beta > 1. (28/1979)
135
Min-Max (L-infinity) estimates for linear multiple regression. (28/1979)
136
A K-means clustering algorithm. (28/1979)
138
Maximum likelihood estimates of the mean and standard deviation of the normal
distribution with censored or confined observations. (28/1979)
139
Maximum likelihood estimation in a linear model from confined and censored normal
data. (28/1979)
140

Clustering the nodes of a directed graph. (28/1979)
141
Inversion of a symmetric matrix ignoring a specified row/column. (28/1979)
142
Exact tests of significance in binary regression. (28/1979)
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StatLib---Applied Statistics algorithms

143
Calculates the median centre. (28/1979)
145
Exact distribution of the largest multinomial frequency. (28/1979)
147
Incomplete gamma function. (29/1980) See also AS 239.
148
Removal of bias in the jackknife procedure. (This is actually ASR 62) (29/1980)
149
Amalgamation of means in the case of simple ordering ('Up-and-Down Blocks'
algorithm for isotonic regression). (29/1980)
150
Computes estimate of spectrum of a point process using a centered moving average of
the periodogram of the counting process. (29/1980)
151
Smoothed spectral estimate for bivariate point processes. (29/1980)
152
Cumulative hypergeometric probabilities. (This is actually AS R77) (29/1980, revised in
38/1989)
153

Distribution of weighted sum of squares of normal variables. (29/1980) Pan's procedure
for the tail probabilities of the Durbin-Watson statistic.
154
Exact maximum likelihood estimation of autoregressive-moving average models by
Kalman filtering. (29/1980) See also AS 182.
155
Distribution function of a linear combination of non-central chi- squared random
variables. (29/1980) See also AS 204. This is a Fortran translation supplied by the
author.
157
The runs-up and runs-down tests. (30/1981)
158
Calculation of probabilities for inferences under simple order restrictions. (30/1981) See
also AS 198.
159
Generate random 2-way table with given marginal totals. (30/1981)
160
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StatLib---Applied Statistics algorithms

Partial and marginal association in multi-dimensional contingency tables. (30/1981)
161
Critical regions of an unconditional non-randomized test of homogeneity in 2 x 2
contingency tables. (30/1981)
162
Multivariate Conditional Logistic Analysis of Stratum-matched Case-control Studies.
(30/1981) Includes a Fortran version of CACM algorithm 382 for generating all
combinations of M out of N items.

163
A Givens Algorithm for Moving from one Linear Model to another without Going back
to the Data. (30/1981)
164
Least squares subject to linear constraints. (30/1981)
165
Discriminant analysis of categorical data. (30/1981)
166
Calculates the entanglement matrix for a specified design. (30/1981)
167
Calculates efficiencies of estimation and their generalized inverse. (30/1981)
168
Calculates 'neat' values for plotting scales. (30/1981)
169
Produces scatter plots. (30/1981)
170
Computation of probability and non-centrality parameter of a non- central chi-square
distribution. (30/1981)
171
Fisher's exact variance test for the Poisson distribution. (31/1982)
172
Generates indices for simulated nested DO-loops. Actually converts (either way)
between a single index and a vector of subscripts. (31/1982)
173
Generates design matrix for balanced factorial experiments. (31/1982)
174
Multivariate rank sum test and median test. (31/1982)
175

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StatLib---Applied Statistics algorithms

Cramer-Wold factorization of self-reciprocal polynomials as the product of two
polynomials. (31/1982)
176
Kernel density estimation using the fast Fourier transform. (31/1982) Also contains an
alternative set of routines for density estimation.
177
Expected values of normal order statistics. (31/1982)
178
Gauss-Jordan sweep operator with multi-collinearity detection. (31/1982)
179
Enumeration of all permutations of multi-sets with fixed repetition numbers. (31/1982)
180
Linear rank estimate of the standard deviation after symmetric trimming. (31/1982)
181
Withdrawn. See R94 below
182
Finite-sample prediction from ARIMA processes. (31/1982) Uses AS 154.
183
The Wichmann & Hill random number generator. An alternative is also provided.
(31/1982)
184
Non-central studentized maximum and related multiple-t probabilities. (31/1982)
185
Backward elimination procedure to find best-fitting log-linear models for contingency
tables. (31/1982) Uses AS 51.
186

Discrete Fast Fourier Transform with data permutation. Series length must be a power of
2. (31/1982)
187
Derivatives of the incomplete gamma integral. (31/1982)
188
Estimation of the order of dependence in sequences. (32/1983)
189
Maximum likelihood estimation for the beta binomial distribution. (32/1983)
190
Distribution function & its inverse, for the studentized range. (32/1983)

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StatLib---Applied Statistics algorithms

191
Approximate likelihood calculation for ARMA and seasonal ARMA models. Includes a
routine for the Banachiewicz (or modified Cholesky) factorization A = LDL'. (32/1983)
192
Calculate approximate percentage points using Pearson curves and the first three or four
moments. (32/1983)
193
The Knuth spectral test for congruential random number generators. (32/1983)
194
Test of goodness of fit of ARMA models. (32/1983)
195
Multivariate normal probabilities for a rectangular region in N- dimensional space. A
version which calls IMSL routines is also included. (33/1984) See AS 251 for a special
case.

196
Conditional multivariate logistic analysis of stratified case-control studies. (33/1984)
197
Likelihood function for an ARMA process. (33/1984)
198
Calculation of level probabilities for order-restricted inference. (33/1984)
199
Branch and bound algorithm to find the subset which maximizes a quadratic form.
(33/1984)
200
Approximate the sum of squares of normal score. (33/1984)
201
Combine predictions about a statistic based on the orderings of a set of means with an
F-test of differences between the means. (33/1984)
202
Data-based nonparametric hazard estimation. (33/1984)
203
Maximum likelihood estimation of mixtures of distributions (normal, exponential,
Poisson and binomial). (33/1984) See also AS221. This is a translation from Algol into
Fortran.
204
Distribution of a sum of non-central chi-squared variables. (33/1984) Translation from
Algol into Pascal.
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StatLib---Applied Statistics algorithms

205
Enumeration of all R x C contingency tables with given row and column totals, and

calculation of hypergeometric probability for each table. (33/1984)
206
Isotonic regression in two independent variables. (33/1984)
207
Fit a generalized log-linear model. (33/1984)
208
Fit multivariate logistic model by the method of moments. (34/1985)
209
The distribution function of skewness and kurtosis. (34/1985)
210
Fit 5-parameter Johnson SB curves by moments. (34/1985)
211
The F-G diagonalization algorithm. (34/1985) Modifications in ASR 71 & ASR 74 have
been added to the file.
212
Fitting the exponential curve by least squares. (34/1985)
213
Generation of correlation matrices with specified eigenvalues. (34/1985)
214
Calculation of Monte Carlo confidence intervals. (34/1985)
215
Maximum likelihood estimation of the parameters of the generalized extreme-value
distribution. (34/1985). Updated on [15/Nov/99]
216
Maximum likelihood fitting when model contains a linear part and auxiliary parameters.
(34/1985)
217
Computation of the Dip statistic to test for unimodality. (34/1985)
218
Elements of the Fisher information matrix for the smallest extreme- value distribution,

also allows for censored data. (35/1986)
219
Height balanced trees. (35/1986)
220

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StatLib---Applied Statistics algorithms

Operating characteristics of James-Stein and Efron-Morris estimation. (35/1986)
221
Maximum likelihood estimation of a mixing distribution. (35/1986)
222
Resistant smoothing using the fast Fourier transform. (36/1987)
223
Optimum ridge parameter selection. (36/1987)
224
Combining component designs to form a design with several orthogonal blocking
factors. (36/1987)
225
Minimizing linear inequality-constrained Mahalanobis distances. (36/1987)
226
Cumulative probabilities for the non-central beta distribution. (36/1987)
227
Generate all possible N-bit binary codes. (36/1987)
228
Finding I-projections (maximizing cross-entropy) subject to a finite set of linear
inequality constraints. (36/1987)
229

Quantile regression. (36/1987)
230
Distribution of customers in M/G/m queues using an approximation due to Hokstad.
(36/1987)
231
Distribution of a noncentral chi-squared variable with non-negative degrees of freedom.
(36/1987) In Pascal.
232
Computation of population and sample correlation and partial correlation matrices in
MARMA(p,q) time series. (37/1988)
233
Branch and bound algorithm for feature subset selection. (37/1988)
234
Approximating the percentage points of simple linear rank statistics with Cornish-Fisher
expansions. (37/1988)
235

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StatLib---Applied Statistics algorithms

Tally frequencies of distinct real values. (37/1988)
236
Recursive enumeration of RxC tables for exact likelihood evaluation. (37/1988) In
Pascal.
237
The corner method for identifying autoregressive moving average models. (37/1988)
238
Recursive procedure for the L1-norm fitting of a straight line. (37/1988) In Pascal.

239
Incomplete gamma function. (37/1988)
240
Updating the inverse of corrected sums of squares and products. (37/1988)
241
Inverse normal - more accurate than AS 111. (37/1988)
242
The exact likelihood of a vector autoregressive moving average model. (38/1989)
243
Cumulative distribution function of the non-central t-distribution. (38/1989)
244
Decomposability and collapsibility for log-linear models. (38/1989) In Pascal.
245
Log of the gamma function. (38/1989) The file includes a slower but more accurate
algorithm which is not an AS algorithm.
246
Repeated measurements analysis of variance with unknown autoregressive parameter.
(38/1989)
247
Updating the sufficient configurations for fitting ZPA (zero partial association) models
to multi-dimensional contingency tables. (38/1989) In Pascal.
248
Empirical distribution function goodness-of-fit tests for the uniform, normal and
exponential distributions. (38/1989)
249
Mean and covariance estimation for the truncated multivariate normal distribution.
(38/1989)
250

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StatLib---Applied Statistics algorithms

Test the equality of dispersion matrices using methods due to Puri & Sen
(distribution-free) and Anderson (normal distribution). (38/1989)
251
Multivariate normal probability integrals with product correlation structure. (38/1989)
252
Generating classes for log-linear models. (39/1990)
253
Maximum likelihood estimation of the RC(M) association model. (39/1990)
254
Fit mixture of normal distributions to grouped & truncated data. (39/1990)
255
Fitting two-way tables by means for rows, columns and cross-term. (39/1990) In Algol.
256
Distribution of a quadratic form in normal variables. (39/1990) In Pascal.
257
Isotonic regression for umbrella orderings. (39/1990)
258
Average run lengths for cumulative sum schemes. (39/1990)
259
Extending confidence intervals by the Robbins-Monro search process. (39/1990)
260
Distribution function of the square (R^2) of the sample multiple- correlation coefficient.
(40/1991)
261
Quantiles of the distribution of square (R^2) of the sample multiple- correlation
coefficient. (40/1991)

262
The Wei-Lachin two-sample test, the generalized Wilcoxon test and the log-rank test for
incomplete multivariate observations with censoring. (40/1991)
263
Construction of irredundant test sets. (40/1991)
264
Printing of bit patterns. (40/1991)
265
Waiting time distribution for the G/G/1 queue using the Fast Fourier transform.
(40/1991)

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StatLib---Applied Statistics algorithms

266
Maximum likelihood estimation of the parameters of the Dirichlet distribution.
(40/1991)
267
Calculate lower bound for the probability of correct selection of a subset containing the
best populations from a set of populations. (40/1991)
268
Calculates statistics for all possible subset regressions using an orthogonal
decomposition. (40/1991)
269
Calculates the Cornish-Fisher adjustment to distribution functions (from the normal
distribution function) using higher cumulants. (41/1992)
270
Maximum likelihood fitting of a 'key' density function, e.g. normal distribution,

multiplied by a series in Hermite polynomials. (41/1992)
271
Optimal (smallest mis-classification probability) joint classification procedure.
(41/1992) See also AS 276.
272
Produce character Box-plots using supplied quartiles. (41/1992)
273
Compare the least-squares fit of two subsets of regression variables using Spj0tvoll's
method. (41/1992)
274
Least squares algorithms to supplement those of Gentleman in AS 75. (41/1992)
274-90
Algorithm 274, translated into Fortran 90.
275
Non-central chi-squared distribution function, number of degrees of freedom must be
positive but not necessarily integer. (41/1992)
276
Optimal combinatoric classification for two normal classes. (41/1992) See also AS 271.
277
The Oja bivariate median. (41/1992)
278
The distribution of quadratic forms of multivariate generalized Student variables.
(41/1992)
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StatLib---Applied Statistics algorithms

279
P-values for the generalized/alternative Durbin-Watson statistic with arbitrary lag using

a Cholesky transformation method. (42/1993)
280
The power function for Fisher's exact test. (42/1993)
281
Scaling and rounding regression coefficients to force them to be integers. (42/1993)
282
High breakdown regression and multivariate estimation. (42/1993)
283
Rapid computation of the permutation paired and grouped t-tests. (42/1993) In ANSI
Standard C.
284
Null distribution of a statistic for testing sphericity and additivity: a Jacobi polynomial
expansion. (42/1993)
285
Calculate multivariate normal probabilities using Monte Carlo methods for a general
region defined by a user-supplied function. (42/1993)
286
Estimation of parameters when there are errors in both the predictor variables and the
dependant variable using least squares with a general model (includes implicit
non-linear regression). (42/1993)
287
Adaptive rejection sampling (to generate random variables) from log-concave density
functions. (42/1993)
288
P-value calculation for the generalized two-sample Smirnov tests. The tests are
conditional on ties in the pooled sample. (43/1994)
289
Convolves hypergeometric distributions generated by several 2x2 tables (43/1994)
290
Generate a grid of variance ratios for contour plotting of confidence regions for a pair of

parameters in non-linear regression. (43/1994)
291
Calculates overlap capability of subsequence SSEQ of length L. (43/1994)
292
Computes Fisher information matrix elements for time (type I) or failure (type II)
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StatLib---Applied Statistics algorithms

censored units from the smallest extreme value (sev), largest extreme value (lev),
normal or logistic distribution. (43/1994)
293
Convolves conditional distributions generated by several 2xK tables (43/1994)
294
Pascal program for ....?
295
Heuristic algorithm to pick N rows of X out of NCAND to maximize the determinant of
X'X, using the Fedorov exchange algorithm. (43/1994)
R94
calculates Shapiro-Wilk normality test and P-value for sample sizes 3 <= n <= 5000 .
Handles censored or uncensored data. Corrects AS 181, which was found to be
inaccurate for n > 50. (Uses function POLY in AS 181) by Patrick Royston (44/1995)
R96
Interval for trapezoidal rule integration to limit error using moment generating function
bound for weighted sum of chi-squared(1) random variables.
297
Nonparametric regression using ultraspherical polynomials.
298
Hybrid minimization routine using simulated annealing and any local minimizer

supplied by the user.
300
Efficient algorithm for determining a simulated percentile point with a fixed degree of
accuracy.
301
Computes the logarithms of F1, P(H) and P(H').
302
Subroutine to compute a multiple isotonic regression for umbrella ordering with known
peak.
303
Generation of ordered multinomial frequencies. Given integers N (N >= 0) and K (K >=
1), generates all possible integer arrays NI of size K, whose elements are non-negative,
non-decreasing and sum to N.
304
Fisher's non-parametric randomization test for two small independent random samples.
305
Compute the ARL of a CUSUM mean chart with linear drift.

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StatLib---Applied Statistics algorithms

306
Calculation of a BIB product operation on any two matrices.
307
Calculation of the simplicial depth and the halfspace depth
310
Computes the cumulative distribution function of a non-central beta random variable.
314

Inverts matrix with contents subject to modulo arithmetic.
319
A program to implement a quasi-newton method. Using new algorithm Varmet July
1994
N.B. This library was maintained by Alan Miller, CSIRO Division of Mathematics and
Statistics, Clayton, Victoria 3169. E-mail: Phone: (03)542-2266 or
(03)542-2253 (Secretary) FAX: (03)542-2474
Recent algorithms are provided by "T.R.Hopkins" <>. The StatLib
collection of Applied Statistics Algorithms is maintained by Pantelis Vlachos,


Credit where credit is due
If you use an algorithm, dataset, or other information from StatLib, please acknowledge both
StatLib and the original contributor of the material.
Last modified: Mon Nov 15 17:25:32 EST 1999 by Pantelis Vlachos

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/>
CSTART OF AS 3
REAL FUNCTION PROBST(T, IDF, IFAULT)
C
C
ALGORITHM AS 3 APPL. STATIST. (1968) VOL.17, P.189
C
C
STUDENT T PROBABILITY (LOWER TAIL)
C
REAL A, B, C, F, G1, S, FK, T, ZERO, ONE, TWO, HALF, ZSQRT, ZATAN

C
C
G1 IS RECIPROCAL OF PI
C
DATA ZERO, ONE, TWO, HALF, G1
$
/0.0, 1.0, 2.0, 0.5, 0.3183098862/
C
ZSQRT(A) = SQRT(A)
ZATAN(A) = ATAN(A)
C
IFAULT = 1
PROBST = ZERO
IF (IDF .LT. 1) RETURN
IFAULT = 0
F = IDF
A = T / ZSQRT(F)
B = F / (F + T ** 2)
IM2 = IDF - 2
IOE = MOD(IDF, 2)
S = ONE
C = ONE
F = ONE
KS = 2 + IOE
FK = KS
IF (IM2 .LT. 2) GOTO 20
DO 10 K = KS, IM2, 2
C = C * B * (FK - ONE) / FK
S = S + C
IF (S .EQ. F) GOTO 20

F = S
FK = FK + TWO
10 CONTINUE
20 IF (IOE .EQ. 1) GOTO 30
PROBST = HALF + HALF * A * ZSQRT(B) * S
RETURN
30 IF (IDF .EQ. 1) S = ZERO
PROBST = HALF + (A * B * S + ZATAN(A)) * G1
RETURN
END
CEND OF AS 3

[3/3/2002 9:48:20 PM]


/>
CSTART OF AS 5
REAL FUNCTION PRNCST(ST, IDF, D, IFAULT)
C
C
ALGORITHM AS 5 APPL. STATIST. (1968) VOL.17, P.193
C
C
COMPUTES LOWER TAIL AREA OF NON-CENTRAL T-DISTRIBUTION
C
REAL ST, D, G1, G2, G3, ZERO, ONE, TWO, HALF, EPS, EMIN, F,
$ A, B, RB, DA, DRB, FMKM1, FMKM2, SUM, AK, FK, FKM1,
$ ALNORM, TFN, ALOGAM, ZSQRT, ZEXP
C
C

CONSTANTS - G1 IS 1.0 / SQRT(2.0 * PI)
C
G2 IS 1.0 / (2.0 * PI)
C
G3 IS SQRT(2.0 * PI)
C
DATA G1, G2, G3 /0.3989422804, 0.1591549431, 2.5066282746/
DATA ZERO, ONE, TWO, HALF,
EPS, EMIN
$
/0.0, 1.0, 2.0, 0.5, 1.0E-6, 12.5/
C
ZSQRT(A) = SQRT(A)
ZEXP(A) = EXP(A)
C
F = IDF
IF (IDF .GT. 100) GOTO 50
IFAULT = 0
IOE = MOD(IDF, 2)
A = ST / ZSQRT(F)
B = F / (F + ST ** 2)
RB = ZSQRT(B)
DA = D * A
DRB = D * RB
SUM = ZERO
IF (IDF .EQ. 1) GOTO 30
FMKM2 = ZERO
IF (ABS(DRB) .LT. EMIN) FMKM2 = A * RB * ZEXP(-HALF * DRB ** 2)
$ * ALNORM(A * DRB, .FALSE.) * G1
FMKM1 = B * DA * FMKM2

IF (ABS(D) .LT. EMIN)
$ FMKM1 = FMKM1 + B * A * G2 * ZEXP(-HALF * D ** 2)
IF (IOE .EQ. 0) SUM = FMKM2
IF (IOE .EQ. 1) SUM = FMKM1
IF (IDF .LT. 4) GOTO 20
IFM2 = IDF - 2
AK = ONE
FK = TWO
DO 10 K = 2, IFM2, 2
FKM1 = FK - ONE
FMKM2 = B * (DA * AK * FMKM1 + FMKM2) * FKM1 / FK
AK = ONE / (AK * FKM1)
FMKM1 = B * (DA * AK * FMKM2 + FMKM1) * FK / (FK + ONE)
IF (IOE .EQ. 0) SUM = SUM + FMKM2
IF (IOE .EQ. 1) SUM = SUM + FMKM1
AK = ONE / (AK * FK)
FK = FK + TWO
10 CONTINUE
20 IF (IOE .EQ. 0) GOTO 40
30 PRNCST = ALNORM(DRB, .TRUE.) + TWO * (SUM + TFN(DRB, A))
RETURN
40 PRNCST = ALNORM(D, .TRUE.) + SUM * G3
RETURN
C

(1 of 2) [3/3/2002 9:48:34 PM]


/>
C

C
C

NORMAL APPROXIMATION - K IS NOT TESTED AFTER THE TWO CALLS
OF ALOGAM, BECAUSE A FAULT IS IMPOSSIBLE WHEN F EXCEEDS 100

50 IFAULT = 1
A = ZSQRT(HALF * F) * ZEXP(ALOGAM(HALF * (F - ONE), K)
$ - ALOGAM(HALF * F, K)) * D
PRNCST = ALNORM((ST - A) / ZSQRT(F * (ONE + D ** 2)
$ / (F - TWO) - A ** 2), .FALSE.)
RETURN
END
CEND OF AS 5

(2 of 2) [3/3/2002 9:48:34 PM]


/>
C This file contains AS6 and the enhanced version ASR44.
See AS7 also.
C
C
SUBROUTINE CHOL (A,N,NN,U,NULLTY,IFAULT)
C
C
Algorithm AS6, Applied Statistics, vol.17, (1968)
C
C
Given a symmetric matrix order n as lower triangle in a( )

C
calculates an upper triangle, u( ), such that uprime * u = a.
C
a must be positive semi-definite. eta is set to multiplying
C
factor determining effective zero for pivot.
C
C
arguments:C
a()
= input, a +ve definite matrix stored in lower-triangula
C
form.
C
n
= input, the order of a
C
nn
= input, the size of the a and u arrays
n*(n+1)/2
C
u()
= output, a lower triangular matrix such that u*u' = a.
C
a & u may occupy the same locations.
C
nullty = output, the rank deficiency of a.
C
ifault = output, error indicator
C

= 1 if n < 1
C
= 2 if a is not +ve semi-definite
C
= 3 if nn < n*(n+1)/2
C
= 0 otherwise
C
C***********************************************************************
C
DOUBLE PRECISION A(NN),U(NN),ETA,ETA2,X,W,ZERO
C
C
The value of eta will depend on the word-length of the
C
computer being used. See introductory text.
C
DATA ETA,ZERO/1.D-9,0.0D0/
C
IFAULT=1
IF (N.LE.0) RETURN
IFAULT=3
IF (NN.LT.N*(N+1)/2) RETURN
IFAULT=2
NULLTY=0
J=1
K=0
ETA2=ETA*ETA
II=0
C

C
Factorize column by column, icol = column no.
C
DO 80 ICOL=1,N
II=II+ICOL
X=ETA2*A(II)
L=0
KK=0
C
C
IROW = row number within column ICOL
C
DO 40 IROW=1,ICOL
KK=KK+IROW
K=K+1
W=A(K)
M=J

(1 of 3) [3/3/2002 9:49:01 PM]


/>
10
20

30
40
50

60

70
80

DO 10 I=1,IROW
L=L+1
IF (I.EQ.IROW) GO TO 20
W=W-U(L)*U(M)
M=M+1
CONTINUE
IF (IROW.EQ.ICOL) GO TO 50
IF (U(L).EQ.ZERO) GO TO 30
U(K)=W/U(L)
GO TO 40
IF (W*W.GT.ABS(X*A(KK))) RETURN
U(K)=ZERO
CONTINUE
IF (ABS(W).LE.ABS(ETA*A(K))) GO TO 60
IF (W.LT.ZERO) RETURN
U(K)=SQRT(W)
GO TO 70
U(K)=ZERO
NULLTY=NULLTY+1
J=J+ICOL
CONTINUE
IFAULT=0
END

C
C
C

C
SUBROUTINE SUBCHL (A,B,N,U,NULLTY,IFAULT,NDIM,DET)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C

REMARK ASR 44

APPL. STATIST. (1982) VOL. 31, NO. 3

A revised and enhanced version of
ALGORITHM AS 6 APPL. STATIST. (1968) VOL. 17, NO. 2
Given a symmetric matrix of order N as lower triangle in A(),
calculates an upper triangle, U(), such that U'U = the sub-matrix
of A whose rows and columns are specified in the integer array
B().

U() may coincide with A().
A() must be +ve semi-definite.
ETA is set to multiplying factor determining effective zero for
a pivot.
NULLTY is returned as number of effective zero pivots.
IFAULT is returned as 1 if N <= 0, 2 if A() is not +ve semidefinite, otherwise 0 is returned.
DOUBLE PRECISION A(NDIM),U(NDIM),DET
INTEGER B(N)

C
C
C

Local variables
DOUBLE PRECISION ETA,ONE,ZERO,W,ETA2

C
C
C
C

The value of ETA below will depend upon the word length of the
computer being used.
DATA ETA/1.D-14/,ONE/1.D0/,ZERO/0.D0/

C
IFAULT=1
IF (N.LE.0) GO TO 90
IFAULT=2
NULLTY=0


(2 of 3) [3/3/2002 9:49:01 PM]


/>
10
20

30
40
50

60
70
80

DET=ONE
J=1
K=0
ETA2=ETA*ETA
DO 80 ICOL=1,N
IJ=B(ICOL)*(B(ICOL)-1)/2
II=IJ+B(ICOL)
X=ETA2*A(II)
L=0
DO 40 IROW=1,ICOL
KK=B(IROW)*(B(IROW)+1)/2
K=K+1
JJ=IJ+B(IROW)
W=A(JJ)

M=J
DO 10 I=1,IROW
L=L+1
IF (I.EQ.IROW) GO TO 20
W=W-U(L)*U(M)
M=M+1
CONTINUE
IF (IROW.EQ.ICOL) GO TO 50
IF (U(L).EQ.ZERO) GO TO 30
U(K)=W/U(L)
GO TO 40
IF (W*W.GT.ABS(X*A(KK))) GO TO 90
U(K)=ZERO
CONTINUE
IF (ABS(W).LE.ABS(ETA*A(KK))) GO TO 60
IF (W.LT.ZERO) GO TO 90
U(K)=SQRT(W)
GO TO 70
U(K)=ZERO
NULLTY=NULLTY+1
J=J+ICOL
DET=DET*U(K)*U(K)
CONTINUE

C
90

IFAULT=0
RETURN
END


(3 of 3) [3/3/2002 9:49:01 PM]


/>
c This file contains AS7 and an enhanced alternative - ASR44. See also AS6.
c
c
subroutine syminv(a, n, nn, c, w, nullty, ifault)
c
c
Algorithm AS7, Applied Statistics, vol.17, 1968, p.198.
c
c
Forms in c( ) as lower triangle, a generalised inverse
c
of the positive semi-definite symmetric matrix a( )
c
order n, stored as lower triangle.
c
c
arguments:c
a()
= input, the symmetric matrix to be inverted, stored in
c
lower triangular form
c
n
= input, order of the matrix
c

nn
= input, the size of the a and c arrays
n*(n+1)/2
c
c()
= output, the inverse of a (a generalized inverse if c is
c
singular), also stored in lower triangular.
c
c and a may occupy the same locations.
c
w()
= workspace, dimension at least n.
c
nullty = output, the rank deficiency of a.
c
ifault = output, error indicator
c
= 1 if n < 1
c
= 2 if a is not +ve semi-definite
c
= 3 if nn < n*(n+1)/2
c
= 0 otherwise
c
c***************************************************************************
c
double precision a(nn), c(nn), w(n), x, zero, one
c

data zero, one /0.0d0, 1.0d0/
c
c
cholesky factorization of a, result in c
c
call chol(a, n, nn, c, nullty, ifault)
if(ifault.ne.0) return
c
c
invert c & form the product (cinv)'*cinv, where cinv is the inverse
c
of c, row by row starting with the last row.
c
irow = the row number, ndiag = location of last element in the row.
c
irow=n
ndiag=nn
10
l=ndiag
if (c(ndiag) .eq. zero) goto 60
do 20 i=irow,n
w(i)=c(l)
l=l+i
20
continue
icol=n
jcol=nn
mdiag=nn
30
l=jcol

x=zero
if(icol.eq.irow) x=one/w(irow)
k=n
40
if(k.eq.irow) go to 50
x=x-w(k)*c(l)
k=k-1
l=l-1

(1 of 4) [3/3/2002 9:49:03 PM]