import numpy as np

from scipy.optimize import minimize

from scipy import optimize

from scipy import stats

$$

$$

$$

np$.$random.seed$(42)$

sample$\_$size $=1000$

data $=$ np$.$random.normal$($loc$=0,$ scale$=1,$ size$=$sample$\_$size$)$

$$

def log$\_$likelihood$($params$,$ data$)$:

mu$,$ sigma$\_$sq $=$ params

n $=$ len$($data$)$

log$\_$likelihood $=-($n $/2)*$ np$.$log$(2*$ np$.$pi $*$ sigma$\_$sq$)-(1/(2*$ sigma$\_$sq$))*$ np$.$sum$(($data $-$ mu$)**2)$

return $-$log$\_$likelihood

$$

def estimate$\_$parameters$($data$)$:

initial$\_$guess $=[$np$.$mean$($data$),$ np$.$var$($data$)]$

result $=$ minimize$($log$\_$likelihood, initial$\_$guess, args$=($data$,),$ method$=\text{'}$L$-$BFGS$-$B$\text{'})$

return result.x

$$

def gradient$\_$matrix$($params$,$ data$)$:

mu$,$ sigma$\_$sq $=$ params

n $=$ len$($data$)$

grad$\_$mu $=(1/$ sigma$\_$sq$)*$ np$.$sum$($data $-$ mu$)$

grad$\_$sigma$\_$sq $=-($n $/(2*$ sigma$\_$sq$))+(1/(2*$ sigma$\_$sq $**2))*$ np$.$sum$(($data $-$ mu$)**2)$

return np$.$array$([$grad$\_$mu$,$ grad$\_$sigma$\_$sq$])$

$$

estimated$\_$params $=$ estimate$\_$parameters$($data$)$

mu$\_$estimate, sigma$\_$sq$\_$estimate $=$ estimated$\_$params

print$("$Estimated mu:$",$ mu$\_$estimate$)$

print$("$Estimated sigma$^2$:$",$ sigma$\_$sq$\_$estimate$)$

$$

gradient $=$ gradient$\_$matrix$($estimated$\_$params, data$)$

print$("$Gradient matrix:"$)$

print$($gradient$)$

$$

from scipy import optimize

from scipy import stats

$$

lr$\_$stat $=2*($log$\_$likelihood$($estimated$\_$params, data$)-$ log$\_$likelihood$([0,1],$ data$))$

lr$\_$p$\_$value $=1-$ stats.chi$2.$cdf$($lr$\_$stat, df$=2)$

$$

inv$\_$hessian $=$ minimize$($log$\_$likelihood, estimated$\_$params, args$=($data$,),$ method$=\text{'}$L$-$BFGS$-$B$\text{'},$ jac$=$True$).$hess$\_$inv

wald$\_$stat $=$ np$.$dot$($gradient$,$ np$.$dot$($inv$\_$hessian, gradient$))$

wald$\_$p$\_$value $=1-$ stats.chi$2.$cdf$($wald$\_$stat, df$=2)$

$$

lm$\_$stat $=$ np$.$dot$($gradient$,$ gradient$)$

lm$\_$p$\_$value $=1-$ stats.chi$2.$cdf$($lm$\_$stat, df$=2)$

$$

print$("$Likelihood Ratio test p$-$value:", lr$\_$p$\_$value$)$

print$("$Wald test p$-$value:", wald$\_$p$\_$value$)$

print$("$Lagrange Multiplier test p$-$value:", lm$\_$p$\_$value$)$

Estimated mu: $0\mathrm{.}01933205582232549$

Estimated sigma$^2$: $0\mathrm{.}9579049897315173$

Gradient matrix:

$[1\mathrm{.}57625582$e$-140\mathrm{.}00000000$e$+00]$

C:$\backslash$Users$\backslash$benoi$\backslash$AppData$\backslash$Local$\backslash$Temp$\backslash$ipykernel$\_22148\backslash 3282395572.$py:$15$: RuntimeWarning: invalid value encountered in log

log$\_$likelihood $=-($n $/2)*$ np$.$log$(2*$ np$.$pi $*$ sigma$\_$sq$)-(1/(2*$ sigma$\_$sq$))*$ np$.$sum$(($data $-$ mu$)**2)$

$---------------------------------------------------------------------------$

IndexError Traceback $($most recent call last$)$

Cell In$[2],$ line $45$

$42$ lr$\_$stat $=2*($log$\_$likelihood$($estimated$\_$params, data$)-$ log$\_$likelihood$([0,1],$ data$))$

$43$ lr$\_$p$\_$value $=1-$ stats.chi$2.$cdf$($lr$\_$stat, df$=2)$

$--->45$ inv$\_$hessian $=$ minimize$($log$\_$likelihood, estimated$\_$params, args$=($data$,),$ method$=\text{'}$L$-$BFGS$-$B$\text{'},$ jac$=$True$).$hess$\_$inv

$46$ wald$\_$stat $=$ np$.$dot$($gradient$,$ np$.$dot$($inv$\_$hessian, gradient$))$

$47$ wald$\_$p$\_$value $=1-$ stats.chi$2.$cdf$($wald$\_$stat, df$=2)$

File ~$\backslash$anaconda$3\backslash$Lib$\backslash$site$-$packages$\backslash$scipy$\backslash$optimize$\backslash \_$minimize.py:$710,$ in minimize$($fun$,$ x$0,$ args, method, jac, hess, hessp, bounds, constraints, tol, callback, options$)$

$707$ res $=\_$minimize$\_$newtoncg$($fun$,$ x$0,$ args, jac, hess, hessp, callback,

$708**$options$)$

$709$ elif meth $==\text{'}$l$-$bfgs$-$b$\text{'}$:

$-->710$ res $=\_$minimize$\_$lbfgsb$($fun$,$ x$0,$ args, jac, bounds,

$711$ callback$=$callback, $**$options$)$

$712$ elif meth $==\text{'}$tnc$\text{'}$:

$713$ res $=\_$minimize$\_$tnc$($fun$,$ x$0,$ args, jac, bounds, callback$=$callback,

$714**$options$)$

File ~$\backslash$anaconda$3\backslash$Lib$\backslash$site$-$packages$\backslash$scipy$\backslash$optimize$\backslash \_$lbfgsb$\_$py$.$py:$307,$ in $\_$minimize$\_$lbfgsb$($fun$,$ x$0,$ args, jac, bounds, disp, maxcor, ftol, gtol, eps, maxfun, maxiter, iprint, callback, maxls, finite$\_$diff$\_$rel$\_$step, $**$unknown$\_$options$)$

$304$ else:

$305$ iprint $=$ disp

$-->307$ sf $=\_$prepare$\_$scalar$\_$function$($fun$,$ x$0,$ jac$=$jac, args$=$args, epsilon$=$eps,

$308$ bounds$=$new$\_$bounds,

$309$ finite$\_$diff$\_$rel$\_$step$=$finite$\_$diff$\_$rel$\_$step$)$

$311$ func$\_$and$\_$grad $=$ sf$.$fun$\_$and$\_$grad

$313$ fortran$\_$int $=\_$lbfgsb$.$types.intvar.dtype

File ~$\backslash$anaconda$3\backslash$Lib$\backslash$site$-$packages$\backslash$scipy$\backslash$optimize$\backslash \_$optimize.py:$383,$ in $\_$prepare$\_$scalar$\_$function$($fun$,$ x$0,$ jac, args, bounds, epsilon, finite$\_$diff$\_$rel$\_$step, hess$)$

$379$ bounds $=(-$np$.$inf, np$.$inf$)$

$381$ # ScalarFunction caches. Reuse of fun$($x$)$ during grad

$382$ # calculation reduces overall function evaluati

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