How to estimate of coefficients of logistic model

Start from defintion $$\text{logit}(p)=\log(\frac p{1-p})$$ So, if you have data $(x_i,p_i)$, compute $$y_i=\log(\frac {p_i} {1-p_i})$$ and the regression is just $$y=a+bx$$ So, standard linear regression will give you estimates of parameters $a,b$.

But, if $p$ is the mesured value, you need to go to nonlinear regression for the model $$p=\frac{e^{a+b x}}{1+e^{a+b x}}$$ because what is measured is $p_i$ and not $\log(\frac {p_i}{1-p_i})$.

If you have a nonlinear regression program, the problem would be simple since the first step gave you at least reasonable estimates of parameters $a,b$.

If you do not have this, start from definition. You want to minimize $$SSQ=\sum _{i=1}^n \left(\frac{e^{a+b x_i}}{1+e^{a+b x_i}}-p_i\right)^2$$ Then $$\frac {d\, SSQ}{da}=\sum _{i=1}^n \frac{1}{1+\cosh (a+b x_i)}\left(\frac{e^{a+b x_i}}{1+e^{a+b x_i}}-p_i\right)$$ $$\frac {d\, SSQ}{db}=\sum _{i=1}^n \frac{x_i}{1+\cosh (a+b x_i)}\left(\frac{e^{a+b x_i}}{1+e^{a+b x_i}}-p_i\right)$$ and you could use Newton Raphson method to solve $$\frac {d\, SSQ}{da}=\frac {d\, SSQ}{db}=0$$ In such a case, to make life simpler, I would suggest numerical derivatives.

To give an example, I generated eleven data points according to $$p=\frac{1}{10} \left\lfloor \frac{10\, e^{0.234+0.456 x}}{1+e^{0.234+0.456 x}}\right\rfloor$$ in order to have some significant noise in the data (the $x_i$ are integers between $-5$ and $+5$).

The first step leads to $y=-0.0737209+0.443856 x$; starting with these values as inital guesses for the nonlinear regression led to $$p=\frac{e^{-0.0343898+0.438486 x}}{1+e^{-0.0343898+0.438486 x}}$$ As you can see, the parameters significantly changed going from first to second step.