The program does not work. Can you help me fix it$?$ # First, we'll simulate a dataset following a centered normal distribution:

import numpy as np

from scipy.optimize import minimize

# Simulate data following a centered normal distribution

np$.$random.seed$(42)$ # Setting seed for reproducibility

sample$\_$size $=1000$

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

#Next, we'll define functions to calculate the log$-$likelihood, estimate parameters $($mu and sigma$^2),$ and the gradient matrix:

# Log$-$likelihood function for centered normal distribution

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

# Estimation of parameters using maximum likelihood estimation

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

# Initial guess for parameters

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

# Calculate the gradient matrix

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$])$

#Now, let's estimate parameters from the simulated data:

# Estimate parameters using MLE

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

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

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

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

#Following that, we'll calculate the gradient matrix:

# Calculate the gradient matrix

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

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

print$($gradient$)$

from scipy import optimize

from scipy import stats

#Finally, we'll conduct the LR$,$ Wald, and LM tests:

# Likelihood ratio test

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

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

# Wald test

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$=2)$ # degrees of freedom $=2$

# Lagrange Multiplier test

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

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

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$)$