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Set $S_n={\sum }_{i=1}^n{Y}_i$ and $a_n={\sum }_{i=1}^n{2}^{-i}=1-2^{-n}$. For the Poisson distribution observations, factors independent of $\theta$ can be removed from the likelihood function, leavingWhen $S_n>0$, the log-likelihood derivative is $S_n/\theta -a_n$ and its second derivative is negative. The unique positive maximum-likelihood estimator is thereforeThere is a genuine boundary qualification: if $S_n=0$, the likelihood decreases strictly on $\theta >0$, so no maximum is attained in the stated open parameter space. On the closure $\theta \ge 0$, its maximizer is $0$. Defining the usual extended maximum-likelihood estimator by the same displayed formula for every $S_n$ allows its statistical consistency to be examined; the all-zero event cannot be ignored because its probability does not tend to zero.

Indeed, sums of independent Poisson distribution variables give $S_n\sim \mathrm{Poi}(\theta a_n)$, soThus the extended maximum-likelihood estimator is inconsistent. More precisely, $S_n$ increases to $S_{\infty }={\sum }_{i\ge 1}{Y}_i$, and $\mathbb{E}{S}_{\infty }=\theta <\infty$ shows that this total count is finite almost surely. Its distribution is Poisson distribution with mean $\theta$, by taking limits of the finite-sum laws. HenceThis finite-exposure Poisson inconsistency leaves a nondegenerate random limit even with infinitely many observations. Correspondingly, the total Fisher information is $a_n/\theta \to 1/\theta$, rather than diverging. The common-distribution hypothesis in the introductory maximum likelihood estimation theorem is absent here.

One sufficient maximum likelihood estimation statistical consistency theorem is the following. Let the observations be independent random variables with a common density $f_{\theta }$ relative to a fixed measure, let $\Theta$ be a compact metric parameter set containing the true value ${\theta }_0$, and suppose that the expected log-likelihoodis finite, continuous, and uniquely maximized at ${\theta }_0$. Assume also the uniform law of large numbersand assume that a measurable global maximum-likelihood estimator exists. Then every such maximum-likelihood estimator converges in probability to the true parameter. A useful sufficient condition for this uniform law of large numbers is almost-sure continuity of $\theta \mapsto \log f_{\theta }(Y_1)$ on the compact parameter set and an integrable envelope ${\sup }_{\theta }|\log f_{\theta }(Y_1)|\le M(Y_1)$, ${\mathbb{E}}_{{\theta }_0}M(Y_1)<\infty$. Under these conditions, identifiability supplies uniqueness through the Kullback-Leibler divergence identitywith equality only at $\theta ={\theta }_0$.

To see the statistical consistency conclusion, fix $\epsilon >0$. Compactness and the unique maximum give a strictly positive separation gapwhenever the set outside the neighbourhood is nonempty. If ${\Delta }_n$ denotes the supremum discrepancy in the uniform law of large numbers, maximization of the empirical log-likelihood yieldsHence the probability of $d({\hat{\theta }}_n,{\theta }_0)\ge \epsilon$ is at most $\mathbb{P}(2{\Delta }_n\ge {\delta }_{\epsilon })$, which tends to zero. This is the argmin consistency under uniform convergence in probability argument applied to the negative log-likelihood. These are sufficient conditions, not necessary ones; parameter-dependent support can require a direct argument, and independent observations need not have a common distribution in other models.

For $1\le x\le n$, conditional binomial probabilities can be writtenThis zero-truncated binomial distribution is an exponential family. Write $q=1-\pi$ and $s=1-q^n$. The cumulant identities $\mathbb{E}X=k^{\prime }$ and $\mathrm{Var}X=k^{\prime \prime }$, with $d\pi /d\varphi =\pi q$, giveThe maximum likelihood estimation score equation is $X=k^{\prime }(\hat{\varphi })$, orFor $n\ge 2$ and $1<X<n$ the increasing mean function gives a unique interior solution; $X=1$ and $X=n$ instead give limiting estimates zero and one, respectively. If $n=1$, $X=1$ always and the parameter is unidentifiable.

For fixed $\pi \in (0,1)$ and large $n$, the excluded event has exponentially small probability $(1-\pi {)}^n$. Conditioning therefore does not change the central-limit approximation for $X/n$, and the score equation differs from $\hat{\Pi }=X/n$ only by an exponentially small correction with high probability. The binomial central limit theorem consequently givesThe suggested inference after observing one is incorrect for the stated sampling model. Its conditional likelihood is the unconditioned likelihood divided by $1-(1-\Pi {)}^n$, a parameter-dependent factor. They are not proportional likelihoods: the unconditioned estimate is $1/n$, whereas the conditional likelihood for $X=1$ is maximized as $\Pi \downarrow 0$ for $n>1$. Knowing that the observed value is positive does not remove the selection mechanism from the model.

Write $m_i=\mu +{\alpha }_i$. Apart from a constant, the log-likelihood isFor each group,Thus maximum likelihood estimation sets ${\hat{m}}_i={\overline{Y}}_i$. The corner-point constraint ${\alpha }_1=0$ identifies $m_1=\mu$, givingThe baseline mean is the first group mean, rather than the grand mean, because of the chosen identifiability constraint.

Allow each subject to have a separate baseline and slope through a correlated random-intercept and random-slope model:The subject random effects are independent of all measurement errors, and subjects are independent. The within-person covariance between times $s,t$ is ${\tau }_0^2+(s+t)\rho {\tau }_0{\tau }_1+st{\tau }_1^2$, with an additional ${\sigma }^2$ when the same reading is used twice.

Prefer the model with a random slope. The maximum likelihood estimation refits improve twice the log-likelihood by $227.9604$ while adding two covariance statistical parameters. Both the Akaike information criterion ($5165.779$ to $4941.819$) and Bayesian information criterion ($5183.367$ to $4968.200$) strongly favour it. The printed nominal likelihood-ratio test is overwhelming. The usual reference chi-squared distribution with two statistical degrees of freedom is not an exact regular calibration, since zero slope variance is a boundary and the intercept–slope correlation is then unidentified; a design-specific parametric bootstrap could calibrate it. This qualification does not undermine the substantial descriptive improvement shown by both information criteria.

The preferred fit estimates population mean baseline strength ${\hat{\beta }}_0=61.09286$ kg and weekly gain ${\hat{\beta }}_1=2.83143$ kg/week. The between-person baseline standard deviation is ${\hat{\tau }}_0=15.785582$ kg, and the between-person slope standard deviation is ${\hat{\tau }}_1=2.738666$ kg/week. Their estimated correlation $\hat{\rho }=0.117$ is weakly positive, giving random-effect covariance about $5.058$ kg${}^2$/week; its uncertainty is not supplied. The measurement-error standard deviation is $\hat{\sigma }=9.923280$ kg. The printed $-0.038$ instead describes the correlation between the estimated fixed effects, not between the subject random effects. These statistical parameter estimates come from restricted maximum likelihood; the model comparison uses the separate maximum likelihood estimation fits.

The population mean fitted trajectory is $61.09286+2.83143t$ kg. HenceUsing the reported slope standard error, an approximate 95% confidence interval for the population mean gain is $28.3143\pm 1.96(2.984464)=(22.46,34.16)$ kg; a Student t confidence interval with 499 statistical degrees of freedom is almost identical.

Successful individuals can improve far more than the average. Their latent ten-week gains have fitted normal distributionA clear estimate for an upper-performing group uses a between-person slope quantile: the 95th percentile is $28.3143+1.645(27.38666)\simeq 73.4$ kg, and the 97.5th percentile is about $82.0$ kg. The upper 5% of fitted underlying gains begin around 73 kg. These are person-to-person performance quantiles, not confidence intervals for the population mean. Identifying the best observed trainee would require that person's data or fitted subject random effects; the aggregate output cannot identify a literal maximum. If performance means the observed difference of endpoint readings, add the measurement-error variance $2{\sigma }^2$ to the latent gain variance.

For a finite Gaussian mixture with a common variance, introduce independent latent labels $Z_i\in \{1,\dots ,k\}$ with probabilities ${\pi }_j>0$, ${\sum }_j{\pi }_j=1$, and conditional responses $Y_i\mid Z_i=j\sim N({\mu }_j,{\sigma }^2)$, with common ${\sigma }^2>0$. The observed density and log-likelihood areThe expectation-maximization algorithm replaces the difficult log of sums by an expected complete-data objective. Choose positive initial weights and variance and separated initial means. At iteration $r$, the E-step computes the mixture responsibilitiesThe common normalizing factor cancels. For numerical stability the probabilities can be evaluated by subtracting the largest log weight before exponentiating.

With the old responsibilities held fixed, the expected complete-data log-likelihood, up to irrelevant constants, isLet $N_j={\sum }_i{\tau }_{ij}^{(r)}$. A Lagrange multiplier for ${\sum }_j{\pi }_j=1$, weighted least squares for the means, and differentiation in the common variance give the M-step:The variance update uses the new means and old responsibilities, with denominator $n$, not a residual degrees-of-freedom adjustment: this is maximum likelihood estimation. Repeat the E- and M-steps until the observed log-likelihood and parameters stabilize.

By EM likelihood monotonicity, exact updates do not decrease the observed likelihood. They need not reach its global maximum, so use several starting configurations and keep the best converged fit, checking for empty or nearly empty components and vanishing variance. The labels are interchangeable; sorting means after fitting supplies an interpretable labeling. Distinct starting means do not guarantee that all fitted components remain distinct. With a common variance and the usual fixed small $k$ relative to distinct observations the model avoids the individual-component variance-collapse pathology of unrestricted Gaussian mixtures, but degenerate data or too many components still require attention.

This is a generalized linear model with independent Bernoulli distributions for the responses and the logit link. Under treatment coding with colour one as the reference level, the linear predictor isColour is a categorical variable, not a numerical trend. Width has the same slope for every colour, since no interaction term is fitted. The parameters are estimated by maximum likelihood estimation.

The approximate Wald confidence interval for the width coefficient is $\boxed{0.46\pm 1.96(0.106)}$, or approximately $(0.2522,0.6678)$. This uses the approximate normal distribution of the maximum-likelihood estimator, rather than a confidence interval for an individual binary response.

For the requested crab, the fitted linear predictor is $\hat{\eta }=-11.38-0.22+0.46(20)=-2.40$. Applying the inverse logit link gives $\boxed{\hat{p}=(1+e^{2.40}{)}^{-1}\simeq 0.0832}$. The rounded displayed coefficients need not reproduce the reported test statistics exactly.

Write $\Omega ={\Sigma }^{-1}$ for a positive-definite matrix. Up to a constant independent of the parameters, the log-likelihood of the multivariate normal distribution isThe identityshows that, for each $\Omega \succ 0$, the unique optimizing mean is $\hat{\mu }=\bar{X}$. Using the matrix trace to rewrite the first quadratic sum, the profile likelihood isTherefore maximum likelihood estimation of the precision matrix, whenever the maximum exists, is exactlyThis is the profile likelihood for a Gaussian precision matrix. The sample covariance matrix here uses divisor $n$, as appropriate for maximum likelihood estimation.

There is an important rank qualification in the printed assumptions. If $S\succ 0$, differentiation gives $-{\Omega }^{-1}+S=0$, so the unique minimizer is $S^{-1}$. Uniqueness follows because the negative log-determinant is a strictly convex function on the positive-definite matrices. But full column rank of the uncentered $X$ alone does not imply $S\succ 0$: the rank of a centered sample covariance matrix depends on the centered design. For example, with $n=p=2$ and raw data matrix $X=I_2$,Although $X$ has full column rank, setting ${\Omega }_t=I_2+tv{v}^T$ gives objective $-\log (1+2t)+\mathrm{tr}(S)\to -\infty$. Thus an unpenalized maximizer need not exist under the literal raw-rank assumption. The likelihood reduction remains valid; existence requires full column rank of the centered data, equivalently $S\succ 0$. For the nonsingular Gaussian model this holds almost surely when $n>p$.

Use the entrywise-penalty convention for the Graphical Lasso:The sum is the entrywise matrix L1 norm. Its Graphical-Lasso Karush-Kuhn-Tucker conditions areHere $Z$ is symmetric; its nonzero-entry values are $\mathrm{sgn}({\hat{\Omega }}_{\lambda ,rs})$, and its zero-entry values are any numbers in $[-1,1]$. These are the subgradient optimality condition for this convex optimization problem. Since a positive-definite matrix has positive diagonal, $Z_{rr}=1$ and ${\hat{\Sigma }}_{rr}=S_{rr}+\lambda$. Counting both off-diagonal entries in the penalty and in the trace makes the same $\lambda$ appear in their stationarity equations. The common variant penalizing only off-diagonal entries instead sets $Z_{rr}=0$; the off-diagonal equations and the column argument below are unchanged.

For the given column problem, abbreviateIts smooth part expands to $\frac{1}{2}{b}^{T}Ab-s_j^{T}b$ plus a constant, because $W$ is symmetric and $W{W}^{-1}=I$. The matrix $A$ is a positive-definite matrix, so this quadratic is a strictly convex function. Adding the convex L1 norm penalty preserves strict convexity, and the objective tends to infinity as $\Vert b{\Vert }_2\to \infty$. The minimizer $b^{\ast }$ exists and is unique. Its Karush-Kuhn-Tucker conditions are

We identify that minimizer directly from the precision matrix. Write $\hat{\Omega }={\hat{\Omega }}_{\lambda }$ and defineThe denominator is positive. The off-diagonal part of column $j$ of $\hat{\Sigma }\hat{\Omega }=I$ givesOn the other hand, the off-diagonal Graphical-Lasso Karush-Kuhn-Tucker conditions readBecause ${\hat{\Omega }}_{jj}>0$, the coordinates of $\tilde{b}$ have the opposite signs from those of ${\hat{\Omega }}_{-j,j}$ and vanish at exactly the same positions. The subgradient of the absolute value therefore satisfiesIt follows thatThese are exactly the Lasso column problem's Karush-Kuhn-Tucker conditions. Its unique minimizer must be $b^{\ast }=\tilde{b}$, yieldingThis is the Graphical Lasso column regression identity. The minus sign is essential: it converts the precision-entry subgradient into the regression-coefficient subgradient. The argument uses the inverse identity directly, so it does not require proving the supplied block inverse formula.

Choose knots independently of the eventual validation responses, and let $b(x)=(b_1(x),\dots ,b_K(x){)}^T$ be a natural cubic spline basis as in part (a). A regression spline writes $m(x)=b(x{)}^T\beta$. Define the design matrix $B_{ij}=b_j(X_i)$. Under the stated normal linear model, maximum likelihood estimation for $\beta$ is the same as ordinary least squares:Differentiation gives the normal equations $B^{T}B\hat{\beta }=B^{T}Y$. If the design has full column rank,If it does not, a Moore-Penrose inverse gives the fitted values, but not every coefficient or off-sample prediction is identifiable. Merely choosing distinct knots does not guarantee full rank for an arbitrary set of observed predictor values.

More knots allow greater curvature and reduce approximation bias at the cost of larger estimation variance. Too many or poorly located knots can cause unstable fits. Equally spaced knots are simple; empirical predictor quantiles put more flexibility where observations are dense; subject knowledge can concentrate knots near likely changes. Both the number and positions can be compared using cross-validation or an appropriate penalized model-selection criterion. Adaptive knot selection must be included within validation, since treating a data-selected basis as prespecified understates its flexibility. Boundary knots should cover the region where inference is intended, and natural linear tails do not by themselves justify far-out extrapolation.

The marginal Akaike information criterion is $\mathrm{AIC}=-2\ell ({\hat{\vartheta }}_{\mathrm{ML}})+2k$, where $k$ counts estimated population parameters. Here they are ${\beta }_0,{\beta }_1,{\sigma }^2,{\tau }^2$, so $k=4$. The realized incubator slopes have been integrated out of the likelihood function; they are not four additional free population coefficients. From the marginal Gaussian log-likelihood, the expression isAll hats here denote the ordinary maximum likelihood estimation fit.

The displayed value $14041.1$ is instead a restricted maximum likelihood criterion. Thus it cannot simply be substituted for the ordinary maximum-likelihood deviance. The logLik method defaults to the fitting convention. If one applies AIC(fly.model) directly to the displayed REML object, the default restricted log-likelihood and the same parameter count give the software's restricted-likelihood-based value $14041.1+8=14049.1$. This number should be labelled as such; it is not the ordinary marginal AIC derived above. To obtain the latter, refit and calculateThe output does not contain that numerical answer. In particular, restricted criteria must not be used to compare models with different fixed-effect designs. A conditional AIC targeting predictions for existing groups is yet another criterion and does not follow by counting each latent slope as an ordinary parameter.

Visits can be missed or rescheduled because of illness, clinic availability or patient choice; follow-up may end through dropout or right censoring. The correct panel-observed multi-state likelihood uses each observed interval length ${\Delta }_{ik}$:Thus the four aggregated one-year counts are generally no longer sufficient; records must retain actual durations. Numerical maximum likelihood estimation uses the corresponding transition probabilities, or a matrix exponential of the generator when a larger model is fitted. One must not treat a two-year interval as a single one-year transition or as two known transitions through an unobserved intermediate state.

This conditional calculation is valid when visit scheduling and loss to follow-up are ignorable given the modeled information. If symptomatic patients are systematically examined earlier or leave the study differently, observation itself may be informative; the visit or dropout process may require a joint statistical model. Simply replacing one year by the actual interval does not correct informative observation.

A potential event time is an allowed support point for the discrete event-time distribution used to derive the Kaplan–Meier estimator. A positive probability mass may be assigned there before fitting, even if no event was observed at that time.

This is a device for nonparametric maximum likelihood estimation from exact event times and right censoring; it does not assert that the underlying biological event time is genuinely discrete. Its estimated mass can be zero at an unobserved potential time.

At an added potential event time with no event and a nonempty risk set, $d_j=0$, so maximum likelihood estimation gives ${\hat{q}}_j=0$. Its factor in the Kaplan–Meier estimator is therefore $1-d_j/r_j=1$.

Adding or deleting such times also leaves the risk sets at actual event times unchanged: those depend on the observed event and censoring histories. Consequently the fitted survivor function depends only on observed event times and their risk sets. Beyond the last observed follow-up, the tail is not identified; the standard flat continuation is a convention, rather than an additional consequence of likelihood maximization.

There are two defects. The mixed fit supplies a restricted maximum likelihood, whereas the ordinary fit supplies an ordinary log-likelihood, so their difference is not a likelihood-ratio test statistic. Also, $H_0:{\tau }^2=0$ lies on the boundary of ${\tau }^2\ge 0$; the usual Wilks theorem does not give a ${\chi }_1^2$ null law.

A valid approach first refits the mixed model by ordinary maximum likelihood estimation, setting REML to false, and fits the same fixed effects under $H_0$. For $V={\tau }^{2}Z{Z}^T+{\sigma }^{2}I$, maximizeover $\beta ,{\sigma }^2>0,{\tau }^2\ge 0$, and separately over $\beta ,{\sigma }^2>0$ with ${\tau }^2=0$. Set $T=2({\hat{\ell }}_1-{\hat{\ell }}_0)\ge 0$.

For finite-sample calibration, use the location-scale invariant simulation test for a Gaussian variance component. With the actual $X,Z$ fixed, simulate $B$ independent vectors $Y^{(b)}\sim N(0,I_{176})$, refit both models by ordinary maximum likelihood estimation to each, and calculate $T_b$ in exactly the same way. Under $H_0$, $Y=X\beta +\sigma z$. Translating by a vector in the column space of $X$ and multiplying by a positive scalar preserves the statistic: both maximized log-likelihoods acquire the same scale constant. Thus this simulation has the correct null law without knowing $\beta$ or $\sigma$. A conservative Monte Carlo p-value isIn the usual regular limit with increasing independent groups, a variance-component likelihood-ratio test at a boundary instead uses $\frac{1}{2}{\delta }_0+\frac{1}{2}{\chi }_1^2$: for $T>0$, its approximate tail probability is $\frac{1}{2}\mathrm{Pr}({\chi }_1^2\ge T)$, and at $T=0$ the nonrandomized p-value is one. This approximation is not an exact guarantee for sixteen rats. Alternatively one can compare consistently defined restricted likelihoods with the same $X$ and simulate their restricted-ratio null distribution, as in the RLRsim documentation.

The risk function of an estimator $\hat{\theta }$ under the stated quadratic loss isThe multivariate normal density is proportional, as a function of $\theta$, toso maximum likelihood estimation gives ${\hat{\theta }}_{\mathrm{MLE}}=X$. Writing $X=\theta +Z$ with $Z\sim N_p(0,I_p)$,for every $\theta \in {\mathbb{R}}^p$.

Apart from terms independent of $\beta$, the Gaussian log-likelihood isDifferentiating with respect to $\beta$ givesThe unique zero, and hence the maximum-likelihood estimate, isThe same value minimizes the residual sum of squares ${\sum }_i(Y_i-\beta x_i{)}^2$, since maximizing the Gaussian likelihood for fixed ${\sigma }^2$ is equivalent to minimizing this sum. It is therefore also the least-squares estimator.

The Gaussian likelihood is proportional toso maximum likelihood estimation is equivalent to ordinary least squares. If $p\le n$ and $X$ has rank $p$, then $X^{T}X$ is invertible and the normal equations giveIf $p>n$, then $\mathrm{rank}X<p$ and $\ker X\ne \{0\}$. A least-squares minimizer always exists, but it is not unique: by rank-deficient ordinary least squares, the complete set isan affine subspace of dimension $p-\mathrm{rank}X>0$. Thus there are uncountably infinitely many likelihood maximizers. In the generic full-row-rank case their dimension is $p-n$.