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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 rz5Argument to ltqnorm %f must be in open interval (0,1))g%1Cg4pFk@g;->qg@ rKa@g͋40>gyTW @)g0rEcCsdddtd|dS)z+function for p-value abcissa transformationgr#g?r)rEr>r2r2r3 _ptransformsrGcCs|dt|d|dt|dd|dt|dd|dt|dd}|dkr|dddt|d7}|d krt|s|nd }|d dd |7}n"t|s|nd }|dd |7}| S)z| calculates f-hat for the coefficients in a, probability p, sample mean difference r, and degrees of freedom v. rr#r5r)r7r8gMb`rg~jt@籡*Gg9ư_?gs@gg@)r;r=rEnpisinf)r?r>rDvfr2r2r3_funcsrMcCst|dkr dS|dkrdS|dkr$dS|dkr0dS|d krrDrKp0p1p2Zy0y1y2Zy_log0Zy_log1Zy_log2Zp_tZp0_tZp1_tZp2_td2d1d0Zy_logyZq0Zq1rCr2r2r3_interpolate_psf   " $ " $ (r`cCs|dkrddtfS|dkrdS|dkr*dS|dkr6d S|d krBd S|d krNd S|dkrd|dkrpdSn |dkrpdStt|}|d||dfS)z-returns the points to use for interpolating vrr.r/r)r-r.r/r)r,r-r.r)r+r,r-r)r+r,g3@)rar+r"g@)r5r)r7g @)r)r7r8r5)infintround)rKr>vir2r2r3 _select_vsWs&  rgcCsJt||\}}}tt||f|||dd}tt||f|||dd}tt||f|||dd}|dkrvd}d|d|d|d|f\} } } } d||| | ||| | | | } | | | | kr||| | d| | | }n ||| | d| | | }|}t| d| | d|| | |}|S)z` interpolates v based on the values in the A table for the scalar value of r and th r#rrHr)rgrMrOr;r<)r>rDrKv0v1v2Zy0_sqZy1_sqZy2_sqv_v0_v1_v2_r\r]r^r_r2r2r3_interpolate_vss$ $" *rocCsB|dks|dkrtd|dkr2|dkrBtdn|dkrBtdt|}t|tjr^|}||ftkrtt||f|||d }n|tkr|t gdgf|dkkrt ||\}}}t |\}}} t |||d} t |||d} t |||d} d |d |d |d |f\} }}}d | | ||| | ||||}||||kr| | ||d |||}n | | ||d |||}| }t |d | |d || ||}n@|t gdgf|dkkrt|||}n|tkrt |||}t|d }t d| tjjd |d |S) zscalar version of qsturngrr(zp must be between .1 and .999r"r)zv must be > 2 when p < .9r5zv must be > 1 when p >= .9r#rrrH)r:r0 isinstancerIZndarrayitemrOrMp_keysv_keysrgrNr`r;r<rorRrSrTrUrV)r>rDrKr_rhrirjrWrXrYZr0_sqZr1_sqZr2_sqrkrlrmrnr\r]r^r2r2r3_qsturngsF   &$" ,   rtzvector version of qsturngcCs,ttt|||gr t|||St|||S)aApproximates 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 )allmapr4rt _vqsturngr>rDrKr2r2r3qsturngs rycsdkrtdfdd}|dkrntd|dkr8dStd|dkrLd Sd t|dd||fd }t|Std||krdStd||krd Sd t|dd||fd }t|Sd S) zscalar version of psturnggzq should be >= 0csttt|||SN)rIsqueezeabsrtrxrCr2r3opt_func8sz_psturng..opt_funcr5r"rr(gMbP?r#)argsN)r:rtrrIZ atleast_1d)rCrDrKr~Zsolnr2r}r3_psturng3s   rcCstt|||Srz)rIr{rrCrDrKr2r2r3_psturng_scalarJsrzvector version of psturngcCs,ttt|||gr t|||St|||S)aJEvaluates 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. )rurvr4r _vpsturngrr2r2r3psturngPs r)__doc__Zstatsmodels.compat.pythonrr;Z scipy.statsrSZnumpyrIZscipy.optimizerrc __version__rOrrrsr4rErGrMrNr`rgrortZ vectorizerwryrrrrr2r2r2r3s                                                    60(  @    T  * Z  9