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According to Gleason this method should be more accurate than the AS190 FORTRAN algorithm of Lund and Lund (1983) and works from .5 <= p <= .999 (The AS190 only works from .9 <= p <= .99). It is more efficient then the Copenhaver & Holland (1988) algorithm (used by the _qtukey_ R function) although it requires storing the A table in memory. (q distribution) approximations in Python. see: Gleason, J. R. (1999). An accurate, non-iterative approximation for studentized range quantiles. Computational Statistics & Data Analysis, (31), 147-158. Gleason, J. R. (1998). 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May 3, 2004. Lower tail quantile for standard normal distribution function. This function returns an approximation of the inverse cumulative standard normal distribution function. I.e., given P, it returns an approximation to the X satisfying P = Pr{Z <= X} where Z is a random variable from the standard normal distribution. The algorithm uses a minimax approximation by rational functions and the result has a relative error whose absolute value is less than 1.15e-9. Author: Peter John Acklam Time-stamp: 2000-07-19 18:26:14 E-mail: pjacklam@online.no WWW URL: http://home.online.no/~pjacklam iis5Argument to ltqnorm %f must be in open interval (0,1)g%1Cg4pFk@g;->qg@ rKa@g͋40>gyTW @g0qg@ rKa@g͋40>gyTW @(g0@ig8@ig3@ig?g@iiig @i(i(i<ix(ii(i<(iii((iii(iii(iii(iii(tinftinttround(RR tvi((sDlib/python2.7/site-packages/statsmodels/stats/libqsturng/qsturng_.pyt _select_vsYs&          cCst||\}}}tt||f|||dd}tt||f|||dd}tt||f|||dd}|dkrd}nd|d|d|d|f\} } } } d||| | ||| | | | } | | | | kr7||| | d| | | }n"||| | d| | | }|}tj| d| | d|| | |}|S(s` interpolates v based on the values in the A table for the scalar value of r and th g?g@g*Gg?(R?RRR R (R RRtv0tv1tv2ty0_sqty1_sqty2_sqtv_tv0_tv1_tv2_R3R4R5R7((sDlib/python2.7/site-packages/statsmodels/stats/libqsturng/qsturng_.pyt_interpolate_vus''' ..%"/cCs|dks|dkr'tdn|dkrQ|dkrltdqln|dkrltdnt|}t|tjr|j}n||ftkrtt||f|||d }n|tkr^|t gdgf|dkkr^t ||\}}}t |\}}} t |||d} t |||d} t |||d} d |d |d |d |f\} }}}d | | ||| | ||||}||||kr| | ||d |||}n"| | ||d |||}| }t j|d | |d || ||}nY|t gdgf|dkkrt|||}n!|tkrt |||}nt|d }t jd| tjjjd |d |S( sscalar version of qsturngg?g+?sp must be between .1 and .999g?isv must be > 2 when p < .9isv must be > 1 when p >= .9g?g@g?g*G(R Rt isinstanceRtndarraytitemRRtp_keystv_keysR?RR:R R RJR!R"R#R$R%(R RRR7R@RARBR&R'R(tr0_sqtr1_sqtr2_sqRFRGRHRIR3R4R5((sDlib/python2.7/site-packages/statsmodels/stats/libqsturng/qsturng_.pyt_qsturngs>    &/..%"2# svector version of qsturngcCs>ttt|||gr.t|||St|||S(sApproximates the quantile p for a studentized range distribution having v degrees of freedom and r samples for probability p. Parameters ---------- p : (scalar, array_like) The cumulative probability value p >= .1 and p <=.999 (values under .5 are not recommended) r : (scalar, array_like) The number of samples r >= 2 and r <= 200 (values over 200 are permitted but not recommended) v : (scalar, array_like) The sample degrees of freedom if p >= .9: v >=1 and v >= inf else: v >=2 and v >= inf Returns ------- q : (scalar, array_like) approximation of the Studentized Range (tallRRRSt _vqsturng(R RR((sDlib/python2.7/site-packages/statsmodels/stats/libqsturng/qsturng_.pytqsturngscsdkrtdnfd}|dkrtd|dkrRdStd|dkrndSd t|ddd ||fStd||krdStd||krdSd t|ddd ||fSd S( sscalar version of psturnggsq should be >= 0cstt|||S(N(tabsRS(R RR(R(sDlib/python2.7/site-packages/statsmodels/stats/libqsturng/qsturng_.pyt;tig?g?g+?gMbP?g?targsN(R RSR(RRRtopt_func((RsDlib/python2.7/site-packages/statsmodels/stats/libqsturng/qsturng_.pyt_psturng6s   svector version of psturngcCs>ttt|||gr.t|||St|||S(sKEvaluates the probability from 0 to q for a studentized range having v degrees of freedom and r samples. Parameters ---------- q : (scalar, array_like) quantile value of Studentized Range q >= 0. r : (scalar, array_like) The number of samples r >= 2 and r <= 200 (values over 200 are permitted but not recommended) v : (scalar, array_like) The sample degrees of freedom if p >= .9: v >=1 and v >= inf else: v >=2 and v >= inf Returns ------- p : (scalar, array_like) 1. - area from zero to q under the Studentized Range distribution. When v == 1, p is bound between .001 and .1, when v > 1, p is bound between .001 and .9. Values between .5 and .9 are 1st order appoximations. 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